On Arithmetic Invariants of Special Families of K3-Type Surfaces Citation Can, Tran Thanh Trung (2024) On Arithmetic Invariants of Special Families of K3-Type Surfaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/z5mc-g704. https://resolver.caltech.edu/CaltechTHESIS:06022024-205427263 Abstract This thesis studies applications of Shimura varieties in positive characteristic to questions on arithmetic invariants of special families of K3-type surfaces. The first main result determines the Newton polygons and Artin invariants of 144 special families of K3-type surfaces. The second is a refinement of a conjecture of Serre for K3 surfaces over number field. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Arithmetic geometry Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Mantovan, Elena Thesis Committee: Graber, Thomas B. (chair) Flach, Matthias Zhao, Roy Mantovan, Elena Defense Date: 15 May 2024 Record Number: CaltechTHESIS:06022024-205427263 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06022024-205427263 DOI: 10.7907/z5mc-g704 ORCID: Author ORCID Can, Tran Thanh Trung 0000-0002-2043-3335 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16471 Collection: CaltechTHESIS Deposited By: Tran Thanh Trung Can Deposited On: 03 Jun 2024 23:48 Last Modified: 12 Jun 2024 22:38 Thesis Files PDF - Final Version See Usage Policy. 479kB Repository Staff Only: item control page

On arithmetic invariants of special families of K3-type surfaces Thesis by Trung Can In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2024 Defended May 15, 2024 ii © 2024 Trung Can ORCID: 0000-0002-2043-3335 All rights reserved except where otherwise noted iii ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor Elena Mantovan. It has been an absolute honor to be your student and learn so much from your expertise. Thank you for taking me under your wing, and for your unwavering support, guidance, and patience in years. I am grateful to all the teachers and mentors who have taught and inspired me to do mathematics. I would also like to thank Tom Graber, Matthias Flach, and Roy Zhao for being part of my thesis committee, and Nam Ung for constant reassurance and kindness. I want to thank my Caltech friends Jeck, Yuxin, Deepesh, Bjarne, Alex, and Abishek for their camaraderie and support on this journey. I will miss our shared food, laughter, and intellectual conversations the most. Thank you Jeck for being you: a goofy roommate, a great cook, and a good-hearted person. I want to thank my Vietnamese friends Linh, Hai, Quang, Chi, Nghia, and The Anh for always being there and reminding me of who I am over the years. I also want to thank my Caltech mentees and PiMA teammates for allowing me to pass on some of my knowledge and wisdom. Finally, I thank my family for their unconditional love, endless support, and constant encouragement. I dedicate this thesis to them. This PhD journey is not easy, and I thank Trung Can for never giving up on your hopes and dreams. iv ABSTRACT This thesis studies applications of Shimura varieties in positive characteristic to questions on arithmetic invariants of special families of K3-type surfaces. In the first part, we consider a K3-type surface 𝑋 over Q that arises from cyclic covers of the projective line P1 and a prime 𝑝 of good reduction. Studying the geometry of the associated Shimura varieties and Hurwitz spaces, our main theorem determines all possible Newton polygons of 𝑋 at 𝑝, at least when 𝑝 is sufficiently large. This establishes several new instances of the Manin problem for K3-type surfaces. When 𝑋 is a generic supersingular K3 surface in the family, we also calculate the corresponding Artin invariant. In the second part, we discuss some applications as corollaries of our calculation and recent works [3, 17]. For instance, let 𝑋 be a K3 surface over a number field 𝐿 for which the endomorphism algebra of its transcendental lattice is an abelian field. Then the set of 𝜇-ordinary primes of 𝐿 has density 1 under some conditions. We remark that the proof is discovered by the authors of [3] for a certain class of abelian varieties. Our contribution, if any, is simply realizing that with minor adjustments their techniques also work for K3 surfaces. For another result, we deduce the existence of infinitely many basic primes for one family of K3 surfaces over Q which can be thought of an analog of Elkies’s supersingular prime theorem in our setting. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The proof strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cyclic covers of the projective line and their Jacobians . . . . . . . . 2.4 k3-type surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Newton polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter III: Calculation of families of k3-type surfaces . . . . . . . . . . . . Chapter IV: Proof of the first main theorem . . . . . . . . . . . . . . . . . . Chapter V: Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v 1 1 2 6 7 8 9 9 9 10 14 17 27 33 43 54 1 Chapter 1 INTRODUCTION 1.1 Motivation Shimura varieties are higher-dimensional analogues of modular curves. They play an important role in the theory of automorphic forms and the Langlands program. In recent years, Shimura varieties have found new applications in arithmetic geometry such as the Schottky problem in positive characteristics [15, 16], generalized Elkies’s supersingular theorem [17], or Serre’s ordinariness conjecture [3]. These questions can be formulated in terms of a discrete invariant in positive characteristics called Newton polygon. Here Shimura varieties enter the picture as moduli spaces of abelian varieties with additional structure in characteristic 𝑝. One of the most beautiful properties of these moduli spaces is that they admit natural stratifications by discrete invariants such as the Newton polygon [4, 26, 34]. Studying the resulting geometry often leads to helpful insights to solve problems. For example, the authors study Newton stratification of certain Shimura varieties of PEL type to calculate all possible Newton polygons that occur for special families of Jacobians of cyclic covers of the projective line [16]. To us, special means the image of the family under the classifying map to the moduli Shimura variety is open and dense in a Shimura subvariety1 (see [20, Section 1] for equivalent definitions). It is a natural question to ask whether their framework can be applied to other settings. And the answer is positive. The first part of this thesis carries out a similar calculation for the occurrence of Newton polygons in special families of K3-type surfaces arising from cyclic covers of the projective line. Our methods are similar to the framework in [15, 16]. As input for our theorem we use the construction of special K3 type families from [21]. An analogue of the Manin problem for abelian varieties asks which Newton polygons occur for a K3-type surface with given degree over a fixed characteristic 𝑝. Earlier work [27] shows that all possible polygons for a K3 surface occur in any given characteristic when the degree is sufficiently large. Our result establishes new instances of Newton polygons for k3-type surfaces that occur in positive characteristic with fixed degrees. 1 In virtue of the recently proven André-Oort conjecture, being special is equivalent to having a Zariski dense set of special points (or CM points). 2 Our main application is a refined version of a conjecture of Serre for K3 surfaces. For an abelian variety 𝐴 over a number field 𝐿, Serre’s conjecture asserts that there exists a finite field extension over which the set of good ordinary primes for the base change of 𝐴 has density 1. The analogue statement for K3 surface over number field is proven in [2]. Recent works of Sawin [28] and [3] show refined versions of Serre’s conjecture for abelian surfaces and certain class of simple abelian varieties of type VI. In particular, they calculate this density explicitly without the passage to a sufficiently large field extension. We follow [3] to get refined results for K3 surfaces over a number field 𝐿 equipped with an action of an abelian field satisfying some mild assumptions. The families from the first part give examples of K3 surfaces defined over Q that satisfy all hypotheses of the theorem. Another application is a generalization of Elkies’s supersingular prime theorem [10] for one special family of K3 surfaces over Q. 1.2 Overview of the results First we briefly recall the definition of Newton polygon and Artin invariant. Let 𝑘 be a perfect field of characteristic 𝑝 and 𝑊 (𝑘) denote the ring of Witt vectors over 𝑘. By a K3-type surface 𝑋 over 𝑘 we mean a smooth projective surface over 𝑘 whose Hodge numbers ℎ𝑖, 𝑗 := dim 𝑘 𝐻 𝑗 (𝑋, Ω𝑖𝑋/𝑘 ) are given by the following Hodge diamond: ℎ2,2 ℎ1,2 ℎ2,0 ℎ1,1 ℎ0,1 ℎ1,0 ℎ0,1 ℎ0,0 1 ℎ2,1 = 0 0 1 2𝑛 − 2 1 0 0 1 . 2 (𝑋/𝑊), one considers a discrete invariTo the second crystalline cohomology 𝐻𝑐𝑟𝑖𝑠 2 (𝑋/𝑊). It is a lower convex ant called the Newton polygon of the 𝐹-crystal 𝐻𝑐𝑟𝑖𝑠 polygon starting at (0, 0) and ending at (2𝑛, 2𝑛) with integer breakpoints. From 2.5, 2 (𝑋/𝑊) either has there exists an integer ℎ = ℎ(𝑋) ∈ {1, 2, . . . , 𝑛} such that 𝐻𝑐𝑟𝑖𝑠 slopes (1 − 1/ℎ, 1, 1 + 1/ℎ) with multiplicities (ℎ, 2𝑛 − 2ℎ, ℎ), 3 or the Newton polygon has slopes 1 only in which case we set ℎ(𝑋) = ∞. We call this invariant the height of 𝑋 2. If 𝑋 is a K3 surface then 𝑛 = 10. 𝑋 is called ordinary if ℎ(𝑋) = 1 and supersingular if ℎ(𝑋) = ∞. Assume that 𝑋 is a supersingular K3 surface. It follows from the Tate conjecture for K3 surfaces over finite fields3 that the Tate module 𝑇𝑋 has rank 22. The bilinear pairing from Poincaré duality restricted to 𝑇𝑋 is a non-degenerate symmetric form ⟨, ⟩ : 𝑇𝑋 × 𝑇𝑋 → Z 𝑝 . One can show that the discriminant of this form has 𝑝-valuation 2𝜎0 for some integer 𝜎0 ∈ {1, 2, . . . , 10}. The invariant 𝜎0 is called the Artin invariant of 𝑋 (see 2.5). Next we describe the families we consider for our main results. In [21], motivated by an analogue of Coleman’s conjecture for K3-type surfaces, Moonen constructs special families of k3-type surfaces from cyclic covers of the projective line P1 . Each family comes from a tuple (𝐺, 𝑎, 𝑏, 𝑁) called a K3 type datum (see 3.0.1) where 𝐺 ≃ Z/𝑚Z is the cyclic group of order 𝑚 ≥ 3, 𝑎 = (𝑎 1 , . . . , 𝑎 𝑁 ) and 𝑏 = (𝑏 1 , 𝑏 2 , 𝑏 3 ) are monodromy data, and 𝑁 is the number of branch points. To be more precise, let 𝐶 and 𝐷 be smooth curves given by the equations 𝐶 : 𝑦 𝑚 = (𝑥 − 𝜆 1 ) 𝑎1 (𝑥 − 𝜆 2 ) 𝑎2 . . . (𝑥 − 𝜆 𝑁 ) 𝑎 𝑁 , 𝐷 : 𝑧 𝑚 = (𝑡 − 𝛾1 ) 𝑏1 (𝑡 − 𝛾2 ) 𝑏2 (𝑡 − 𝛾3 ) 𝑏3 , and 𝜇𝑚 be the group scheme of 𝑚 𝑡ℎ roots of unity acting on 𝐶 × 𝐷 via (𝑥, 𝑦, 𝑡, 𝑧) ↦→ (𝑥, 𝜁 𝑦, 𝑡, 𝜁 −1 𝑧), where 𝜁 is a choice of a primitive 𝑚 𝑡ℎ root. Then the minimal model resolution of singularities of the GIT quotient (𝐶 × 𝐷)/𝐺 is a smooth projective k3-type surface. Varying the base point 𝜆 = (𝜆1 , . . . , 𝜆 𝑁 ) one obtains a family of k3-type surfaces of dimension 𝑁 − 3. Let X = X(𝐺, 𝑎, 𝑏, 𝑁) be the family associated to the K3 type datum (𝐺, 𝑎, 𝑏, 𝑁). 1 Then X admits a smooth model over Z[ 2𝑚 ] (see 3). Let 𝑝 ∤ 2𝑚 be a prime and 𝑋 be a k3-type surface in X. By abuse of notation, we use 𝑋 to denote the reduction of 𝑋 at 𝑝 which is a smooth proper k3-type surface in positive characteristic. Let 𝐸 𝑚 be the 𝑚 𝑡ℎ cyclotomic field and 𝐸 𝑚0 be its totally real field. We write 𝑑 for the order of 𝑝 mod 𝑚. Our main theorem is the following. 2 The invariant ℎ(𝑋) is also the height of the formal Brauer group of 𝑋. 3 Established by the works of several authors. 4 Theorem (Theorem 4.0.9). Let X, 𝑋, 𝑝 and 𝑑 be as above. 1. If 𝑝 is not inert in 𝐸 𝑚 /𝐸 𝑚0 , then 𝑑 divides ℎ(𝑋) and 1≤ ℎ(𝑋) ≤ 𝑁 − 2. 𝑑 Furthermore, for any 1 ≤ 𝑖 ≤ 𝑁 − 2, there exists a k3-type surface 𝑋 ′ in X such that ℎ(𝑋 ′) = 𝑖𝑑. 2. If 𝑝 is inert in 𝐸 𝑚 /𝐸 𝑚0 , we assume further that 𝑝 is sufficient large when 𝑁 is even, then 𝑑 divides ℎ(𝑋) and 1≤ ℎ(𝑋) 𝑁 −2 ≤⌊ ⌋ or ℎ(𝑋) = ∞. 𝑑 2 ′ In addition, for any 1 ≤ 𝑖 ≤ ⌊ 𝑁−2 2 ⌋, there exists a k3-type surface 𝑋 in X such that ℎ(𝑋 ′) = 𝑛𝑑. Moreover, if X corresponds to a family of K3 surfaces, then there exists a supersingular K3 surface 𝑋 ′ in X with 𝜎0 (𝑋 ′) = (𝑁 − 2 − ⌊ 𝑁−2 2 ⌋)𝑑. Section 4 discusses how to obtain an effective bound for 𝑝 in the second part of the theorem. Table 5.1 lists all special families for 𝑚 ≤ 100 from a brute-force search done by a computer program. Note that there are no examples for 23 ≤ 𝑚 ≤ 100. It is an open question whether Table 5.1 is complete. While we are not able to prove this for k3-type surfaces in general, we can easily show the following. Proposition 1.2.1. There are no other special families of K3 surfaces arising from cyclic covers of the projective line except those listed in Table 5.1. We remark that our examples cover all the possible heights 1, . . . , 10, and ∞ of a K3 surface. Example 1.2.1. The K3 type datum (4, (2, 3, 3), (1, 1, 2), 3) gives rise to a single K3 surface 𝑋 with complex multiplication by Q[𝑖]. Furthermore, 𝑋 is defined over Z and has supersingular reduction for all primes 𝑝 ≡ 3 (mod 4). The first example of this phenomenon goes back to Tate’s K3 surface: 𝑥 4 + 𝑦 4 + 𝑧4 + 𝑡 4 = 0 which also has supersingular reduction for all primes 𝑝 ≡ 3 (mod 4). 5 Example 1.2.2. Let X be the 1-dimensional family of K3 surfaces associated to (7, (2, 4, 4, 4), (1, 2, 4), 4) and 𝑋 ∈ X. At primes 𝑝 ≡ 1 (mod 7) we have ℎ(𝑋) = 1 or ℎ(𝑋) = 2. At primes 𝑝 ≡ 2, 4 (mod 7) we have ℎ(𝑋) = 3 or ℎ(𝑋) = 6. At primes 𝑝 ≡ 3, 5 (mod 7) we have ℎ(𝑋) = 6 or ℎ(𝑋) = ∞ (and 𝜎0 (𝑋) = 6). Finally at primes 𝑝 ≡ 6 (mod 7) we have ℎ(𝑋) = 2 or ℎ(𝑋) = ∞ (and 𝜎0 (𝑋) = 2). For a fix datum (𝐺, 𝑎, 𝑏, 𝑁) and a prime 𝑝 of good reduction, the set of possible Newton polygons that occur on the family X has a natural "lying above" order. With respect to this order, the lowest polygon is called 𝜇-ordinary and the highest is called basic. Using these notions, the authors show a refined version of Serre’s ordinariness conjecture for certain type of abelian varieties [3] and a generalized Elkies’s theorem for a specific family of cyclic covers of the projective line [17]. As a consequence of their results and our calculation, we can deduce the following results that are new even for K3 surfaces (see Chapter 5 for details and discussion). For a K3 surface 𝑋 over a number field 𝐿, we write E (𝑋) for the endomorphism algebra of the transcendental lattice 𝑇 (𝑋) of 𝑋 (see 2.4.6) and 𝐿 conn for the extension of 𝐿 that maps to the connected component of the 𝑙-adic monodromy group (see Section 5, Chapter 5). Theorem (Theorem 5.0.9). Let 𝑋 be a K3 surface over a number field 𝐿 such that E (𝑋) is an abelian field 𝐹 and 𝐿 conn ⊆ 𝐹 𝐿. If 𝐹 is totally real, we assume further that 𝑇 (𝑋)Q has even dimension over E (𝑋). Then the set of primes of 𝐿 of 𝜇-ordinary reduction has density 1. Remark 1.2.1. If X(𝐺, 𝑎, 𝑏, 𝑁) corresponds to a family of K3 surfaces, then a generic surface 𝑋 in the family satisfies the conditions of the theorem. Theorem (Theorem 5.0.12). Let 𝑋𝛼 be a K3 surface in the family (5, (1, 1, 1, 2), (1, 1, 3), 4). In particular, 𝑋𝛼 arises from the pair of cyclic covers 𝐶𝛼 : 𝑦 𝑚 = 𝑥(𝑥 − 1)(𝑥 − 𝛼) and 𝐷 : 𝑧 𝑚 = 𝑡 (𝑡 − 1). 2 3 −𝛼+1) Assume in addition that 𝑗 := (𝛼 ∈ Q ∩ [0, 27 4 ] and the reduction of 𝐶𝛼 at 5 is 𝛼2 (𝛼−1) 2 singular. Then 𝑋𝛼 can be defined over Q and there exists infinitely many primes at which the reduction of 𝑋𝛼 has basic Newton polygon. Remark 1.2.2. The two results together imply that a set of density 0, namely the set of basic primes, is infinite. 6 1.3 The proof strategy Theorem 4.0.9 The main idea is to reduce the problem to the calculation of discrete invariants that occur on an intermediate special family of abelian varieties, which we describe next. Assume that 𝑝 ∤ 2𝑚 is a prime and we work in characteristic 𝑝. By abuse of notation, a moduli space M also denotes its geometric fiber M ⊗ F 𝑝 . The Hurwitz space H (𝐺, 𝑎) is the moduli space of 𝐺-covers of P1 with inertia type 𝑎. The K3 type family X(𝐺, 𝑎, 𝑏, 𝑁) can be parametrized over H (𝐺, 𝑎) in the following way. Let 𝜆 ∈ H (𝐺, 𝑎) be a basepoint. By 𝑋𝜆 and 𝐶𝜆 we mean the corresponding k3-type surface and cyclic cover of P1 above 𝜆. The Jacobian 𝐽 (𝐶𝜆 ) of 𝐶𝜆 admits a decomposition up to isogenies into the new part and the old part: 𝐽 (𝐶𝜆 ) ∼ 𝐽 (𝐶𝜆 ) 𝑛𝑒𝑤 ⊕ 𝐽 (𝐶𝜆 ) 𝑜𝑙𝑑 . We can show that the Newton polygon of 𝐽 (𝐶𝜆 ) 𝑛𝑒𝑤 determines the Newton polygon of 𝑋𝜆 . Hence it remains to determine the variation of Newton polygons for 𝐽 (𝐶𝜆 ) 𝑛𝑒𝑤 when 𝜆 varies in H (𝐺, 𝑎). Let 𝜃 denote the map 𝐶𝜆 ↦→ 𝐽 (𝐶𝜆 ) 𝑛𝑒𝑤 so we have an induced morphism 𝜃 : H (𝐺, 𝑎) → A𝑔𝑛𝑒𝑤 , where 𝑔𝑛𝑒𝑤 = 𝜙(𝑚) 2(𝑁−2) is the genus of 𝐽 (𝐶𝜆 ) 𝑛𝑒𝑤 (see 2.3.2 ) and A𝑔𝑛𝑒𝑤 is the moduli space of polarized abelian varieties of genus 𝑔𝑛𝑒𝑤 . From 3.0.2, the image 𝑍 (𝐺, 𝑎) := 𝜃 (H (𝐺, 𝑎)) is open and dense in a certain Shimura variety of PEL type which we denote by 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 )4. Here 𝑓 𝑛𝑒𝑤 is the signature of the action of 𝐺 on the tangent space of 𝐽 (𝐶𝜆 ) 𝑛𝑒𝑤 . It follows that the Zariski closure 𝑍 (𝐺, 𝑎) is 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ). The Newton polygons that occur on the Shimura variety side are well-known. From the moduli point of view, they are the Newton polygons of the corresponding universal abelian schemes over 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ). In [14], Kottwitz gives a combinatorial description of the so-called Kottwitz set 𝐵(𝐺, 𝑓 𝑛𝑒𝑤 ) 5 of admissible Newton polygons equipped with additional structure from the group-theoretic data at an unramified prime 𝑝. In our setting, the signature 𝑓 𝑛𝑒𝑤 is simple so the Kottwitz set admits a total order 𝜈ord =: 𝜈0 ≻ 𝜈1 ≻ · · · ≻ 𝜈𝑙 := 𝜈basic 4 These varieties were studied by Deligne and Mostow [7]. 5 A more common notation is 𝐵(𝐺, 𝜇) where 𝜇 is the local cocharacter describing the polarized Hodge structure at 𝑝. 7 where the maximal element 𝜈ord is called 𝜇-ordinary and minimal 𝜈basic called basic. This order agrees with the "lying above" order of lower convex polygons. Let 𝑆ℎ[𝜈] denote the locus of 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) with Newton polygon 𝜈. For the PEL-type Shimura varieties, Viehman and Wedhorn [34] show that all admissible polygons occur, i.e., 𝑆ℎ[𝜈] is nonempty ∀𝜈 ∈ 𝐵(𝐺, 𝑓 𝑛𝑒𝑤 ). Thus, the question can be reformulated as which Newton strata has nonempty intersection with the open image 𝑍 (𝐺, 𝑎). Let 𝑍 (𝐺, 𝑎) [𝜈] := 𝑍 (𝐺, 𝑎) ∩ 𝑆ℎ[𝜈]. Via the geometry of Newton stratification, we can reduce the problem to showing that 𝑍 (𝐺, 𝑎) [𝜈 𝑏𝑎𝑠𝑖𝑐 ] ≠ ∅. This last step can be done by a detailed comparison between boundary components of the Hurwitz space H (𝐺, 𝑎) and the basic locus 𝑆ℎ[𝜈 𝑏𝑎𝑠𝑖𝑐 ], at least when 𝑝 is large. A discussion on an effective bound for 𝑝 is given in 4. Theorems 5.0.9 and 5.0.12 For Theorem 5.0.9, we borrow the authors’ proof from [3] with minor adjustments. For completeness, we summarize the main steps as following: 1. We construct a conjugacy-invariant algebraic function on the 𝑙-adic monodromy group 𝐺 𝑋,𝑙 that detects 𝜇-ordinariness and use Serre’s results to reduce density question to a trace calculation on the connected components of 𝐺 𝑋,𝑙 ; 2. We use the conditions 𝐹 is abelian and 𝐿 𝑐𝑜𝑛𝑛 ⊆ 𝐹 𝐿 to realize the group of connected components of 𝐺 𝑋,𝑙 as a permutation subgroup the Weyl group of the orthogonal group; 3. A direct calculation with matrices shows that the trace is non-constant on each connected component. We remark that unlike the case of abelian varieties, for K3 surfaces, the techniques also work when E 𝑋 is totally real and 𝑇 (𝑋) has even rank over E 𝑋 . Theorem 5.0.12 follows from the main theorem of [17] and the calculations in 4. 1.4 Future work This section discusses further directions in increasing order of difficulty. 8 Non-cyclic covers The group 𝐺 can be non-cyclic. Because of the K3 type condition, 𝐷 has to be branched above 3 points. In particular, 𝐺 has at most two generators. Thus the next cases to consider will be products or semidirect products of two cyclic groups. The calculations of the signature via Hurwitz characters and minimal resolution of singularity, though slightly more complicated, could be done to obtain to new examples of K3 type families. Complete occurrence of the Artin invariants Similar to the Newton polygon, if we could show that the lowest Artin invariant occurs, then from purity of the stratification, all admissible Artin invariants occur. The former step would require an analysis of the number of superspecial points in 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) to verify whether it grows with 𝑝. Density of primes for other invariants such as the Picard rank As an analogue to Serre’s conjecture, an interesting direction is studying the density of primes at which the Picard rank has to be of some "generic type". The strategy we could carry can be: first calculate of admissible Picard ranks for a K3 surface with extra endomorphism, then find a conjugacy-invariant algebraic function that detects genericity of the Picard rank. Since the Picard group consists of Tate classes, a potential characterization can be related to the eigenvalue 𝑝 of the Frobenius Fr𝔭 . Completeness of all K3-type families arising from cyclic covers of P1 In [20], the author shows that there are precisely 20 special families of Jacobians that arise from cyclic covers of P1 . In a similar light, we can conjecture that there are no further family of k3-type surfaces arising from cyclic covers of P1 except those listed in Table 5.1. The nonexistence of further families reduces to the nonexistence of further integer solutions to the equation involving fractional functions from 3.0.1. This is perhaps the most open and difficult question. 1.5 Structure of the thesis In Chapter 2, we introduce general notations and recall some preliminaries. Chapter 3 reviews Moonen’s construction of special families of k3-type surfaces and prove relevant features. In Chapter 4, we prove our first main theorem. Chapter 5 presents other theorems as applications. 9 Chapter 2 BACKGROUND 2.1 General notations Throughout, let 𝑘 be a perfect field of positive characteristic 𝑝. We write 𝑊 (𝑘) for ring of Witt vectors over 𝑘 and 𝐾 (𝑘) for its faction field 𝑊 (𝑘) [ 1𝑝 ], or just 𝑊 and 𝐾 for short. We use 𝜎 to denote the Frobenius automorphism of 𝑘 as well as its lifts to 𝑊 and 𝐾. We fix the following notations: • Q 𝑝 is an algebraic closure of Q 𝑝 ; • C 𝑝 is the 𝑝-adic completion of Q 𝑝 and we fix an identification C 𝑝 ≃ C; 𝑢𝑛 • Q𝑢𝑛 𝑝 is the maximal unramified extension Q 𝑝 and Z 𝑝 is its ring of integers; • For a positive integer 𝑚 ≥ 2, let 𝜇𝑚 denote the group scheme of 𝑚 𝑡ℎ roots of unity, 𝐸 𝑚 denote the 𝑚 𝑡ℎ cyclotomic field of degree 𝜑(𝑚), and fix a choice 𝜁𝑚 ∈ C of a primitive 𝑚 𝑡ℎ root of unity; • For a ring 𝑅, 𝑅 × is the subset of nonzero elements and 𝑅 ∗ is the subset of invertible elements. • For a real number 𝑥, ⟨𝑥⟩ = 𝑥 − ⌊𝑥⌋ is the fractional part of 𝑥. 2.2 The group algebra Let 𝐺 ≃ Z/𝑚Z be the cyclic group of order 𝑚. The group algebra Q[𝐺] ≃ Q[𝑋]/(𝑋 𝑚 − 1) is the same as the coordinate ring of the group scheme 𝜇𝑚 of 𝑚 𝑡ℎ roots of unity. Let T := homQ (Q[𝐺], C). Then T can be identified with 𝐺 via 𝑛 ∈ Z/𝑚Z ↦→ 𝜏𝑛 ∈ 𝑇 where 𝜏𝑛 maps the generator of 𝐺 to (𝜁𝑚 ) 𝑛 ∈ C∗ . We write 𝜏0 for the trivial character and 𝜏 ∗ for the image of 𝜏 under complex conjugation in C; in particular, 𝜏𝑛∗ = 𝜏−𝑛 . 10 Grouping the characters with the same kernel, we have yet another way to describe the group algebra Ö Q[𝐺] ≃ 𝐸𝑑 . 𝑑|𝑚 Each 𝜏 is given by the composition of the projection Q[𝐺] to 𝐸 𝑑 with a complex Î character of 𝐸 𝑑 of order 𝑑. Since Q[𝐺] ⊗ C = 𝜏∈T C, any module Q[𝐺] ⊗ Cmodule 𝑊 admits a decomposition 𝑊 = ⊕𝜏∈T 𝑊𝜏 where each 𝑊𝜏 is the Eigenspace on which 𝑔 ⊗ 1 ∈ Q[𝐺] ⊗ C acts as multiplication by 𝜏(𝑔). Assume that 𝑝 does not divide the order 𝑚 of 𝐺. Then each character 𝜏 : Q[𝐺] → C 𝑝 (under the identification C 𝑝 ≃ C) is unramified at 𝑝 and factors through Q𝑢𝑛 𝑝 . 𝑢𝑛 Let us by abuse of notation also use T for homQ (Q[𝐺], Q 𝑝 ). The Frobenius 𝜎 of Q𝑢𝑛 𝑝 induces an automorphism 𝜎 : 𝜏𝑛 ↦→ 𝜏𝑝𝑛 of T . We write 𝔒 for the set of 𝜎-orbits 𝔬 in T ∗ := T − {𝜏0 }. The elements in the same orbit 𝔬 have the same kernel 𝑅𝜏 = 𝑅𝔬 ⊂ 𝐺 and order 𝑑 𝜏 = 𝑑 (𝔬) = [𝐺 : 𝑅(𝔬)] (if the orbit contains 𝜏𝑛 then this order is the same as the additive order of 𝑛 in Z/𝑚Z). Each orbit 𝔬 is in bijection with a prime 𝑝 𝔬 of 𝐸 𝑑𝔬 above 𝑝 such that the following decomposition holds Ö Q[𝐺] ⊗Q Q 𝑝 ≃ 𝐸 𝑑𝔬 ,𝑝𝔬 𝔬∈𝔒 where 𝐸 𝑑𝔬 ,𝑝𝔬 is the completion of 𝐸 𝑑𝔬 at the prime 𝑝 𝔬 . 2.3 Cyclic covers of the projective line and their Jacobians Let 𝐺 be as in the previous section. We fix the following numerical data: a positive integer 𝑁 ≥ 3 and an 𝑁-tuple 𝑎 = (𝑎 1 , . . . , 𝑎 𝑁 ) of elements in 𝐺. Definition 2.3.1. The datum (𝐺, 𝑎, 𝑁) is called a monodromy datum with inertia type 𝑎 if it satisfies the following conditions: 1. 𝑎 1 , ..., 𝑎 𝑁 are nontrivial elements that generate the whole group 𝐺; 2. their product Î 𝑗 𝑎 𝑗 is equal to the identity of 𝐺. We remark that the above conditions mean 𝑎𝑖 . 0 (mod 𝑚) for all 𝑖, gcd(𝐺, 𝑎 1 , . . . , 𝑎 𝑁 ) = 1, and 𝑎 1 + · · · + 𝑎 𝑚 ≡ 0 (mod 𝑚). 11 Fix an 𝑁-tuple 𝜆 = (𝜆 1 , . . . , 𝜆 𝑁 ) where 𝜆𝑖 ≠ 𝜆 𝑗 . A monodromy datum (𝐺, 𝑎, 𝑁) defines a smooth irreducible curve 𝐶 = 𝐶𝜆 (𝐺, 𝑎, 𝑁) by the equation 𝑦 𝑚 = (𝑥 − 𝜆 1 ) 𝑎1 (𝑥 − 𝜆 2 ) 𝑎2 . . . (𝑥 − 𝜆 𝑁 ) 𝑎 𝑁 . In addition, 𝐶 has the structure of a Galois cover of P1 branched above 𝑁 points with Galois group 𝐺 (or 𝐺-cover for short) via the covering map (𝑥, 𝑦) ↦→ 𝑥 and the group scheme 𝜇𝑚 acting via 𝜁𝑚 · (𝑥, 𝑦) := (𝑥, 𝜁𝑚 𝑦). The monodromy conditions ensure that the 𝜇𝑚 action on 𝐶 over P1 is Galois. When 𝜆 is fixed, according to Riemann’s Existence Theorem, the isomorphism classes of 𝐺-covers are in bijection with branching data up to permutations of the 𝑎𝑖 ’s and automorphisms of 𝐺. The later group is isomorphic to (Z/𝑚Z) ∗ . Remark 2.3.1. The action of the group scheme 𝜇𝑚 is étale over Z[ 𝑚1 , 𝜁𝑚 ]. For convenience, all schematic constructions will be assumed to be over the base scheme 𝑆 := Spec(Z[ 𝑚1 , 𝜁𝑚 ]). Let 𝐽 (𝐶) = 𝐽 (𝐺, 𝑎) denote the Jacobian of 𝐶. The dimension of 𝐽 (𝐶) is equal to the genus of 𝐶 and can be computed from the monodromy datum via Riemann-Hurwitz formula: Í𝑁 gcd(𝐺, 𝑎𝑖 ) (𝑁 − 2)𝑚 − 𝑖=1 . 𝑔(𝐶) = 1 + 2 The tangent space 𝑉 (𝐶) := Lie(𝐽 (𝐶)) can be identified with the sheaf of holomorphic differentials 𝐻 0 (𝐶, Ω𝐶1 ). The 𝜇𝑚 -action on 𝐶 induces an action of the group algebra Q[𝐺] on 𝐻 0 (𝐶, Ω𝐶1 ). Then 𝑉 decomposes as a Q[𝐺] ⊗ C-module into Eigenspaces Ê 𝑉= 𝑉𝜏 𝜏∈T where 𝜇𝐺 acts on 𝑉 via multiplication by 𝜏(𝐺). The complex dimension 𝑓 (𝜏) of 𝑉𝜏 , or multiplicity of 𝜏, can be computed by the Hurwitz-Chevalley-Weil formula (see [23, Theorem 3.3]): 𝑓 (𝜏𝑛 ) =   0    −1 +  if 𝑛 = 0 . −𝑛𝑎 𝑖 𝑖=1 ⟨ 𝑚 ⟩ Í𝑁 otherwise. (2.1) 12 The tuple 𝑓 = ( 𝑓 (𝜏))𝜏∈T is called the signature and only depends on the monodromy datum, not the base point 𝜆. Proposition 2.3.1. 𝑓 (𝜏) + 𝑓 (𝜏 ∗ ) = 𝑁 − 2 − 𝑙 (𝜏) where 𝑙 (𝜏) = #{𝑖 : 𝑑 (𝜏) | 𝑎𝑖 }. 𝑛𝑎 𝑖 𝑖 Proof. This follows from Hurwitz-Chevalley-Weil formula and that ⟨ −𝑛𝑎 𝑚 ⟩+⟨ 𝑚 ⟩ is 0 if 𝑑 (𝜏) | 𝑎𝑖 and is 1 otherwise. □ Remark 2.3.2. As a corollary, 𝑓 (𝜏) + 𝑓 (𝜏 ∗ ) only depends on the orbit 𝔬 of 𝜏. It is convenient to write 𝑔(𝔬) := 𝑓 (𝜏) + 𝑓 (𝜏 ∗ ) for this number. Each character 𝜏 gives rise to a cyclic quotient 𝐺/𝑅𝜏 ≃ Z/𝑑 𝜏 Z. We write 𝑅 for ker 𝜏 and 𝑑 for 𝑑 𝜏 . The datum (Z/𝑑Z, 𝑎 (mod 𝑑)) is a monodromy datum for a cyclic Z/𝑑Z-cover of P1 . There is a surjective map 𝐶 (𝐺, 𝑎) → 𝐶 (Z/𝑑Z, 𝑎 (mod 𝑑)), (𝑥, 𝑦) ↦→ (𝑥, 𝑦 𝑚/𝑑 ) which by functoriality induces an injection 𝐽 (Z/𝑑Z, 𝑎 (mod 𝑑)) ↩→ 𝐽 (𝐺, 𝑎) as an abelian subvariety. We define the new part of 𝐽 (𝐺, 𝑎) to be ∑︁ 𝐽 (𝐺, 𝑎) 𝑛𝑒𝑤 := 𝐽 (𝐺, 𝑎)/( 𝐽 (Z/𝑑Z, 𝑎 (mod 𝑑)))). 𝑑|𝑚,𝑑≠𝑚 Up to isogeny, we have a decomposition Ê 𝐽 (𝐺, 𝑎) ∼ 𝐽 (Z/𝑑Z, 𝑎 (mod 𝑑)) 𝑛𝑒𝑤 𝑑|𝑚 that is compatible with the decomposition of the group algebra into the product of cyclotomic fields 𝐸 𝑑 for 𝑑 | 𝑚. Proposition 2.3.2. The genus of the new part 𝐽 (𝐺, 𝑎) 𝑛𝑒𝑤 is 𝑔 𝑛𝑒𝑤 = (𝑁−2)𝜑(𝑚) . 2 Proof. The genus of 𝐽 (𝐺, 𝑎) 𝑛𝑒𝑤 is the same as the dimension of Ê 𝑉 𝑛𝑒𝑤 = 𝑉𝜏 . 𝜏:𝑑 (𝜏)=𝑚 ∗ ∗ There are 𝜑(𝑚) 2 pairs (𝜏, 𝜏 ) for which 𝑙 (𝜏) = 0, and thus, 𝑓 (𝜏) + 𝑓 (𝜏 ) = 𝑁 − 2. □ 13 Example 2.3.1. The monodromy datum (Z/5Z, (1, 1, 1, 2), 4) defines a family of curves 𝐶𝜆 by the affine equation 𝑦 5 = 𝑥(𝑥 − 1)(𝑥 − 𝜆). When 𝜆 varies, we get a 1-dimensional family of 𝐺-covers of P1 . A basis for 𝐻 0 (𝐶𝜆 , Ω1 ) (see [20, Lemma 2.7]) is given by 𝑑𝑥 , 𝑦2 𝑑𝑥 , 𝑦3 𝑑𝑥 , 𝑦4 𝑥𝑑𝑥 . 𝑦4 The action of 𝐺 is defined via 𝑦 ↦→ 𝜁5 𝑦 so 𝐺 acts via 𝜏2 on 𝑑𝑥 𝑑𝑥 , 𝜏 on , and 𝜏1 on 3 𝑦3 𝑦2 𝑥𝑑𝑥 𝑑𝑥 and . It follows that 𝑦4 𝑦4 𝑓 (𝜏2 ) = 𝑓 (𝜏3 ) = 1, 𝑓 (𝜏1 ) = 2, 𝑓 (𝜏4 ) = 0. These multiplicities of the eigenspaces agree with Hurwitz-Chevalley-Weil formula. The Hurwitz space H (𝐺, 𝑎) Fix a monodromy datum (𝐺, 𝑎) with inertia type 𝑎. Let H (𝐺) be the moduli space of 𝐺-covers of P1 and H (𝐺) denote its Deligne-Mumford compactification. The later is the moduli space of stable 𝐺-covers of P1 . The moduli space H (𝐺, 𝑎) with ordered 𝑎 is an irreducible component of H (𝐺) and has a model over Z[ 𝑚1 ]. It classifies 𝐺-covers with ordered inertia type 𝑎. Let 𝑈 ⊂ (A1 ) 𝑁 be the complement of the weak diagonal, i.e., the scheme of 𝑁2 whose affine tuples of distinct points of A. Consider the projective curve 𝐵 ⊂ P𝑈 curve over a point 𝜆 ∈ 𝑈 is given by 𝑦 𝑚 = (𝑥 − 𝜆 1 ) 𝑎1 (𝑥 − 𝜆 2 ) 𝑎2 . . . (𝑥 − 𝜆 𝑁 ) 𝑎 𝑁 . There exists an open subscheme 𝑇 ⊂ 𝑈 for which there exists a smooth proper 𝐺-cover 𝐶𝑇 over 𝑇 with inertia type 𝑎. Then H (𝐺, 𝑎) together with its universal family can be constructed as the stable quotient of 𝐶𝑇 → P𝑇1 by the action of Aut(P1 ) × Sym(𝑁) (see [20], 2.2). From dim 𝑇 = dim 𝑈 = 𝑁 and dim Aut(P1 ) = 3, we deduce that dim H (𝐺, 𝑎) = 𝑁 − 3. (2.2) If 𝑝 ∤ 𝑚 is a prime then both H (𝐺, 𝑎) and H (𝐺, 𝑎) have good reduction at 𝑝. 14 The Shimura varieties 𝑆ℎ(𝐺, 𝑓 ) and 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) Let 𝐵 be the group algebra Q[𝐺]. Let G := GSp2𝑔 ∩GL𝐵 be the connected reductive algebraic group over Q consisting of the endomorphisms of 𝐻 1 (𝐽𝐶 , Z) that commute with 𝐵. Let 𝑋 be a G(R)-conjugacy class of the homomorphism ℎ : S1 → 𝐺 (R) parametrizing the Hodge structure on 𝐻 1 (𝐽𝐶 , Z) with an action of 𝐵 prescribed by 𝑓 . Then (G, 𝑋) is a Shimura datum in the sense of Deligne. This datum gives rise to a moduli variety denoted by 𝑆ℎ(𝐺, 𝑓 ) whose reflex field is contained in the cyclotomic field 𝐸 𝑚 . Furthermore, 𝑆ℎ(𝐺, 𝑓 ) has a good integral model over Z[ 𝑚1 , 𝜁𝑚 ]. For a prime 𝑝 not dividing 𝑚, by abuse of notation, we use 𝑆ℎ(𝐺, 𝑓 ) to denote the reduction modulo 𝑝 and 𝑆ℎ(𝐺, 𝑓 )F 𝑝 to denote its geometric fiber. When the basepoint 𝜆 varies over H (𝐺, 𝑎), the Jacobian 𝐽𝜆 (𝐺, 𝑎) is a principally polarized abelian variety equipped with the action of Q[𝐺] on the tangent space of 𝐽𝜆 (𝐺, 𝑎) specified by 𝑓 . There exists a classifying morphism defined over Z[ 𝑚1 , 𝜁𝑚 ] 𝜙 : H (𝐺, 𝑎) → 𝑆ℎ(𝐺, 𝑓 ), 𝜆 ↦→ 𝐽 (𝐶𝜆 ). From [20, (3.3.1)], the dimension of 𝑆ℎ(𝐺, 𝑓 ) is the same as the dimension of G which can be computed as dim 𝑆ℎ(𝐺, 𝑓 ) = ∑︁ 𝑓 (𝜏𝑙 ) 𝑓 (𝜏−𝑙 ) +    𝑓 (𝜏𝑘 ) ( 𝑓2(𝜏𝑘 )+1) if 𝑚 = 2𝑘 is even,;  (2.3)   0 if 𝑚 is odd (𝑙,−𝑙)  where the sum is over all pairs (𝑙, −𝑙) such that 2𝑙 ≠ 𝑚. Similarly, we write 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) for the Shimura variety that is the moduli space of the new part 𝐽 (𝐺, 𝑎) 𝑛𝑒𝑤 . 2.4 k3-type surface In this section we work over C. Definition 2.4.1. An K3 surface over C is a smooth projective surface 𝑋 such that ℎ1 (𝑋, O 𝑋 ) = 0 and 𝜔 𝑋 ≃ O 𝑋 , that is, 𝑋 is simply connected and has trivial canonical bundle. Example 2.4.1. If 𝑋 is a smooth quartic in P3 , then 𝜔 𝑋 ≃ O𝑋 via the adjunction formula, and ℎ1 (𝑋, O 𝑋 ) = 0 from taking cohomology of the short exact sequence 0 → OP3 (−4) → OP3 → O 𝑋 → 0. In particular, 𝑋 is a K3 surface. 15 Example 2.4.2. If 𝐴 is a abelian surface, then the surface quotient 𝐴/±𝑖𝑑 has 16 singularities of type 𝐴1 . Its minimal resolution of singularities Km( 𝐴) is a K3 surface, called the Kummer surface associated to 𝐴. Definition 2.4.2. A Hodge structure of (pure) weight 𝑛 is a free Z-module 𝐻Z of finite rank together with a (Hodge) decomposition of complex vector spaces Ê 𝐻 := 𝐻Z ⊗ C = 𝐻 𝑝,𝑞 𝑝+𝑞=𝑛 such that 𝐻 𝑝,𝑞 = 𝐻 𝑝,𝑞 . A rational Hodge structure 𝐻Q is a finite dimensional vector space over Q with a Hodge decomposition of 𝐻Q ⊗ C. If 𝐻Z is a Hodge structure then 𝐻Z ⊗ Q is a rational Hodge structure. The Hodge number ℎ 𝑝,𝑞 is the complex dimension dimC 𝐻 𝑝,𝑞 and the set S ⊂ Z2 of ( 𝑝, 𝑞) with ℎ 𝑝,𝑞 ≠ 0 is called the type of 𝐻. Example 2.4.3. If 𝑋 is a compact Kähler manifold, e.g., the analytic variety associated to a smooth projective variety, its 𝑛𝑡ℎ Betti cohomology group 𝐻 𝑛 (𝑋, Z) modulo torsion is a pure Hodge structure. By Chow’s theorem, among the compact Kähler manifolds, the algebraic ones are those that admit an ample line bundle 𝐿. The embedding by 𝐿 into some projective space gives rise to a bilinear form 𝑄(L) on 𝐻 𝑛 (𝑋, Z) satisfying certain compatibility and positivity conditions. Such linear algebra data are captured in the notion of a polarized Hodge structure. Definition 2.4.3. A polarized Hodge structure of weight 𝑛 is a Hodge structure 𝐻Z together with a non-degenerate integer bilinear pairing 𝑄 on 𝐻Z whose linear extension to 𝐻 satisfies the following conditions: 1. 𝑄(𝑥, 𝑦) = (−1) 𝑛 𝑄(𝑦, 𝑥); 2. 𝑄(𝑥, 𝑦) = 0 ′ ′ for all 𝑥 ∈ 𝐻 𝑝,𝑞 , 𝑦 ∈ 𝐻 𝑝 ,𝑞 and 𝑝 ≠ 𝑞′; 3. 𝑖 𝑝−𝑞 𝑄(𝑥, 𝑥) ¯ >0 for all 𝑥 ∈ 𝐻 𝑝,𝑞 and 𝑥 ≠ 0. Definition 2.4.4. A K3 type Hodge structure is a polarized Hodge structure of weight 2 with Hodge numbers ℎ2,0 = 1, ℎ1,1 = 𝑛, and ℎ0,2 = 1 (here 𝑛 is necessarily even). 16 Definition 2.4.5. A k3-type surface 𝑋 is a smooth projective surface such that ℎ1 (𝑋, O 𝑋 ) = 0 and its second Betti cohomology 𝐻 2 (𝑋, Z) is a K3 type Hodge structure. Example 2.4.4. Let (𝑋, L) be an algebraic K3 surface where L is a primitive ample line bundle (necessarily of degree 2𝑑 for some positive integer 𝑑). The Chern class 𝑐(L) is a (1, 1)-class and determines the isomorphism class of 𝐿 since ℎ1,0 (𝑋) = 0. Using 𝑐(L), we can change the sign of the intersection pairing to obtain a polarization form 𝑄(L) on 𝐻 2 (𝑋, Z) so that (𝐻 2 (𝑋, Z), 𝑄(L)) is a polarized K3 type Hodge structure of rank 22. To construct the moduli of algebraic K3 surfaces, one fixes the degree of 𝐿 and considers the primitive cohomology 𝐻 2𝑝𝑟𝑖𝑚 (𝑋, Z) := 𝑐(L) ⊥ ⊂ 𝐻 2 (𝑋, Z) with the restricted bilinear form of signature (2, 19) and rank 21. (𝑋, L) and (𝑋 ′, L ′) are isomorphic polarized K3 surfaces if and only if there exists a Hodge isometry between (𝐻 2 (𝑋, Z), 𝑄(L)) and (𝐻 2 (𝑋 ′, Z), 𝑄(L ′)) taking 𝑐(L) to 𝑐(𝐿 ′). We recall the definition of complex multiplication, or CM, for K3 surface analogous to CM abelian varieties. Definition 2.4.6. Let 𝑇 (𝑋) denote the orthogonal complement of 𝑁 𝑆(𝑋) inside 𝐻 2 (𝑋 (C), Z) with respect to the Poincaré pairing. The Z-module 𝑇 (𝑋) equipped with the restricted Poincaré pairing is called the transcendental lattice of 𝑋. From 𝑁 𝑆(𝑋) ⊗ Q = 𝐻 2 (𝑋, Q) ∩ 𝐻 1,1 (𝑋), we deduce that 𝑇 (𝑋)Q is a rational Hodge structure of K3 type, i.e., 𝑇 (𝑋) ⊗ C = 𝑇 2,0 ⊕ 𝑇 1,1 ⊕ 𝑇 0,2 where 𝑇 1,1 is self-conjugate and 𝑇 2,0 = 𝑇 0,2 has complex dimension 1. Let E (𝑋) denote the endomorphism algebra EndHg (𝑇 (𝑋)) of the Hodge structure 𝑇 (𝑋)Q . One can show that E (𝑋) is commutative via a natural embedding E (𝑋) ↩→ EndC (𝐻 2,0 (𝑋))  C. In particular, E (𝑋) is either a totally real or an imaginary number field. Note that [𝐸 : Q] ≤ 20 for dimension reason. Definition 2.4.7. 𝑋 is said to have complex multiplication if E (𝑋) is equal to a CM field 𝐹 and in addition [𝐹 : Q] = rankZ𝑇 (𝑋). In particular, there exists a unique complex embedding 𝜏 of 𝐿 such that 𝐹 acts on 𝑇 (𝑋) 2,0 via multiplication by 𝜏. The datum of (𝐹, 𝜏) is called the CM-type of 𝑋. 17 2.5 Newton polygon We recall the definition 𝐹-crystal over 𝑘 and define its Newton polygon. 𝐹-crystals Definition 2.5.1. An 𝐹-crystal (𝐺, 𝜑) is a free 𝑊 (𝑘)-module of finite rank together with an injective map 𝜑 : 𝑀 → 𝑀 that is both 𝜎-linear and additive. An 𝐹-isocrystal is a finite dimensional vector space over 𝐾 together with a bijective 𝜎-linear map 𝜑. We write 𝑀 for an 𝐹-crystal and omit 𝜑 from the notation without any confusion. Example 2.5.1. For a smooth projective variety 𝑋 over 𝑘, its 𝑛𝑡ℎ crystalline cohomology group modulo torsion 𝐻 := 𝐻 𝑛 (𝑋/𝑊)/torsion is a free 𝑊-module of finite rank. The absolute Frobenius morphism of 𝑋 induces a 𝜎-linear map 𝜑 : 𝐻 → 𝐻 by functoriality. Poincaré duality induces a perfect pairing ⟨, ⟩ : 𝐻 𝑖 × 𝐻 2 dim(𝑋)−𝑖 → 𝐻 2 dim(𝑋) (𝑋/𝑊) ≃ 𝑊 satisfying ⟨𝜑(𝑥), 𝜑(𝑦)⟩ = 𝑝 dim(𝑋) 𝜎(⟨𝑥, 𝑦⟩). Since ⟨, ⟩ is perfect and 𝜎 : 𝑊 → 𝑊 is injective, 𝜑 is injective, and thus, (𝐻, 𝜑) is an 𝐹-crystal. Example 2.5.2. Let 𝑊 ⟨𝑇⟩ be the non-commutative polynomial ring in one variable over 𝑊 (𝑘) subject to the relations 𝑇𝑥 = 𝜎(𝑥)𝑇 for all 𝑥 ∈ 𝑊 (𝑘). Let 𝛼 = 𝑟𝑠 ∈ Q≥0 be a rational number in reduced form. Then the 𝑊 (𝑘)-module 𝑀𝛼 := 𝑊 ⟨𝑇⟩/(𝑇 𝑠 − 𝑝𝑟 ) equipped with 𝜑 : 𝑥 ↦→ 𝑇𝑥 is an 𝐹-crystal (𝑀𝛼 , 𝜑) of rank 𝑠. The rational value 𝛼 is called the slope of this 𝐹-crystal. The next result classifies 𝐹-crystal up to isogeny when 𝑘 is algebraically closed. Theorem 2.5.1 (Dieudonné-Manin). If 𝑘 is algebraically closed, then the category of 𝐹-crystals over 𝑘 up to isogeny is semi-simple and the simple objects are of form (𝑀𝛼 , 𝜑) for 𝛼 ∈ Q>0 . 18 Definition 2.5.2. Let 𝑀 be an 𝐹-crystal 𝑀 over an algebraically closed field 𝑘 and consider the decomposition of 𝑀 into the simple objects up to isogeny Ê 𝑀∼ 𝑀𝛼𝑛 𝛼 . 𝛼∈Q ≥0 The elements in the set {𝛼 ∈ Q≥0 | 𝑛𝛼 ≠ 0} are called the Newton slopes of 𝑀. For each slope 𝛼 of 𝑀, its multiplicity is defined as 𝑚 𝛼 := 𝑛𝛼 · rank𝑊 (𝑘) .(𝑀𝛼 ). The slopes in ascending order 0 ≤ 𝛼1 < · · · < 𝛼𝑙 together with their multiplicities 𝑚 1 , . . . , 𝑚 𝑙 define the Newton polygon of 𝑀. It can be equivalently encoded in the lower convex polygon connecting the 𝑙 + 1 points with integer coordinates {(0, 0), (𝑚 1 , 𝑚 1 𝛼1 ), (𝑚 1 + 𝑚 2 , 𝑚 1 𝛼1 + 𝑚 2 𝛼2 ), . . . , ( 𝑙 ∑︁ 𝑖=1 𝑚𝑖 , 𝑙 ∑︁ 𝑚𝑖 𝛼𝑖 )}. 𝑖=1 For an 𝐹-crystal 𝑀 over 𝑘, its Newton polygon is defined as the Newton polygon of ¯ which is an 𝐹-crystal over 𝑘. ¯ Let 𝜈(𝑀) denote this polygon. 𝑀 ⊗𝑊 (𝑘) 𝑊 ( 𝑘) Next, as 𝜑 is injective, 𝑀/𝜑(𝑀) is an Artinian 𝑊-module. Using the structure theorem for Artinian modules over the PID 𝑊, there exists 𝑠 non-negative integers ℎ𝑖 and an isomorphism Ê 𝑀/𝜑(𝑀)  (𝑊/𝑝𝑖 𝑊) ℎ𝑖 . 𝑖≥1 Í In addition, we define ℎ0 := rank𝑊 (𝑀) − 𝑖≥1 ℎ𝑖 . The integers 𝑖 ≥ 0 such that ℎ𝑖 ≠ 0 are called the Hodge slopes of 𝑀. The Hodge slopes 𝑖 in ascending order together with their multiplicities ℎ𝑖 define the Hodge polygon of 𝑀. Let 𝜇(𝑀) denote this polygon. The Newton polygon always lies on or above the Hodge polygon. Furthermore, for smooth proper varieties 𝑋 satisfying certain conditions, the Hodge polygon of its crystalline cohomology are equivalently encoded in the Hodge numbers. 19 Proposition 2.5.2. Let 𝑀 be an 𝐹-crystal over 𝑘. Then 𝜈(𝑀) lies on or above 𝜇(𝑀). Theorem 2.5.3 (Mazur, Nygaard, Ogus). Let 𝑋 be a smooth and proper variety 𝑛 (𝑋/𝑊) is torsion-free and the Frölicher spectral sequence of 𝑋 over 𝑘. If 𝐻𝑐𝑟𝑖𝑠 degenerates at 𝐸 1 , then ℎ𝑖 = dim 𝐻 𝑛−𝑖 (𝑋, Ω𝑖𝑋/𝑘 ) for all 0 ≤ 𝑖 ≤ 𝑛. The varieties of interest in this thesis are good reductions of smooth projective varieties in characteristic 0 and hence satisfy the conditions of the theorem. Example 2.5.3. Let 𝐴 be an abelian variety of dimension 𝑔 over 𝑘. The crys1 ( 𝐴/𝑊) is an 𝐹-crystal of rank 2𝑔. Its Hodge polygon is talline cohomology 𝐻𝑐𝑟𝑖𝑠 {(0, 0), (𝑔, 0), (2𝑔, 𝑔)}. Its Newton polygon lies above the Hodge polygon, has the same endpoints, and has integer breakpoints such that if 𝛼 is a slope with multiplicity 𝑛, then 1 − 𝛼 is also a slope with multiplicity 𝑛. Definition 2.5.3. A K3 𝐹-crystal of rank 𝑛 is an 𝐹-crystal (𝐻, 𝜑) of rank 𝑛 together with a symmetric bilinear form ⟨, ⟩ : 𝐻 ⊗𝑊 𝐻 → 𝑊 such that 1. 𝜑 ⊗ 𝑘 has rank 1; 2. 𝑝 2 𝐻 ⊂ 𝜑(𝐻); 3. ⟨, ⟩ is a perfect pairing; 4. ⟨𝜑(𝑥), 𝜑(𝑦)⟩ = 𝑝 2 ⟨𝑥, 𝑦⟩. The K3 crystal (𝐻, 𝜑) is called supersingular if in addition it is purely of slope 1. 2 (𝑋/𝑊) equipped with Example 2.5.4. If 𝑋 is a K3 surface over 𝑘, then 𝐻 := 𝐻𝑐𝑟𝑖𝑠 the map 𝜑 induced by the absolute Frobenius and the bilinear pairing ⟨, ⟩ from Poincaré duality is a K3 𝐹-crystal of rank 22. Proposition 2.5.4. There exists an integer ℎ := ℎ(𝑋) ∈ {1, 2, . . . , 10} such that the 2 (𝑋/𝑊) has slopes Newton polygon of 𝐻𝑐𝑟𝑖𝑠 (1 − 1/ℎ, 1, 1 + 1/ℎ) 20 with multiplicities (ℎ, 22 − 2ℎ, ℎ). Or the Newton polygon has slopes 1 only in which case we set ℎ(𝑋) = ∞. Proof. Poincaré duality implies that ⟨𝐹 (𝑥), 𝐹 (𝑦)⟩ = 𝑝 2 𝜎⟨𝑥, 𝑦⟩ which means the Newton slopes are closed under the map 𝜆 ↦→ 2 − 𝜆. As the Hodge numbers of 𝑋 are 1, 20, 1, and the Newton polygon has to lie above or on the Hodge polygon with integer break points, 𝜈(𝑋) has slopes 1 only or there exists an integer ℎ that is the 𝑥-coordinate of the first break point (ℎ, ℎ − 1). This point corresponds to the slope 1−1/ℎ with multiplicity ℎ and so 1+1/ℎ must also be a slope with multiplicity ℎ. The only remaining slope possible is 1 with multiplicity 22 − 2ℎ. A priori, 1 ≤ ℎ ≤ 11. One considers an ample line bundle 𝐿 on 𝑋. Its first Chern 2 (𝑋/𝑊) satisfies class 𝑐 1 (𝐿) ∈ 𝐻𝑐𝑟𝑖𝑠 𝐹 (𝑐 1 (𝐿) = 𝑐 1 (𝐹 ∗ (𝐿)) = 𝑐 1 (𝐿 ⊗ 𝑝 ) = 𝑝𝑐 1 (𝐿). 2 (𝑋/𝑊) has at least one slope 1 and ℎ < 11. Hence 𝐻𝑐𝑟𝑖𝑠 □ Remark 2.5.1. We call ℎ(𝑋) the height of 𝑋. The terminology stems from the fact that ℎ(𝑋) is also the height of the formal Brauer group of 𝑋. 2 ( 𝐴/𝑊) ≃ Λ2 𝐻 1 ( 𝐴/𝑊) Example 2.5.5. If 𝐴 an abelian surface over 𝑘, then 𝐻𝑐𝑟𝑖𝑠 𝑐𝑟𝑖𝑠 is a K3 𝐹-crystal of rank 6. Definition 2.5.4. A k3-type surface over 𝑘 is a smooth proper surface 𝑋 such that ℎ1 (𝑋, O 𝑋 ) = 0, the Frölicher spectral sequence of 𝑋 degenerates at 𝐸 1 , and the 2 (𝑋/𝑊) is a K3 𝐹-crystal. second crystalline cohomology 𝐻𝑐𝑟𝑖𝑠 Artin invariant There exists a crystalline Chern map 2 𝑐 1 : Pic(𝑋) → 𝐻𝑐𝑟𝑖𝑠 (𝑋/𝑊) which is a homomorphism of abelian groups. We call the image of this map the Neron-Severi group NS(𝑋). For any 𝐿 ∈ Pic(𝑋), 𝑐 1 (𝐹 ∗ (𝐿)) = 𝑐 1 (𝐿 ⊗ 𝑝 ) = 𝑝𝑐 1 (𝐿) where 𝐹 is the absolute Frobenius morphism of 𝑋. This means the image NS(𝑋) in 2 (𝑋/𝑊) is contained in the Z -submodule of elements 𝑥 such that 𝜑(𝑥) = 𝑝𝑥. 𝐻𝑐𝑟𝑖𝑠 𝑝 21 Definition 2.5.5. Let 𝐻 be a K3 𝐹-crystal. The Tate module of 𝐻 is defined as 𝑇𝐻 := {𝑥 ∈ 𝐻 : 𝜑(𝑥) = 𝑝𝑥}. Remark 2.5.2. The Tate conjecture (established for K3 surfaces over finite fields due to works of several authors) implies that 𝑐 1 induces an isomorphism after tensoring with Q 𝑝 . Hence the Picard rank of 𝑋 is given by 𝜌(𝑋) := rankZ 𝑝 Pic(𝑋) = rankZ 𝑝 𝑇𝐻 . Proposition 2.5.5 (Proposition 4.7, [18]). Let (𝐻, 𝜑, ⟨, ⟩) be a supersingular K3 𝐹-crystal. Then rank𝑊 𝐻 = rankZ 𝑝 𝑇𝐻 and the bilinear form ⟨, ⟩ restricted to 𝑇𝐻 is non-degenerate form ⟨, ⟩ : 𝑇𝐻 ⊗Z 𝑝 𝑇𝐻 → Z 𝑝 which is not perfect. In particular, 1. ord 𝑝 (𝑇𝐻 , ⟨, ⟩) = 2𝜎0 > 0 for some integer 𝜎0 called the Artin invariant of 𝐻. 2. (𝑇𝐻 , ⟨, ⟩) is determined up to isometry of polarized Z 𝑝 -lattices by 𝜎0 . 3. There exists an orthogonal decomposition (𝑇𝐻 , ⟨, ⟩) = (𝑇0 , 𝑝⟨, ⟩) ⊕ (𝑇1 , ⟨, ⟩) where (𝑇0 , ⟨, ⟩) and (𝑇1 , ⟨, ⟩) are Z 𝑝 -lattices with perfect bilinear forms such that rankZ 𝑝 𝑇0 = 2𝜎0 and rankZ 𝑝 𝑇1 = rank𝑊 𝐻 − 2𝜎0 , respectively. Thus 𝜎0 is an integer between 1 and 10. The 𝜇-ordinary polygon and Kottwitz method The main references of this section are [14, 19, 25]. Let 𝑘 be a perfect algebraically closed field in characteristic 𝑝. We briefly recall Kottwitz method to compute the set of admissible Newton polygons of an 𝐹-isocrystals with compatible bilinear form and an action of a split unramified semi-simple algebra 𝐵 over Q 𝑝 . Via Morita equivalence, it suffices to consider unramified extension 𝐵 of Q 𝑝 . 22 Let 𝐵 be a degree 𝑑 unramified extension of Q 𝑝 and O𝐵 be a maximal order of 𝐵 with residue field F𝑞 . By an 𝐹-crystal with O𝐵 structure we mean an 𝐹-crystal (𝑀, 𝜑, ⟨.⟩) together with an injection O𝐵 ↩→ End(𝑀, 𝜑) and an involution ∗ : O𝐵 → O𝐵 such that for all 𝑏 ∈ O𝐵 ⟨𝑏−, −⟩ = ⟨−, 𝑏 ∗ −⟩. Let 𝐼 := Hom(O𝐵 , 𝑊 (F 𝑝 )). The group 𝐼 can be identified with Gal(F𝑞 /F 𝑝 ) and the cyclic group Z/𝑑Z. Since 𝑀 is a module over O𝐵 ⊗ 𝑊 (𝑘) and 𝑘 contains F 𝑝 , we have a decomposition into character spaces 𝑀 = ⊕𝑖∈𝐼 𝑀𝑖 . For each 𝑖, the map 𝜑 restricts to a 𝜎-linear map 𝜑𝑖 : 𝑀𝑖 → 𝑀𝑖+1 . Then 𝜑 𝑑 restricts to a 𝜎 𝑑 -linear map on each 𝑀𝑖 and 𝜙 is a morphism between the 𝐹-crystals (𝑀𝑖 , 𝜙 𝑑 ) and (𝑀𝑖+1 , 𝜙 𝑑 ). In particular, the (𝑀𝑖 , 𝜙 𝑑 ) are isogenous summands of (𝑀, 𝜙) - a slope 𝛼 appears in 𝜈(𝑀, 𝜙) with multiplicity 𝑚 if and only if the slope 𝑑𝛼 appears in 𝜈(𝑀𝑖 , 𝜙 𝑑 ) with multiplicity 𝑚/𝑑. Next, we consider the decomposition of Artinian 𝑊-modules 𝑀/𝜙(𝑀) = ⊕𝑖 𝑀𝑖 /𝜙𝑖 (𝑀𝑖−1 ). Let 𝜇𝑖 be the Hodge polygon of 𝑀𝑖 /𝜙𝑖 (𝑀𝑖−1 ) so that 𝜇(𝑀) is the amalgamate sum of the 𝜇𝑖 . The 𝜇-ordinary polygon (also called 𝜎-invariant Hodge polygon) of 𝑀 is defined as 1 ∑︁ 𝑖 𝜎 (𝜇(𝑀)). 𝜈𝑜𝑟 𝑑 := 𝑑 𝑖∈Z/𝑑Z Here the sum is entry-wise where we organize 𝜇(𝑀) into 𝑑 blocks corresponding to the 𝜇𝑖 and 𝜎𝑖 (𝜇(𝑀)) is obtained from 𝜇(𝑀) by replacing 𝜇 𝑗 with 𝜇 𝑗+𝑖 for all 𝑗. Let us compute the 𝜇-ordinary polygons for the Shimura variety 𝑆ℎ(𝐺, 𝑓 )F 𝑝 from 2.3. Example 2.5.6. Let 𝑥 ∈ 𝑆ℎ(𝐺, 𝑓 )(F 𝑝 ) and 𝐴𝑥 be the corresponding abelian variety 1 ( 𝐴/𝑊 (F )) is an 𝐹-crystal together over F 𝑝 . The crystalline cohomolgy 𝑀 := 𝐻𝑐𝑟𝑖𝑠 𝑝 with endomorphism by the split unramified semi-simple algebra Q[𝐺] ⊗ Q 𝑝 . The action of the maximal order Ö O𝑑𝔬 ,𝔬 Z[𝐺] ⊗ Z 𝑝 = 𝔬∈𝔒 23 induces a decomposition 𝑀 = ⊕𝔬∈𝔒 𝑀𝔬 into 𝐹-crystals with O𝑑𝔬 ,𝔬 structure 𝑀𝔬 . The 𝜇-ordinary polygon of 𝑀 is defined as the amalgamate sum of the 𝜇-ordinary polygons of the 𝑀𝔬 . Let 𝑠(𝔬) be the number of distinct values of { 𝑓 (𝜏) | 𝜏 ∈ 𝔬} in [1, 𝑔(𝔬) − 1]. Let 𝐸 (1), . . . , 𝐸 (𝑠(𝔬) denote these distinct values such that 𝐸 (0) := 𝑔(𝔬) > 𝐸 (1) > · · · > 𝐸 (𝑠(𝔬)) > 𝐸 (𝑠(𝔬) + 1) := 0. Lemma 2.5.1. With notations as above, the 𝜇-ordinary polygon 𝜇𝔬 of 𝑀𝔬 is given by the 𝑠(𝔬) + 1 distinct slopes 0 ≤ 𝛼(0) < 𝛼(1) < · · · < 𝛼(𝑠(𝔬)) ≤ 1 such that for all 0 ≤ 𝑙 ≤ 𝑠(𝔬) 𝑡 1 ∑︁ 𝛼(𝑡) = #{𝜏 ∈ 𝔬 | 𝑓 (𝜏) = 𝐸 (𝑖)} #𝔬 0 with multiplicity 𝑚(𝑡) = #𝔬 · (𝐸 (𝑙) − 𝐸 (𝑙 + 1)). Proof. For each orbit 𝔬, we identify 𝔬 with Hom(O𝑑𝔬 ,𝔬 , 𝑊 (𝐹 𝑝 )) and Z/#𝔬Z. Consider the decomposition into character spaces 𝑀𝔬 = ⊕𝜏𝑖 ∈𝔬 𝑀𝑖 . By a comparison theorem between Betti and algebraic de Rham cohomology, 𝜇𝑖 has the form {0, . . . , 0, 1, . . . , 1} with multiplicities 𝑔(𝔬) − 𝑓 (𝜏𝑖 ) and 𝑓 (𝜏𝑖 ). The formula then follows from averaging over the 𝜇𝑖 . □ By [34, Theorem 1.6], the Newton polygons that occur on 𝑆ℎ(𝐺, 𝑓 )F 𝑝 are precisely of the forms ⊕𝔬∈𝔒 𝜈𝔬 such that the following conditions hold: 1. If 𝔬∗ ≠ 𝔬, then 𝑀𝔬  𝑀𝔬∨∗ (1), i.e., if 𝜈𝔬 has slopes 𝛼1 , . . . , 𝛼𝑠 with multiplicities 𝑚 1 , . . . , 𝑚 𝑠 then 𝜈𝔬∗ has slopes 1 − 𝛼1 , . . . , 1 − 𝛼𝑠 with multiplicities 𝑚 1 , . . . , 𝑚 𝑠 . If 𝔬∗ = 𝔬, then 𝜈𝔬 is symmetric, i.e. if 𝛼 is a slope with multiplicity 𝑚 then 1 − 𝛼 is also a slope with multiplicity 𝑚. 24 2. 𝜈𝔬 ⪰ 𝜇𝔬 . 3. The multiplicities of slopes in 𝜈𝔬 are divisible by #𝔬. The polygons satisfying all these conditions are also called admissible. The set of admissible polygons is called the Kottwitz set and denoted by 𝐵(𝐺, 𝑓 ). The highest polygon in this set is called the basic polygon. Let 𝐹 be an abelian extension of Q of degree 𝑚 and 𝑝 be an unramified rational prime for 𝐹. As 𝐹 is abelian, we have a well-defined Frobenius element Fr 𝑝 in Gal(𝐹/Q). We identify the set of primes of 𝐹 above 𝑝 with the set 𝔒 of Fr 𝑝 -orbits in Gal(𝐹/Q). Let 𝐾 be the subfield of 𝐹 over which 𝑝 splits completely, i.e., 𝐾 = 𝐹 ⟨Fr 𝑝 ⟩ , and set 𝑑 := [𝐹 : 𝐾]. We have Ö 𝐹 ⊗ Q𝑝 ≃ 𝐹𝔬 𝔬∈𝔒 where each 𝐹𝔬 is an unramified extension of degree 𝑑 of Q 𝑝 . Let 𝜎 ∗ denote the image of 𝜎 under the action of complex conjugation in Gal(𝐹/Q). In particular, 𝜎 ∗ = 𝜎 if 𝐹 is totally real. Lemma 2.5.2. Let (𝑀, 𝜑, ⟨, ⟩) be a K3 F-crystal over Z 𝑝 of rank 𝑛 together with endomorphism by a maximal order O of 𝐹 ⊗ Q 𝑝 such that ⟨𝑏−, −⟩ = ⟨−, 𝑏 ∗ −⟩ for all 𝑏 ∈ O. Then the following hold: 1. If 𝔬 ≠ 𝔬∗ or 𝑛 > 𝑚, then 𝜇 𝑜𝑟 𝑑 (𝑀) has slopes 1 − 𝑑1 , 1, 1 + 𝑑1 with multiplicities 𝑑, 𝑛 − 2𝑑, 𝑑. 2. If 𝔬 = 𝔬∗ and 𝑛 = 𝑚, then 𝜇 𝑜𝑟 𝑑 (𝑀) is supersingular, i.e., has slopes 1 only. Proof. Consider the decomposition of (𝑀, 𝜑) into the 𝐹-crystals with O𝐹𝔬 structure 𝑀𝔬 of rank 𝑛/#𝔒 𝑀 ≃ ⊕𝔬∈𝔒 𝑀𝔬 . The compatibility condition implies that the Hodge slopes of 𝑀𝔬 and 𝑀𝔬∗ are in bijection under the map 𝜆 ↦→ 2 − 𝜆. As the Hodge polygon of 𝑀 has the form (0, 1, . . . , 1, 2), there exists a unique 𝔢 ∈ 𝔒 such that 𝑀𝔢 has Hodge slope 0. Now a routine case by case calculation shows that 1. If 𝔬 ≠ 𝔬∗ then 𝜇 𝑜𝑟 𝑑 (𝑀𝔢 ) is obtained by averaging over the 𝑑 Hodge polygons of length 𝑛/𝑚, one of the form {0, 1, . . . , 1} and 𝑑 −1 of the form {1, 1, . . . , 1}. 25 Similarly, 𝜇 𝑜𝑟 𝑑 (𝑀𝔢∗ ) is the average of the 𝑑 Hodge polygons of length 𝑛/𝑚: one of the form {1, . . . , 1, 2} and 𝑑 − 1 of the form {1, 1, . . . , 1}. The remaining 𝑀𝔬 have slopes 1 only. 2. If 𝔬 = 𝔬∗ and 𝑛 > 𝑚 then 𝜇 𝑜𝑟 𝑑 (𝑀𝔢 ) is obtained by averaging over the 𝑑 Hodge polygons of length 𝑛/𝑚: one of the form {0, 1, . . . , 1}, one of the form {1, . . . , 1, 2}, and 𝑑 − 2 of the form {1, 1, . . . , 1}. The remaining 𝑀𝔬 have slopes 1 only. 3. If 𝔬 = 𝔬∗ and 𝑛 = 𝑚 then 𝜇 𝑜𝑟 𝑑 (𝑀𝔢 ) is obtained by averaging over the 𝑑 Hodge polygons of length 1: one of the form {0}, one of the form {2}, and 𝑑 − 2 of the form {1}, which results in slopes 1 only. □ The Artin invariant and Weyl group element The main references for this section are [9, 22]. In [22], the authors define orthogonal 𝐹-zips and prove that they are classified by certain elements of the Weyl group of the associated orthogonal group called the final types. The example of interest to us is the primitive de Rham cohomology 𝐻 2𝑝𝑟𝑖𝑚 (𝑋, 𝑘) of a K3 surface over 𝑘 which is the reduction mod 𝑝 of its primitive K3 F-crystal 𝐻 2𝑝𝑟𝑖𝑚 (𝑋/𝑊). We describe to state the main result from [9] that describes the correspondence from final types to Newton polygons and Artin invariants. Let (𝑀, 𝜑, ⟨, ⟩) be a K3 F-crystal of even rank 2𝑛. Let (𝑀, 𝜑, ⟨, ⟩) denote the reduction mod 𝑝 which is a vector space over 𝑘 equipped with an orthogonal filtration . Let 𝑆𝑂 (2𝑛) denote the standard special orthogonal group. With a suitable choice of basis, the Weyl group 𝑊𝑆𝑂 (2𝑛) can be identified with the subgroup of 𝔖2𝑛 of permutations 𝜋 such that 𝜋(𝑖) + 𝜋(2𝑛 + 1 − 𝑖) = 2𝑛 + 1 and the number of 𝑖 with 𝜋(𝑖) > 𝑛 + 1 is even. The 𝑛 simple reflections of 𝑊𝑆𝑂 (2𝑛) are the permutations 𝑠𝑖 = (𝑖, 𝑖 + 1)(2𝑛 − 𝑖, 2𝑛 + 1 − 𝑖) for 𝑖 = 1, . . . , 𝑛 − 1, and 𝑠𝑛 = (𝑛 − 1, 𝑛 + 1)(𝑛, 𝑛 + 2). The length of a permutation 𝜋 is defined as 𝑙 (𝜋) = #{1 ≤ 𝑖 < 𝑗 ≤ 𝑛 : 𝜋(𝑖) > 𝜋( 𝑗)} + #{1 ≤ 𝑖 < 𝑗 ≤ 𝑛 : 𝜋(𝑖) + 𝜋( 𝑗) > 2𝑛 + 1}. We set 𝐽 := {𝑠2 , . . . , 𝑠𝑛 } so that the set of cosets 𝑊𝐽 ⧹𝑊𝑆𝑂 (2𝑛) consists of 2𝑛 elements 𝜔1 ≻ 𝜔2 ≻ · · · ≻ 𝜔2𝑛 with the total Bruhat order. Proposition 2.5.6 (Theorem 7.1. [9]). With notations as above, we have: 26 1. 𝑀 has finite height ℎ ≤ 𝑛 if and only if 𝑀 has final type 𝜔 ℎ . 2. 𝑀 is supersingular with Artin invariant 𝜎0 ≤ 𝑛 if and only if 𝑀 has final type 𝜔2𝑛+1−𝜎0 . Lemma 2.5.3. Let 𝐵 be a degree 𝑑 unramified extension of Q 𝑝 and (𝑀, 𝜑, ⟨, ⟩) be K3 crystal with O𝐵 structure of rank 𝑚 over O𝐵 . Assume that we are in the situation where 𝑏 ∗ ≠ 𝑏, then 1. If 𝑀 has finite height ℎ, then 𝑑 divides ℎ. 2. If 𝑀 is supersingular with Artin invariant 𝜎0 , then 𝑑 divides 𝜎0 . Proof. 1. We recall the decomposition into character spaces that are isogenous 𝐹-crystals 𝑀 = ⊕𝜏∈𝐼 𝑀𝜏 . In particular, in the case of finite height, the multiplicity ℎ is divisible by the cardinality 𝑑 of 𝐼. 2. From ⟨𝑏𝑥, 𝑦⟩ = ⟨𝑥, 𝑏 ∗ 𝑦⟩ and 𝑏 ≠ 𝑏 ∗ , we deduce that the bilinear form admits an orthogonal decomposition (𝑀, ⟨, ⟩) = ⊕𝜏∈𝐼 (𝑀𝜏 , ⟨, ⟩𝜏 ). By definition, the Artin invariant is the 𝑝-ordinal of the discriminant of ⟨, ⟩ restricted to a basis of Tate classes of 𝑀. Let 𝑥 = (𝑥 𝜏𝑖 )𝜏𝑖 ∈𝐼 and 𝑦 = (𝑦 𝜏𝑖 )𝜏𝑖 ∈𝐼 be two Tate classes, i.e., 𝜑(𝑥) = 𝑝𝑥 and 𝜑(𝑦) = 𝑝𝑦. Since 𝜑 : 𝑀𝜏𝑖 → 𝑀𝜏𝑖+1 , we nave 𝜑(𝑥 𝜏𝑖 ) = 𝑝𝑥 𝜏𝑖+1 and 𝜑(𝑦 𝜏𝑖 ) = 𝑝𝑦 𝜏𝑖+1 . It suffices to show that the discriminant of each ⟨, ⟩𝜏 is the same, i.e., for all 𝜏𝑖 ∈ 𝐼, we have ⟨𝑥 𝜏𝑖 , 𝑦 𝜏𝑖 ⟩𝜏𝑖 = ⟨𝑥 𝜏𝑖+1 , 𝑦 𝜏𝑖+1 ⟩𝜏𝑖+1 . Indeed we can verify that 𝑝 2 ⟨𝑥 𝜏𝑖 , 𝑦 𝜏𝑖 ⟩𝜏𝑖 = ⟨𝜑(𝑥 𝜏𝑖 ), 𝜑(𝑦 𝜏𝑖 )⟩𝜏𝑖 = ⟨𝑝𝑥 𝜏𝑖+1 , 𝑝𝑦 𝜏𝑖+1 ⟩𝜏𝑖 = 𝑝 2 ⟨𝑥 𝜏𝑖+1 , 𝑦 𝜏𝑖+1 ⟩𝜏𝑖+1 . □ 27 Chapter 3 CALCULATION OF FAMILIES OF K3-TYPE SURFACES In this chapter, we review the construction and show important features of the families for which our main theorems apply. Inventory of examples Let 𝐶 = 𝐶 (𝐺, 𝑎, 𝑁𝐶 ) and 𝐷 = 𝐷 (𝐺, 𝑏, 𝑁 𝐷 ) be two monodromy data. The group scheme 𝜇𝐺 acts diagonally on the product 𝐶 × 𝐷 via (𝑥, 𝑦, 𝑧, 𝑡) ↦→ (𝑥, 𝜁 𝑦, 𝑧, 𝜁 −1 𝑡). Let C[𝐶] and C[𝐷] denote the affine coordinate rings over C. We consider the GIT quotient 𝑆 := (𝐶 × 𝐷)/𝜇𝐺 which has isolated singularities of cyclic-quotient type. After blowing up all these singularities, we get a smooth surface 𝑆˜ that may not be minimal in general. One can compute the minimal model 𝑋 := 𝑆˜min by searching and blowing down the (-1)-divisors. Definition 3.0.1. We call the monodromy datum (𝐺, 𝑎, 𝑏, 𝑁𝐶 , 𝑁 𝐷 ) a K3 type pair if the resulting surface 𝑋 is of K3 type. Using the birational invariance of ℎ2,0 and Kunneth formula, we can show that ∑︁ ℎ2,0 (𝑋) = ℎ2,0 ((𝐶 × 𝐷)/𝜇𝐺 ) = dimC 𝐻 2,0 (𝐶 × 𝐷) 𝐺 = 𝑓𝐶 (𝜏) 𝑓 𝐷 (𝜏 ∗ ). (𝜏,𝜏 ∗ ) Proposition 3.0.1. 𝑋 is a K3 type if and only if there exists a unique 𝜏 ∈ T such that 𝑓𝐶 (𝜏) 𝑓 𝐷 (𝜏 ∗ ) = 1 and 𝑓𝐶 (𝜏) 𝑓 𝐷 (𝜏 ∗ ) = 0 otherwise. Proof. This follows directly from the definition of k3-type surface and the Kunneth formula for ℎ2,0 (𝑋). □ Corollary 3.0.2. If the monodromy data 𝐶 (𝐺, 𝑎, 𝑁𝐶 ) and 𝐷 (𝐺, 𝑏, 𝑁 𝐷 ) is a K3 type pair, then 𝑁𝐶 = 3 or 𝑁 𝐷 = 3. Assume that 𝑁 𝐷 = 3, then 𝑁𝐶 − 3 = dim 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ). 𝐶 Proof. From Proposition 2.3.1, 𝑓𝐶 (𝜏𝑙 )+ 𝑓𝐶 (𝜏−𝑙 ) = 𝑁𝑐 −2 and 𝑓𝐶 (𝜏𝑙 )+ 𝑓𝐶 (𝜏−𝑙 ) = 𝑁 𝐷 − 2 for any 𝑙 ∈ (Z/𝑚Z) ∗ . This means the unique index 𝑙 such that 𝑓𝐶 (𝜏𝑙 ) 𝑓 𝐷 (𝜏−𝑙 ) = 1 28 is in (Z/𝑚Z) ∗ . Assume on the contrary that both 𝑁𝐶 and 𝑁 𝐷 are greater than 3 so that 𝑁𝑐 − 2 and 𝑁 𝐷 − 2 are both greater than 1. Thus, 𝑓𝐶 (𝜏−𝑙 ) and 𝑓 𝐷 (𝜏𝑙 ) are both at least 1 so 𝑓𝐶 (𝜏−𝑙 ) 𝑓 𝐷 (𝜏𝑙 ) is at least 1, which is a contradiction. Without loss of generality, assume that 𝑁 𝐷 = 3 so { 𝑓 𝐷 (𝜏𝑙 ), 𝑓 𝐷 (𝜏−𝑙 )} = {0, 1} for every 𝑙 ∈ (Z/𝑚Z) ∗ . Let 𝑙0 be the unique index such that 𝑓 𝐷 (𝜏𝑙0 ) = 𝑓𝐶 (𝜏−𝑙0 ) = 1 so 𝑓𝐶 (𝜏𝑙0 ) 𝑓𝐶 (𝜏−𝑙0 ) = 𝑁𝐶 − 3. For other 𝑙 ∈ (Z/𝑚Z) ∗ , 𝑓𝐶 (𝜏𝑙 ) 𝑓𝐶 (𝜏−1 ) = 0 since if both are nonzero then 𝑓 𝐷 (𝜏𝑙 ) = 𝑓 𝐷 (𝜏−𝑙 ) = 0 but their sum is 1. Í It follows from 2.3 that dim 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) = 𝑓𝐶 (𝜏𝑙 ) 𝑓𝐶 (𝜏−𝑙 ) = 𝑁𝐶 − 1. □ 𝐶 The unique 𝜏 in 3.0.1 can be chosen to be 𝜏−1 without changing the isomorphism class of the monodromy datum and this choice corresponds to normalizing the inertia data such that ∑︁ ∑︁ 𝑎𝑖 = 2𝑚 and 𝑏𝑖 = 𝑚. To summarize, we shall work with normalized K3 type datum (𝐺, 𝑎, 𝑏, 𝑁) satisfying the following conditions: 1. 𝑎 1 + · · · + 𝑎 𝑁 = 2𝑚 and gcd(𝑎 1 , . . . , 𝑎 𝑁 , 𝑚) = 1; 2. 𝑏 1 + 𝑏 2 + 𝑏 3 = 𝑚 and gcd(𝑏 1 , 𝑏 2 , 𝑏 3 , 𝑚) = 1; 3. 𝑓𝐶 (𝜏−1 ) 𝑓 𝐷 (𝜏1 ) = 1 and 𝑓𝐶 (𝜏−𝑖 ) 𝑓 𝐷 (𝜏𝑖 ) = 0 for 𝑖 ≠ 1. Remark 3.0.1. Moonen only considered the case 𝑁𝐶 ≥ 4 to obtain examples of positive dimensional families but the construction also works for 𝑁𝐶 = 3. The following example illustrates how to compute the minimal model of the resolution of singularities. Example 3.0.1. Take (5, (3, 3, 4), (1, 1, 3), 3) to be the datum. This datum defines affine curves 𝐶 : 𝑦 5 = 𝑥 3 (𝑥 − 1) 3 and 𝐷 : 𝑧5 = 𝑡 (𝑡 − 1). They correspond to a K3 type pair since 𝑓𝐶 = (0, 1, 0, 1) and 𝑓 𝐷 = (1, 1, 0, 0). The action of 𝜇5 on the product is given by (𝑥, 𝑦, 𝑡, 𝑧) ↦→ (𝑥, 𝜁5 𝑦, 𝑡, 𝜁5−1 𝑧). The ring of invariants is generated by 𝑥, 𝑡, and 𝑦𝑧. 29 The surface 𝑆 = (𝐶 × 𝐷)/𝜇5 has affine model (𝑦𝑧) 5 = 𝑥 3 (𝑥 − 1) 3 𝑡 (𝑡 − 1). On 𝑆 the singularities are of the form (𝑃𝑖 , 𝑄 𝑗 ) where 𝑃𝑖 ∈ {0, 1, ∞} and 𝑄 𝑗 ∈ {0, 1, ∞}. Let 𝑌¯𝑖 be the image of {𝑃𝑖 } × 𝐷 in 𝑆 and 𝑍¯ 𝑗 be the image of 𝐶 × {𝑄 𝑗 } under the quotient map 𝐶 × 𝐷 → 𝑆. Then 𝑌¯𝑖 ∩ 𝑍¯ 𝑗 = {(𝑃𝑖 , 𝑄 𝑗 )} and each point (𝑃𝑖 , 𝑄 𝑗 ) is a cyclic-quotient singularity of type (𝑛, 𝑟) = (𝑛(𝑖, 𝑗), 𝑟 (𝑖, 𝑗)) as in the following table: (5, 3) (5, 3) (5, 4) 𝑍¯ 1 (5, 3) (5, 3) (5, 4) 𝑍¯ 2 (5, 1) (5, 1) (5, 3) 𝑍¯ 3 𝑌¯1 𝑌¯2 𝑌¯3 Resolving these singularities we obtain 𝑆˜ → 𝑆 that looks like: −1 −1 −2 −1 −5 𝑍3 −2 −5 −3 𝑆˜ −2 −2 −3 −2 −2 −2 −3 𝑍2 −2 −2 −2 −2 −3 𝑌1 −2 −2 −3 𝑌2 −2 𝑌3 𝑍1 −2 −2 ˜ The continued fraction repwhere 𝑌𝑖 , 𝑍 𝑗 are the strict transforms of 𝑌¯𝑖 , 𝑍¯ 𝑗 to 𝑆. resentation of 𝑛/𝑟 determines the string of exceptional divisors 𝐸𝑖,(𝑘) 𝑗 connecting 𝑌𝑖 and 𝑍 𝑗 together with their self-intersection numbers. We can also compute the self-intersection number of the canonical sheaf 𝐾𝑆2˜ = −4. To get a minimal surface, (2) we first blow down the (-1)-curves 𝑌1 , 𝑌2 , and 𝑍3 . After blowing down 𝑍3 , 𝐸 3,3 becomes a (-1)-curve so we blow down this divisor and arrive at a surface 𝑋 with 𝐾 𝑋2 = 0, or in other words, 𝑋 is a K3 surface. Minimal model of resolution of singularities We now describe the well-known procedure in general which involves 2 main steps: ˜ then perform first resolve all singularities on 𝑆 to get a non-singular resolution 𝑆; 30 a search for (-1)-exceptional divisors on 𝑆˜ and blown them down to get a minimal model 𝑋. Resolving singularities (see [24, Section 1.2]). Let 𝑠 : 𝑆˜ → 𝑆 be the total blowup of singularities associated to the K3 type data (𝐺, 𝑎, 𝑏, 𝑁). Let 𝑃𝑖 ∈ {𝜆1 , . . . , 𝜆 𝑁 } and 𝑄 𝑗 ∈ {0, 1, ∞}. Let 𝑌¯𝑖 be the image of {𝑃𝑖 } × 𝐷 in 𝑆 and 𝑍¯ 𝑗 be the image of 𝐶 × {𝑄 𝑗 } under the quotient map 𝐶 × 𝐷 → 𝑆. We set ℎ = 𝑙𝑐𝑚(gcd(𝐺, 𝑎𝑖 ), gcd(𝐺, 𝑏 𝑗 )), 𝑚 𝑛(𝑖, 𝑗) = , ℎ 𝑎𝑖 𝑐(𝑖) = ( ) −1 ∈ (Z/gcd(𝐺, 𝑎𝑖 )Z) ∗ , gcd(𝐺, 𝑎𝑖 ) 𝑏𝑗 𝑑 ( 𝑗) = ( ) −1 ∈ (Z/gcd(𝐺, 𝑏 𝑗 )Z) ∗ , gcd(𝐺, 𝑏 𝑗 ) 𝑟 (𝑖, 𝑗) = 𝑐(𝑖)𝑑 ( 𝑗) −1 ∈ (Z/𝑛(𝑖, 𝑗)Z) ∗ . Then 𝑌¯𝑖 and 𝑍¯ 𝑗 intersect in gcd(𝑎𝑖 , 𝑏 𝑗 , 𝑚) singular points of cyclic-quotient type ˜ and let given by the pair (𝑛, 𝑟). Let 𝑌𝑖 , 𝑍 𝑗 be the strict transforms of 𝑌¯𝑖 , 𝑍¯ 𝑗 to 𝑆, 𝑛/𝑞 = [𝑟 1 , . . . , 𝑟 𝑡 ] be the continued fraction expansion. For each 𝑅 ∈ 𝑌¯𝑖 ∩ 𝑍¯ 𝑗 , the exceptional fiber 𝑠−1 (𝑅), so-called the Hirzbrunch-Jung string, consists of a chain 𝐸 1 , . . . , 𝐸 𝑡 of rational curves such that in the sequence 𝑍 𝑗 , 𝐸 1 , . . . , 𝐸 𝑡 , 𝑌𝑖 each curve transversally intersects the next and the self-intersection number is given by 𝐸 𝑘2 = −𝑟 𝑘 for 𝑘 = 1, . . . , 𝑡. Furthermore, if 𝑍𝑖 contains 𝑣 singularities of types (𝑛 𝑘 , 𝑟 𝑘 ) for 𝑘 = 1, . . . , 𝑣 then Í its self-intersection number 𝑍𝑖2 = 𝑣𝑘=1 −𝑟 𝑘 /𝑛 𝑘 . On the other hand, if 𝑌 𝑗 contains 𝑣 singularities of types (𝑛 𝑘 , 𝑟 𝑘 ) for 𝑘 = 1, . . . , 𝑣 then the self-intersection number Í 𝑌 𝑗2 = 𝑣𝑘=1 −𝑟 ′𝑘 /𝑛 𝑘 where 𝑟 ′𝑘 is the inverse of 𝑟 𝑘 (mod 𝑛 𝑘 ) (see [24, Proposition 2.8]). Finally, the self-intersection number of 𝑆˜ is given by 𝐾𝑆2˜ = 8(𝑔𝐶 − 1)(𝑔 𝐷 − 1) + 𝑚 ∑︁ ℎ𝑅 𝑅∈Sing(𝑆) with 𝑡 2 + 𝑟 + 𝑟 ′ ∑︁ − (𝑟 𝑘 − 2) ℎ𝑅 = 2 − 𝑛 𝑘=1 where 𝑅 has singularity type (𝑛, 𝑟) = [𝑟 1 , . . . , 𝑟 𝑡 ] and 𝑟 ′ is the inverse of 𝑟 (mod 𝑛) (see [24, Proposition 2.5]). 31 Getting the minimal model. We first write down the intersection matrix whose rows correspond to all strict transforms and exceptional divisors in the HirzbrunchJung strings. In each iteration, we simply look for the next divisor whose selfintersection number is -1 and whose coefficient in 𝐾𝑆˜ is positive, remove the corresponding row, and update our matrix accordingly. It follows that the only curves that may be blown down must come from following types: 𝑚−𝑎 𝑖 1. the (-1)-strict transforms 𝑌𝑖 for which gcd(𝐺,𝑎 > 1; 𝑖) 𝑏 𝑗 2. the (-1)-strict transforms 𝑍 𝑗 for which gcd(𝐺,𝑏 > 1; 𝑗) 3. the exceptional components 𝐸 𝑘 in the Hirzbrunch-Jung strings such that either 𝑌𝑖 or 𝑍 𝑗 got blown down. Once we have exhausted all such divisors, we arrive at the minimal model 𝑋 with selfintersection number 𝐾 𝑋2 = 𝐾𝑆2˜ + 𝑤 where 𝑤 is the number of (-1)-curves which were blown down. This process can be efficiently done via a computer program: given as input the data (𝐺, 𝑎, 𝑏, 𝑁), the output 𝐾 𝑋2 determines the Kodaira dimension of 𝑋. Moonen shows that 𝑋 is a K3 surface if 𝐾 𝑋2 = 0 and is of general type otherwise. Ring of definition. Let Z ⊂ 𝑅 ⊂ C be a ring. We shall show that if 𝐶 and 𝐷 are defined over 𝑅, so is 𝑋. Proposition 3.0.3. Assume that 𝐶 and 𝐷 are defined over 𝑅. Then the affine subring of invariants C[𝐶 × 𝐷] 𝐺 ⊂ C[𝐶 × 𝐷] is defined over 𝑅. Î Î Proof. We have C[𝐶 × 𝐷] ≃ C[𝑥, 𝑦, 𝑧, 𝑡]/[𝑦 𝑚 = 𝑖 (𝑥 − 𝜆𝑖 ) 𝑎𝑖 , 𝑡 𝑚 = 𝑗 (𝑧 − 𝜇 𝑗 ) 𝑏 𝑗 ], where 𝜆 and 𝜇 ∈ 𝑅. Since the 𝐺-action does not change the degree of a monomial, it suffices to find the ring of invariant monomials. This set consists of precisely the monomials whose 𝑦’s degree and 𝑡’s degree are equal, i.e., Ö Ö C[𝐶 × 𝐷] 𝐺 = C[𝑥, 𝑧, 𝑦𝑡]/[(𝑦𝑡) 𝑚 = (𝑥 − 𝜆𝑖 ) 𝑎𝑖 (𝑧 − 𝜇 𝑗 ) 𝑏 𝑗 ], 𝑖 and this descends to 𝑅[𝑥, 𝑧, 𝑦𝑡]/[(𝑦𝑡) 𝑚 = defined over 𝑅. Î 𝑖 (𝑥 − 𝜆𝑖 ) 𝑗 𝑎 𝑖 Î (𝑧 − 𝜇 ) 𝑏 𝑗 ] which is 𝑗 𝑗 □ As a corollary from GIT, the quotient 𝑆 := (𝐶 × 𝐷)/𝐺 is defined over 𝑅. Furthermore, the (-1)-divisors are rational, i.e., isomorphic to P1 , so the minimal model 𝑋 32 is defined over 𝑅. In particular, if 𝐶 and 𝐷 are defined over Z, then 𝑋 is defined over Z. The reduction mod 𝑝 of 𝑋 when 𝑝 ∤ 𝑚 is smooth a projective k3-type surface over F 𝑝 . Polarization. We construct a polarization on 𝑋 as following. On 𝐶 × 𝐷 we consider the tensor product 𝐿 := 𝐿 1 ⊗ 𝐿 2 of principal ample line bundles on 𝐶 and 𝐷. Then the tensor power 𝐿 𝑚 is a 𝐺-invariant ample line bundle on 𝐶 × 𝐷 where 𝐺 acts via 𝑚 𝑡ℎ roots of unity. Then we take the pushforward of 𝐿 𝑚 under the finite map 𝐶 × 𝐷 → 𝑆 to obtain an ample line bundle 𝐿 ′ on 𝑆. By the projection formula and the birational invariance of ℎ2,0 (𝑆, 𝐿 ′), we get an ample line bundle of degree 2𝑚 on 𝑋. One can extend this construction to a family by considering an ample line bundle over the base H (𝐺, 𝑎). Corollary 3.0.4. For any K3 type datum (𝐺, 𝑎, 𝑏, 𝑁), the family X(𝐺, 𝑎, 𝑏, 𝑁) has 1 a model over Z[ 2𝑚 ]. 33 Chapter 4 PROOF OF THE FIRST MAIN THEOREM Let X = X(𝐺, 𝑎, 𝑏, 𝑁) be the family associated to the K3 type datum (𝐺, 𝑎, 𝑏, 𝑁). Let 𝑝 ∤ 2𝑚 be a prime and 𝑋 be a k3-type surface in X. By abuse of notation, we use 𝑋 to denote the reduction of 𝑋 at 𝑝 which is a smooth proper k3-type surface in positive characteristic. We calculate the Newton polygon variation in the family X when 𝐶 moves in the family parametrized by H (𝑚, 𝑎) via two steps: 1. We first show how the Newton polygon of 𝑋 can be determined from that of 𝐶 and 𝐷. In particular, when 𝐷 is fixed and 𝐶 varies in a family, each Newton polygon of 𝑋 corresponds one-to-one to a Newton polygon of 𝐽 (𝐶) 𝑛𝑒𝑤 . 2. We find all possible Newton polygons of 𝐽 (𝐶) 𝑛𝑒𝑤 . The technical tools for our calculation are crystalline cohomology and 𝐹-crystals with additional structure. We first recall the following wonderful result due to Ekedahl. Proposition 4.0.1. [8, Proposition 4.1] Let 𝑆 be smooth projective surface over 𝑘. Suppose 𝐺 is a finite group acting on 𝑆 whose order is prime to 𝑝. Let 2 (𝑋/𝑊 (𝑘)) = 𝑋 be the minimal resolution of singularities of 𝑆/𝐺. Then 𝐻𝑐𝑟𝑖𝑠 𝐻 2 (𝑆/𝑊 (𝑘)) 𝐺 ⊕ 𝐶 where 𝐶 is the 𝑊 (𝑘)-submodule generated by the Chern classes of the exceptional divisors of 𝑋 → 𝑆/𝐺. Furthermore, the decomposition is orthogonal with respect to Poincaré pairing product and the pairing restricted to 𝐶 is perfect. As Chern classes all have slope 1, Ekedahl’s result reduces the problem to calculating 2 (𝐶 × 𝐷/𝑊 (𝑘)) 𝐺 . The Kunneth formula and 𝐺 acting trivially on the slopes of 𝐻𝑐𝑟𝑖𝑠 0 2 𝐻𝑐𝑟𝑖𝑠 and 𝐻𝑐𝑟𝑖𝑠 implies that 2 0 2 0 2 1 1 𝐻𝑐𝑟𝑖𝑠 (𝐶×𝐷) 𝐺 = 𝐻𝑐𝑟𝑖𝑠 (𝐶)⊗𝐻𝑐𝑟𝑖𝑠 (𝐷)⊕𝐻𝑐𝑟𝑖𝑠 (𝐷)⊗𝐻𝑐𝑟𝑖𝑠 (𝐶)⊕(𝐻𝑐𝑟𝑖𝑠 (𝐶)⊗𝐻𝑐𝑟𝑖𝑠 (𝐷)) 𝐺 . 0 (𝐶) (resp. 𝐻 0 (𝐷)) is 0 and 𝐻 2 (𝐷) (resp. 𝐻 2 (𝐶)) is 1 (by The slope of 𝐻𝑐𝑟𝑖𝑠 𝑐𝑟𝑖𝑠 𝑐𝑟𝑖𝑠 𝑐𝑟𝑖𝑠 compatibility of Poincaré duality), and thus, their tensor product has slope 1 and 34 Artin invariant 0 (coming from the determinant of the Kronecker product of the two pairings). The action of 𝐺  𝜇𝑚 on 𝐶 and 𝐷 induces an action of Ö Z[𝜇𝑚 ] ⊗ Z 𝑝 = O𝑑𝔬 ,𝔬 𝔬∈𝔒 on crystalline cohomology by functoriality and thus decompositions of 𝐹-crystals Ê Ê 1 1 𝐻𝑐𝑟𝑖𝑠 (𝐶) = 𝑀𝔬 (𝐶) and 𝐻𝑐𝑟𝑖𝑠 (𝐷) = 𝑀𝔬 (𝐷). 𝔬∈𝔒 𝔬∈𝔒 In particular 1 1 (𝐻𝑐𝑟𝑖𝑠 (𝐶) ⊗ 𝐻𝑐𝑟𝑖𝑠 (𝐷)) 𝐺 = Ê 𝑀𝔬 (𝐶) ⊗O𝑑𝔬 ,𝔬 𝑀𝔬∗ (𝐷), 𝔬∈𝔒 where ⊗O𝑑𝔬 ,𝔬 is the tensor product between 𝐹-crystals with O𝑑𝔬 ,𝔬 -module structure. Now let 𝜈𝔬 (𝐶) and 𝜈𝔬 (𝐷) denote the Newton polygon of 𝑀𝔬 (𝐶) and 𝑀𝔬 (𝐷) respectively. Let 𝔒′ denote the set of orbits 𝔬 such that 𝑑𝔬 ∤ 𝑏 𝑗 for all 𝑗. We show the following: Proposition 4.0.2. Each 𝜈𝔬∗ (𝐷) is isoclinic with slope 𝜆 𝑜∗ (𝐷). Furthermore 1. if 𝔬 ∈ 𝔒′, the Newton polygon of 𝑀𝔬 (𝐶) ⊗O𝑑𝔬 ,𝔬 𝑀𝔬∗ (𝐷) is obtained by adding 𝜆 𝑜∗ (𝐷) to the slopes of 𝜈𝔬 (𝐶); 2. if 𝔬 ∉ 𝔒′, the component 𝑀𝔬 (𝐶) ⊗O𝑑𝔬 ,𝔬 𝑀𝔬∗ (𝐷) is trivial. Proof. From the multiplication type of 𝐷, 𝑀𝔬∗ (𝐷)  O𝑑𝔬 ,𝔬 if 𝔬 ∈ 𝔒′ and is 0 otherwise, hence the second part. Suppose that we are now in the case 𝔬 ∈ 𝔒′. Then a calculation using Kottwitz method implies that 𝑀𝔬∗ (𝐷) is isoclinic with 𝑛 ∗ (𝐷) slope 𝜆 𝔬∗ (𝐷) = 𝔬#𝔬∗ where 𝑛𝔬∗ (𝐷) = #{𝑖 ∈ 𝔬∗ | 𝑓 𝐷 (𝑖) = 1}. Hence the first part follows. □ Proposition 4.0.3. Suppose that 𝔬 ∈ 𝔇′ and 1 ∉ 𝔬, then 𝑀𝔬 (𝐶) ⊗O𝑑𝔬 ,𝔬 𝑀𝔬∗ (𝐷) is isoclinic with slope 1. Proof. We first show that 𝑓𝐶 (𝔬) has a single nonzero multiplicity, i.e., 𝑓𝐶 (𝑖) ∈ {0, 𝑔𝐶 (𝔬)} for all 𝑖 ∈ 𝔬. Indeed, assume on the contrary that there exists 𝑖 ≠ 𝑗 ∈ 𝔬 such that 𝑔𝐶 (𝔬) ≥ 𝑓𝐶 (𝑖) > 𝑓𝐶 ( 𝑗) > 0. Hence, 𝑓𝐶 ( 𝑗 ∗ ) = 𝑔𝐶 (𝔬) − 𝑓𝐶 ( 𝑗) > 0. It follows from 𝑓𝐶 (𝑙) 𝑓 𝐷 (𝑙 ∗ ) = 0 for all 𝑙 ≠ 1 that both 𝑓 𝐷 ( 𝑗) and 𝑓 𝐷 ( 𝑗 ∗ ) must be 0: this cannot happen since 𝑓 𝐷 ( 𝑗) + 𝑓 𝐷 ( 𝑗 ∗ ) = 𝑔 𝐷 (𝔬) = 1. 35 A calculation using Kottwitz method for 𝜈𝔬 (𝐶) implies that it has a single slope 𝑛𝔬 (𝐶) ∗ #𝔬 where 𝑛𝔬 (𝐶) = #{𝑖 ∈ 𝔬 | 𝑓 (𝑖) = 𝑔𝐶 (𝔬)}. It also follows from 𝑓𝐶 (𝑙) 𝑓 𝐷 (𝑙 ) = 0 for all 𝑙 ≠ 1 that there is a bijection between the set of nonzero multiplicities in 𝑓𝐶 (𝔬) and the set of zero multiplicities in 𝑓 𝐷 (𝔬∗ ). Hence 𝑛𝔬 (𝐶) + 𝑛𝔬∗ (𝐷) = #𝔬 = #𝔬∗ and from the previous proposition 𝜈𝔬 (𝐶) + 𝜈𝔬∗ (𝐷) has slope 1 only. □ Corollary 4.0.4. When 𝐷 is fixed and 𝐶 varies in a family, the Newton polygon of 𝑋 varies in one-to-one correspondence to the Newton polygon of 𝐽 (𝐶) 𝑛𝑒𝑤 . 1 (𝐶), it follows from Proof. Since the orbit of 1 is contained in the new part of 𝐻𝑐𝑟𝑖𝑠 2 (𝑋) are uniquely determined by the two previous propositions that the slopes of 𝐻𝑐𝑟𝑖𝑠 1 (𝐶). For a smooth and proper curve, 𝐻 1 (𝐶) = the slopes of the new part of 𝐻𝑐𝑟𝑖𝑠 𝑐𝑟𝑖𝑠 1 𝐻𝑐𝑟𝑖𝑠 (𝐽𝐶 ) so their new parts are isomorphic as well. □ Deligne and Mostow [5] show that the classifying morphism 𝜃 : H (𝑚, 𝑎) → 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) for the family 𝐽 (𝐶) 𝑛𝑒𝑤 is dominant and quasi-finite onto 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ). In particular, the Zariski closure 𝑍 new (𝐺, 𝑎) is 𝑆ℎ(𝐺, 𝑓 ). On one hand, Kottwitz [14] gives a combinatorial description of a set 𝐵(𝐺, 𝑓 𝑛𝑒𝑤 ) of admissible Newton polygons equipped with the extra structure from 𝐸 𝑚 and multiplication type 𝑓 𝑛𝑒𝑤 . The Kottwitz set in the case of interest to us admits a total order 𝜈ord =: 𝜈0 ≻ 𝜈1 ≻ · · · ≻ 𝜈𝑙 := 𝜈basic where the maximal element 𝜈ord is called 𝜇-ordinary and minimal 𝜈basic called basic. Let 𝑆ℎ[𝜈𝑖 ] denote the locus of 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) with Newton polygon 𝜈𝑖 - from the moduli point of view, the locus of abelian schemes with Newton polygon 𝜈𝑖 . On the other hand, Viehman-Wedhorn [34] show that all admissible polygons occur, i.e., 𝑆ℎ[𝜈𝑖 ] is nonempty ∀𝑖, for the PEL type Shimura variety 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ). Furthermore, these loci give rise to the so-called Newton stratification by locally closed subsets 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) = 𝑆ℎ[𝜈0 ] ⊔ 𝑆ℎ[𝜈1 ] ⊔ · · · ⊔ 𝑆ℎ[𝜈𝑙 ] that satisfies 1. 𝑆ℎ[𝜈𝑖 ] is equidimensional of codimension 𝑖 and open in the Zariski closure 𝑆ℎ[𝜈𝑖 ]; 36 2. 𝑆ℎ[𝜈𝑖 ] = ⊔ 𝑗 ≥𝑖 𝑆ℎ[𝜈 𝑗 ]; 3. codim(𝑆ℎ[𝜈𝑖+1 ], 𝑆ℎ[𝜈𝑖 ]) = 1; 4. (de Jong’s purity) for a smooth irreducible subvariety inside 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ), the induced Newton stratification goes up in codimension 1. Let 𝑍 [𝜈] := 𝑍 ∩ 𝑆ℎ(𝐺, 𝑓 ) [𝜈] for any admissible 𝜈. The next result reduces our calculation to showing 𝑍 new (𝐺, 𝑎) [𝜈basic ] ≠ ∅. Proposition 4.0.5. If 𝑍 new (𝐺, 𝑎) [𝜈basic ] ≠ ∅, then 𝑍 new (𝐺, 𝑎) [𝜈𝑖 ] ≠ ∅ for all 𝑖. Proof. Firstly, 𝑍 new (𝐺, 𝑎) [𝜈ord ] ≠ ∅ since both 𝑍 new (𝐺, 𝑎) and 𝑆ℎ(𝐺, 𝑓 ) [𝜈ord ] are open and dense inside 𝑆ℎ(𝐺, 𝑓 ). Then de Jong’s purity and 𝑍 new (𝐺, 𝑎) [𝜈basic ] ≠ ∅ implies that the induced Newton stratification on 𝑍 new (𝐺, 𝑎) goes up exactly 𝑙 times (otherwise the basic locus would have had incorrect codimension inside 𝑆ℎ(𝐺, 𝑓 )). In other words, 𝑍 new (𝐺, 𝑎) [𝜈𝑖 ] ≠ ∅ for all 𝑖. □ As 𝑍 new (𝐺, 𝑎) = 𝑆ℎ(𝐺, 𝑓 ), the basic polygon 𝜈basic occurs on 𝑍 new (𝐺, 𝑎). But it is not straightforward to see why 𝜈basic already occurs on the smooth locus 𝑍 new (𝐺, 𝑎). From the moduli point of view, this means there exists a smooth curve 𝐶 (𝐺, 𝑎) for which 𝐽𝐶new has basic polygon. The next section shows that this is indeed the case, at least when 𝑝 is sufficiently large. Boundary of Hurwitz space and basic locus The following relies on the boundary behavior of the Hurwitz space H (𝐺, 𝑎) (see [35, Chapter 4] and [16, Section 5.2]). Since 𝑍 new is the image of H (𝐺, 𝑎) under the classifying map, a moduli point [𝐽𝐶new ] on the boundary 𝜕𝑍 new := 𝑍 new (𝐺, 𝑎) − 𝑍 new (𝐺, 𝑎) comes from the new part of the Jacobian of a singular curve 𝐶 = 𝐶 (𝐺, 𝑎) of compact type. In particular, 𝐶 is obtained via clutching a pair of compatible curves 𝐶1 = 𝐶 (𝐺, 𝑎 1 ) and 𝐶2 = 𝐶 (𝐺, 𝑎 2 ). Such compatible (𝑎 1 , 𝑎 2 ) is called a degeneration type of 𝐶 (𝐺, 𝑎). By definition a degeneration type satisfies 𝑓1 + 𝑓2 = 𝑓 and the Newton polygon of 𝐶 is the amalgamate sum of that of 𝐶1 and 𝐶2 . For a fixed monodromy datum (𝐺, 𝑎), we denote by 𝑇 = 𝑇 (𝐺, 𝑎) the set of all degeneration types. What is important to us is that 𝑇 is finite and 𝑓 𝑛𝑒𝑤 is of a particularly simple form so the compatibility condition ensures that 𝑓1 𝑛𝑒𝑤 and 𝑓2 𝑛𝑒𝑤 must have simple forms. Recall that the new part of the signature records the multiplicities of the action of 𝐸 𝑚 . 37 Definition 4.0.1. Let 𝑛 ≥ 3. A signature 𝑓 for an 𝐸 𝑚 -module is called simple of rank 𝑛 − 2 if { 𝑓 (𝑖), 𝑓 (𝑖 ∗ )} = {1, 𝑛 − 3} for a unique 𝑖 ∈ (Z/𝑚Z) ∗ and { 𝑓 (𝑖), 𝑓 (𝑖 ∗ )} = {0, 𝑛 − 2} otherwise. If 𝑓 is simple of rank 𝑛 − 2 then dim 𝑆ℎ(𝐺, 𝑓 ) = 𝑛 − 3. In our case, 𝑓 𝑛𝑒𝑤 is simple of rank 𝑁 − 2 so it follows from 2.3 that dim 𝑆ℎ(𝐺, 𝑓 𝑛𝑒𝑤 ) = 𝑁 − 3. By definition, if (𝑎 1 , 𝑎 2 ) is a degeneration type of (𝐺, 𝑎) then we can assume that 𝑓1 new + 𝑓2 new = 𝑓 new and 𝑓1 new is simple. That is, 𝑓1 new is simple of rank 𝑛1 − 2, 𝑓2 new satisfies { 𝑓2 (𝑖), 𝑓2 (𝑖 ∗ )} = {0, 𝑛2 − 2} for all 𝑖 ∈ (Z/𝑚Z) ∗ , and 𝑛 = 𝑛1 + 𝑛2 − 2. In what follows, we look that the new parts of the corresponding Jacobians and Shimura varieties only so we can drop new from the notation of the signature for abbreviation without any confusion. Proposition 4.0.6. Suppose that 𝑓 is simple of rank 𝑁 − 2 and let 𝔢 ∈ 𝔒 be the orbit of the unique 𝑖 where 𝑓 (𝑖) = 1. Let 𝑑 := #𝔢 and 𝑛 := #{ 𝑗 ∈ 𝔢 : 𝑓 ( 𝑗) = 𝑁 − 2}. 1. If 𝔢 ≠ 𝔢∗ , or equivalently, 𝔭 is not inert in 𝐸 𝑚 /𝐸 𝑚0 , then    #𝐵(𝐺, 𝑓 ) = 𝑁 − 2;     1 𝜈basic,𝔢 is isoclinic of slope 𝑑𝑛 + 𝑑 (𝑁−2) ;      dim 𝑆ℎ(𝐺, 𝑓 ) [𝜈basic ] = 0.  2. If 𝔢 = 𝔢∗ , or equivalently, 𝔭 is inert in 𝐸 𝑚 /𝐸 𝑚0 , then    #𝐵(𝐺, 𝑓 ) = ⌊ 𝑁2 ⌋;     𝜈basic,𝔢 is isoclinic of slope 21 ;      dim 𝑆ℎ(𝐺, 𝑓 ) [𝜈basic ] = 𝑁 − 2 − ⌊ 𝑁2 ⌋.  Proof. This is a direct calculation from the combinatorial description of 𝐵(𝐺, 𝑓 ). Recall that any 𝜈 ∈ 𝐵(𝐺, 𝑓 ) has a decomposition 𝜈 = ⊕𝔬 𝜈𝔬 . For any 𝔬 ≠ 𝔢, since 𝑓 is simple, there is a single nonzero multiplication in 𝑓 (𝔬). This means 𝜈ord,𝔬 = 𝜈basic,𝔬 . Hence, it suffices to find all possible polygons for 𝜈𝔢 . Case 1. 𝔢 ≠ 𝔢∗ 38 𝜈ord,𝔢 has slopes: 𝑑𝑛 with multiplicity 𝑑 (𝑁 − 3) and 𝑛+1 𝑑 with multiplicity 𝑑. The Newton polygon of 𝜈ord,𝔢 can be drawn in the plane with endpoints {(0, 0), (𝑑 (𝑁− 2), 𝑛(𝑁 − 2) + 1)} and an integer breakpoint (𝑑 (𝑁 − 3), 𝑛(𝑁 − 3)). An admissible polygon corresponds to a choice of an integer breakpoint on the segment connecting (0, 0) and (𝑑 (𝑁 − 3), 𝑛(𝑁 − 3)) whose first coordinate is a multiple of 𝑑. There are 𝑁 − 2 such choices and the basic one corresponds to a straight line of slope 1 𝑛 𝑑 + 𝑑 (𝑁−2) . By de Jong’s purity, dim 𝑆ℎ(𝐺, 𝑓 ) [𝜈basic ] = 𝑁 − 2 − #𝐵(𝐺, 𝑓 ) = 0. Case 2. 𝔢 = 𝔢∗ so 𝑑 has to be even and 𝑛 = 𝑑2 − 1 𝜈ord,𝔢 has slopes: 12 − 𝑑1 with multiplicity 𝑑, 12 with multiplicity 𝑑 (𝑁 − 4), and 21 + 𝑑1 with multiplicity 𝑑. The Newton polygon of 𝜈ord,𝔢 is symmetric with endpoints {(0, 0), (𝑑 (𝑁 −2), 𝑑2 (𝑁 − 2))} and 2 integer break points {(𝑑, 𝑑2 −1), (𝑑 (𝑁 −3), 𝑑2 (𝑁 −3)−1)}. Any admissible polygon corresponds to a choice of an integer breakpoint on the left-half of the segment connecting 𝑑2 (𝑁 − 2)) and (𝑑, 𝑑2 − 1) whose first coordinate is a multiple of 𝑑. There are ⌊ 𝑁2 ⌋ such choices and the slopes of the basic one are all 12 (a supersingular polygon). By de Jong’s purity, dim 𝑆ℎ(𝐺, 𝑓 ) [𝜈basic ]) = 𝑁 − 2 − #𝐵(𝐺, 𝑓 ) = 𝑁 − 2 − ⌊ 𝑁2 ⌋. □ Suppose that ( 𝑓1 , 𝑓2 ) is a degeneration of 𝑓 = 𝑓 𝑛𝑒𝑤 . There is a natural inclusion of Shimura varieties 𝑆ℎ(𝐺, 𝑓1 ) × 𝑆ℎ(𝐺, 𝑓2 ) ↩→ 𝑆ℎ(𝐺, 𝑓 ) compatible with taking the amalgamate sum of Newton polygons. In what follows, we drop 𝐺 from the notation for abbreviation. Theorem 4.0.7. If 𝑁 is odd or 𝑝 is sufficiently large, the basic locus 𝑆ℎ( 𝑓 ) [𝜈basic ] is not contained in Ø 𝑆ℎ( 𝑓1 ) × 𝑆ℎ( 𝑓2 ). (𝑎 1 ,𝑎 2 )∈𝑇 Proof. Assume on the contrary that 𝑆ℎ( 𝑓 ) [𝜈basic ] ⊂ Ø (𝑎 1 ,𝑎 2 )∈𝑇 𝑆ℎ( 𝑓1 ) × 𝑆ℎ( 𝑓2 ). 39 Then the compatibility with Newton stratification implies that in fact Ø 𝑆ℎ( 𝑓1 ) [𝜈1,basic ] × 𝑆ℎ( 𝑓2 ) [𝜈2,basic ]. 𝑆ℎ( 𝑓 ) [𝜈basic ] ⊂ (4.1) (𝑎 1 ,𝑎 2 )∈𝑇 In particular, 𝜈basic = 𝜈1,basic ⊕ 𝜈2,basic . The following follows directly from Kottwitz method: 𝜈2,basic is isoclinic of slope 𝑑𝑛 and dim 𝑆ℎ( 𝑓2 ) = 0. 1 of 𝜈basic Case 1. 𝔢 ≠ 𝔢∗ : From the previous proposition, the basic slope 𝑑𝑛 + 𝑑 (𝑁−2) in 𝔢 is not the same as the basic slope 𝑑𝑛 + 𝑑 (𝑁11 −2) of 𝜈1,basic in 𝔢: a contradiction. Case 2. 𝔢 = 𝔢∗ : In this case, all basic polygons are supersingular. Comparing dimensions of both sides of the inclusion yields 𝑁 −2−⌊ 𝑁1 𝑁 ⌋ ≤ 𝑁1 − 2 − ⌊ ⌋. 2 2 This holds only when 𝑁 is even, 𝑁1 = 𝑁 − 1, and 𝑁2 = 3. On one hand, Li et el ([16, Theorem 8.1]) show that when 𝑁 is even and 𝑓 is simple, the number of irreducible components of 𝑆ℎ( 𝑓 ) [𝜈basic ] grows to infinity with 𝑝. On the other hand, Xiao and Zhu ([36, Theorem 1.1.4]) show that if 𝑁1 is odd, 𝑓1 is simple, and 𝔭 | 𝑝 is inert in 𝐸 𝑚 /𝐸 𝑚0 , the number of irreducible components of 𝑆ℎ( 𝑓1 ) [𝜈1,basic ] is bounded at all unramified primes. Thus, we have arrived at a contradiction when 𝑝 is sufficiently large. □ Proposition 4.0.8. Assume that 𝑁 is odd or 𝑝 ≫ 0, then 𝑍 new (𝐺, 𝑎) [𝜈basic ] ≠ ∅. Proof. Assume on the contrary that 𝜕𝑍 new (𝐺, 𝑎) ⊃ 𝑆ℎ(𝐺, 𝑓 ) [𝜈basic ]. The discussion on boundary of Hurwitz space implies that Ø 𝑆ℎ(𝐺, 𝑓1 ) × 𝑆ℎ(𝐺, 𝑓2 ) ⊃ 𝜕𝑍 new (𝐺, 𝑎). (𝑎 1 ,𝑎 2 )∈𝑇 We arrive at a contradiction to the previous theorem Ø 𝑆ℎ(𝐺, 𝑓1 ) × 𝑆ℎ(𝐺, 𝑓2 ) ⊃ 𝑆ℎ(𝐺, 𝑓 ) [𝜈basic ]. (𝑎 1 ,𝑎 2 )∈𝑇 □ 40 Stratification of the supersingular locus by Artin invariant Assume that X(𝐺, 𝑎, 𝑏, 𝑁) is a family of K3 surfaces and we are in Case 2 of Proposition 4.0.6 and Theorem 4.0.7. In particular, the basic polygon 𝜈 𝑏𝑎𝑠𝑖𝑐 is the supersingular polygon. As a unitary Shimura variety with simple signature, 𝑆ℎ(𝐺, 𝑓 ) is fully Hodge-Newton decomposable, i.e., 𝑍 new (𝐺, 𝑎) [𝜈 𝑠𝑠 ] = 𝑆ℎ(𝐺, 𝑓 ) [𝜈 𝑠𝑠 ] is a union of strata by the final type of 𝑀𝔢 (𝐶), each of codimension 1 in the previous one ([12, Theorem 3.3]). From Proposition 4.0.6, there are 𝑁 − 2 − ⌊ 𝑁−2 2 ⌋ such strata. The final type of 𝑀𝔢 (𝐶) determines the final type of 𝑋 via 𝑀𝔢 (𝐶) ⊗O𝑑,𝔢 𝑀𝔢 (𝐷), or equivalently, its Artin invariant from Proposition 2.5.6. Hence, on X [𝜈 𝑠𝑠 ] we have a further stratification into 𝑁 − 2 − ⌊ 𝑁−2 2 ⌋ strata by the Artin invariant for which the inclusion order agrees with the Bruhat order of final type of 𝑀𝔢 (𝐶) ⊗O𝑑,𝔢 𝑀𝔢 (𝐷). Main theorem: proof Theorem 4.0.9. Let X, 𝑋, 𝑝 and 𝑑 be as above. 1. If 𝑝 is not inert in 𝐸 𝑚 /𝐸 𝑚0 , then 𝑑 divides ℎ(𝑋) and 1≤ ℎ(𝑋) ≤ 𝑁 − 2. 𝑑 Furthermore, for any 1 ≤ 𝑖 ≤ 𝑁 − 2, there exists a k3-type surface 𝑋 ′ in X such that ℎ(𝑋 ′) = 𝑖𝑑. 2. If 𝑝 is inert in 𝐸 𝑚 /𝐸 𝑚0 , we assume further that 𝑝 is sufficient large when 𝑁 is even. Then 𝑑 divides ℎ(𝑋) and 1≤ ℎ(𝑋) 𝑁 −2 ≤⌊ ⌋ or ℎ(𝑋) = ∞. 𝑑 2 ′ In addition, for any 1 ≤ 𝑖 ≤ ⌊ 𝑁−2 2 ⌋, there exists a k3-type surface 𝑋 in X such that ℎ(𝑋 ′) = 𝑖𝑑. Moreover, if X corresponds to a family of K3 surfaces, then there exists a supersingular K3 surface 𝑋 ′ in X with 𝜎0 (𝑋 ′) = (𝑁 − 2 − ⌊ 𝑁−2 2 ⌋)𝑑. Proof. 1. Because 𝑝 is not inert in 𝐸 𝑚 /𝐸 𝑛0 if and only if 𝔢 ≠ 𝔢∗ , we are in Case 1 of Proposition 4.0.6 and Theorem 4.0.7. In particular, there are 𝑁 − 2 admissible polygons for 𝑀𝔢 (𝐶). For each 𝑖 = 1, . . . , 𝑁 − 2, the 𝑖-th admissible 𝑛 𝑛 polygon has slopes 𝑖𝑛+1 𝑖𝑑 and 𝑑 . On the other hand, 𝑀𝔢 (𝐷) has slope 1 − 𝑑 41 only. Thus the Newton polygon of 𝑋 has slopes 𝑖𝑑+1 𝑖𝑑 and 1, i.e., ℎ(𝑋) = 𝑖𝑑. The result follows from the fact that all admissible polygons occur for some 𝐶 in the family of cyclic covers (Proposition 4.0.5 and 4.0.8). 2. We are in Case 2 of Proposition 4.0.6 and Theorem 4.0.7. In particular, there are ⌊ 𝑁−2 2 ⌋ admissible polygons for 𝑀𝔢 (𝐶). A similar arguments holds: for 1 1 1 each 𝑖 = 1, . . . , ⌊ 𝑁−2 2 ⌋, the 𝑖-th admissible polygon has slopes 2 + 𝑖𝑑 and 2 , and 𝑀𝔢 (𝐷) has slope 12 . Regarding the Artin invariant, from Lemma 2.5.3, the ⌊ 𝑁−2 2 ⌋ strata with finite heights correspond to the final elements 𝜔 𝑑 , . . . , 𝜔 ⌊ 𝑁 −2 ⌋𝑑 . The supersingular 2 locus X [𝜈 𝑠𝑠 ] is further stratified by the Artin invariant into 𝑁 − 2 − ⌊ 𝑁−2 2 ⌋ strata. These correspond to final elements 𝜔 (𝑁−2)𝑑+1−𝜎0 such that 𝑑 divides 𝜎0 and 𝑁 −2 ⌋𝑑 < (𝑁 − 2)𝑑 + 1 − 𝜎0 ≤ (𝑁 − 2)𝑑. ⌊ 2 It follows that the open Artin stratum in X [𝜈 𝑠𝑠 ] is the largest admissible one ′ 𝜎0 = (𝑁 − 2 − ⌊ 𝑁−2 2 ⌋)𝑑 and has to occur for some 𝑋 in X. □ Sufficient bound for 𝑝 Assume that we are in Case 2 of Theorem 4.0.7. From 4.1, we want 𝑝 such that the number of irreducible components of 𝑆ℎ( 𝑓 ) [𝜈basic ] is larger than the number of Ð irreducible components of 𝑆ℎ( 𝑓1 ) [𝜈1,basic ] × 𝑆ℎ( 𝑓2 ) [𝜈2,basic ]. (𝑎 1 ,𝑎 2 )∈𝑇 Fix 𝑥 ∈ 𝑆ℎ( 𝑓 ) [𝜈basic ] and let 𝐴𝑥 denote the corresponding abelian variety endowed with the PEL structure. We write 𝐼 = 𝐼𝑥 for the group of quasi-isogenies of 𝐴𝑥 that are compatible with the 𝐸 𝑚 action and the polarization. From [16, Proposition 8.5] and [11, Proposition 2.13], a lower bound for the number of irreducible components of 𝑆ℎ( 𝑓 ) [𝜈basic ] can be given by 𝜙 (𝑚) 2 −[(𝑁−2) 𝜙 (𝑚) 2 +1] 𝐿 (𝑀𝐼 )𝜏(𝐼) 𝑝 (𝑁−2) 2 − 1 𝜙 (𝑚) 𝑝 2 +1 where • 𝑀𝐼 is the Artin-Tate motive attached to 𝐼 by Gross; • 𝐿(𝑀𝐼 ) is the value of the 𝐿-function of 𝑀𝐼 at 0; 42 • 𝜏(𝐼) is the Tamagawa number of 𝐼.
The number of degeneration types |𝑇 | can be bounded above by 𝑁2 . For each (𝑎 1 , 𝑎 2 ), the Shimura variety 𝑆ℎ( 𝑓2 ) is a single special point. From [36, Theorem 1.1.4(1)], the number of irreducible components of 𝑆ℎ( 𝑓1 ) [𝜈basic ] is equal to the dimension of the middle Borel-Moore homology group dim 𝐻 𝑁𝐵𝑀 (𝑆ℎ( 𝑓1 )) 1 −3 (note that 𝑁1 = 𝑁 − 1). It follows that the number of irreducible components of Ð 𝑆ℎ( 𝑓1 ) [𝜈1,basic ] × 𝑆ℎ( 𝑓2 ) [𝜈2,basic ] can be bounded above by (𝑎 1 ,𝑎 2 )∈𝑇   𝑁 𝐵𝑀 max (dim 𝐻 𝑁−4 (𝑆ℎ( 𝑓1 )). 2 𝑎1 ∈𝑇 The following result is an easy consequence of the discussion above. Proposition 4.0.10. Suppose that 𝑝 is inert in 𝐸 𝑚 /𝐸 𝑚0 . With the above notations, 𝑍 new (𝐺, 𝑎) [𝜈basic ] ≠ ∅ when 𝑝 satisfies 𝜙 (𝑚) 2 −[(𝑁−2) 𝜙 (𝑚) 2 +1] 𝐿(𝑀𝐼 )𝜏(𝐼) 𝑝 (𝑁−2) 2 − 1 𝜙 (𝑚) 𝑝 2 +1   𝑁 𝐵𝑀 > (𝑆ℎ( 𝑓1 )). max (dim 𝐻 𝑁−4 2 𝑎1 ∈𝑇 43 Chapter 5 APPLICATIONS Table 5.1 list all special families together with all the possible Newton polygons. Our examples have all the possible heights in {1, . . . , 10, ∞}. Proposition 5.0.1. All monodromy data (𝐺, 𝑎, 𝑏, 𝑁) such that X(𝐺, 𝑎, 𝑏, 𝑁) is a family of K3 surfaces are from in Table 5.1. Proof. If X(𝐺, 𝑎, 𝑏, 𝑁) is a family of K3 surface then the transcendental lattice 𝑇 (X) has rank at most 22. The rank of 𝑇 (X) is 𝜙(𝑚)(𝑁 − 2) so that 𝜙(𝑚)(𝑁 − 2) ≤ 22. √︁ From the bound 𝜙(𝑚) ≥ 𝑚2 , we deduce that 𝑚 ≤ 968 and a brute-force computer search yields all 𝑚 together with the monodromy data given in Table 5.1. □ Example 5.0.1. The examples with 𝑁 = 3 are K3 surfaces with complex multiplication by the 𝑚-th cyclotomic field 𝐸 𝑚 . Our theorem implies that they are supersingular with Artin invariant 𝜎0 = 𝑑/2 iff 𝑝 is inert in 𝐸 𝑚 /𝐸 𝑚0 . The first example of this phenomenon is due to Tate: the K3 surface 𝑥 4 + 𝑦 4 + 𝑧 4 + 𝑡 4 = 0 is supersingular over any field of characteristic 𝑝 ≡ 3 (mod 4). Example 5.0.2. When 𝑁 = 3 and 𝑑 = 2, or equivalently, 𝑝 ≡ −1 (mod 𝑚), 𝑋 is a K3 surface with Artin invariant 1. It is known that such surface has a unique isomorphism class over F 𝑝 and admits a model over F 𝑝 . Here as an application of our computations, we have found an explicit equation for 𝑋. Remark 5.0.1. The Manin problem for K3 surfaces can be formulated as which Newton strata are nonempty in the moduli stack M2𝑙 ⊗ F 𝑝 of K3 surfaces equipped with a line bundle of degree 2𝑙 in characteristic 𝑝. The most general result due to Rizov [27] asserts that all polygons occur when 𝑑 and 𝑝 are sufficiently large. Here our families provide a positive answer to many explicit pairs of invariants 𝑙 = 𝑚 and ℎ as listed in Table 5.1. A refinement of Serre’s conjecture for K3 surfaces Let 𝑋 be a K3 surface over a number field 𝐿. We assume the following: (A1) 𝐹 := E (𝑋) is an abelian extension of Q; 44 (A2) 𝑋 does not have CM and 𝑛 = rank𝐹 (𝑇 (𝑋)) is even if 𝐹 is totally real. We consider the 𝑆 of primes 𝔭 of 𝐿 such that: • 𝑋 has good reduction at 𝔭; • 𝔭 has degree 1. Then 𝑆 has density 1 since the set of primes of degree 1 has density 1 and 𝑋 has good reduction at all but finitely many primes of 𝐿. Let 𝔭 ∈ 𝑆 and denote by 𝑋𝔭 the K3 surface over the residue field F 𝑝 . Let 𝑝 be the rational prime in Q below 𝔭. Let 𝐾 be the largest subfield of 𝐹 over which 𝑝 splits completely and set 𝑑 := [𝐹 : 𝐾]. The Newton polygon of 𝑋𝔭 lies above a certain polygon called 𝜇-ordinary, denoted by 𝜇𝔭 , that only depends on the splitting behavior of 𝑝 in 𝐹. Remark 5.0.2. (A1) allows us to realize the group connected components of the 𝑙-adic monodromy group as a subgroup of certain Weyl group. On (A2), if 𝑋 has complex multiplication by 𝐹, then an analogue of the Shimura-Taniyama formula for CM abelian varieties shows that the Newton polygon of 𝑋𝔭 is uniquely determined by the splitting of 𝑝 in 𝐹 (see [13, Theorem 1.1]) and coincides with 𝜇𝔭 . So in this case, every good prime of 𝐿 is 𝜇-ordinary. We fix an embedding of 𝐿 into C and write 𝑋C for the corresponding complex K3 surface. Let 𝑋 𝐿 denote the base change of 𝑋 to 𝐿. Proposition 5.0.2. 𝜇𝔭 has slopes 1 − 𝑑1 , 1, 1 + 𝑑1 with multiplicities 𝑑, 22 − 2𝑑, 𝑑. Proof. We first consider the comparison theorem between Betti and 𝑝-adic cohomologies 𝐻 2 (𝑋C , Z) ⊗Z Q 𝑝 ≃ 𝐻 2 (𝑋 𝐿 , Q 𝑝 ) compatiple with Poincare pairing and Chern class map. In particular 𝐻 2 (𝑋 𝐿 , Q 𝑝 ) ≃ (𝑇 (𝑋C ) ⊕ 𝑁 𝑆(𝑋C )) ⊗ Q 𝑝 . Next, let 𝐵𝑐𝑟𝑖𝑠 denote Fontaine’s crystalline period field - an extension of Q 𝑝 with = Q 𝑝 . There exists a functor an action of Gal(𝐿/𝐿) such that 𝐵Gal(𝐿/𝐿) 𝑐𝑟𝑖𝑠 𝐷 𝑐𝑟𝑖𝑠 (𝐻 2 (𝑋 𝐿 , Q 𝑝 )) := (𝐻 2 (𝑋 𝐿 , Q 𝑝 ) ⊗ 𝐵𝑐𝑟𝑖𝑠 ) Gal(𝐿/𝐿) 45 such that there exists an isomorphism 2 𝐷 𝑐𝑟𝑖𝑠 (𝐻 2 (𝑋 𝐿 , Q 𝑝 )) ≃ 𝐻𝑐𝑟𝑖𝑠 (𝑋𝔭 /Q 𝑝 ) compatible with the Frobenius action, Poincaré pairing, and Chern class map. Since Gal(𝐿/𝐿) acting on 𝑋 𝐿 preserves the space spanned by algebraic classes 𝑁 𝑆(𝑋C ) and the Frobenius acting on the Chern class of a line bundle is multiplication 2 (𝑋 /Q ). As by 𝑝, the image 𝐷 𝑐𝑟𝑖𝑠 (𝑁 𝑆(𝑋C ) ⊗ Q 𝑝 ) lies in the Tate module of 𝐻𝑐𝑟𝑖𝑠 𝔭 𝑝 2 these classes have slope 1 only, the complement 𝐷 𝑐𝑟𝑖𝑠 (𝑇 (𝑋C ) ⊗Q 𝑝 ) ⊂ 𝐻𝑐𝑟𝑖𝑠 (𝑋𝔭 /Q 𝑝 ) is a K3 F-isocrystal with an action of the endomorphism algebra 𝐹 ⊗ Q 𝑝 and 2 (𝑋 /Q ). Thus, the slopes of 𝜇 can be determines the Newton polygon of 𝐻𝑐𝑟𝑖𝑠 𝔭 𝑝 𝔭 computed to be of the desired form following Proposition 2.5.2. □ The condition (A2) also ensures that we are in the case 𝑛 ≠ 𝑚 of Proposition 2.5.2. Fix a prime 𝑙 and consider the 𝑙-adic representation on the étale cohomology 𝜌 𝑋,𝑙 : Gal(𝐿/𝐿) → 𝐺 𝐿(𝐻 2 (𝑋 𝐿 , Q𝑙 )). The 𝑙-adic monodromy group, denoted by 𝐺 𝑋,𝑙 , is defined as the Zariski closure of the image of 𝜌 𝑋,𝑙 in 𝐺 𝐿(𝐻 2 (𝑋 𝐿 , Q𝑙 )). Let 𝐺 ◦𝑋,𝑙 be the identity component of 𝐺 𝑋,𝑙 viewed as an algebraic group over Q𝑙 . Let 𝜋0 (𝐺 𝑋,𝑙 ) be the group of connected components of 𝐺 𝑋,𝑙 which is a finite group. By Galois theory, there exists a finite field extension 𝐿 conn of 𝐿 such that Gal(𝐿 conn /𝐿) ≃ 𝜋0 (𝐺 𝑋,𝑙 ). We assume one further condition: (A3) 𝐿 𝑐𝑜𝑛𝑛 ⊆ 𝐹 𝐿. Next, let us choose a polarization of degree 2𝑚 on 𝑋, or equivalently, a polarization form 𝜓 on 𝐻 2 (𝑋C , Z). Let S denote the Deligne torus and let ℎ : S → 𝐺𝑂 (𝜓)(R) be the structure homomorphism of the rational polarized K3 Hodge structure on (𝐻 2 (𝑋C , Q), 𝜓) where 𝐺𝑂 (𝜓) denotes the group of orthogonal similitudes of (𝐻 2 (𝑋C , Q), 𝜓). The Mumford-Tate group of 𝑋 is the smallest algebraic subgroup 𝑀𝑇𝑋 of 𝐺𝑂 (𝜓) over Q such that 𝑀𝑇𝑋 (R) contains the image of ℎ in 𝐺𝑂 (𝜓)(R). The Mumford-Tate conjecture asserts that there exists a canonical isomorphism 𝐺 ◦𝑋,𝑙 ≃ 𝑀𝑇𝑋 ⊗ Q𝑙 . 46 The Mumford-Tate conjecture holds for K3 from the works of Tankeev [32, 33]. On the other hand, considering the decomposition 𝐻 2 (𝑋, Q) = 𝑁 𝑆(𝑋)Q ⊕ 𝑇 (𝑋)Q , one has 𝑀𝑇𝑋 fix 𝑁 𝑆(𝑋) as these are precisely the Q-span of Hodge classes. Since 𝑇 (𝑋)Q with the restricted polarization 𝜓 is again a K3 polarized Hodge structure, we can identify 𝑀𝑇𝑋 with the Mumford-Tate group of (𝑇 (𝑋)Q , 𝜓 |𝑇 (𝑋) ) which is contained in 𝐺𝑂 (𝑇 (𝑋), 𝜓). From the definition of E (𝑋) = 𝐹, we deduce that the Mumford-Tate group of 𝑇 (𝑋) is contained in the centralizer of 𝐹 in 𝐺𝑂 (𝑇 (𝑋), 𝜓). Theorem 5.0.3 (Zarhin, [37]). With notations as above, the Mumford-Tate group of the polarized Hodge structure (𝑇 (𝑋), 𝜓) is the centralizer of 𝐹 in 𝐺𝑂 (𝑇 (𝑋), 𝜓). Let 𝑇𝑙 := 𝑇 (𝑋) ⊗ Q𝑙 = 𝑁 𝑆(𝑋) ⊥ ⊗ Q𝑙 ⊂ 𝐻 2 (𝑋 𝐿 , Q𝑙 ) together with the bilinear form 𝜓𝑙 := 𝜓 ⊗ Q𝑙 . Because Hodge classes of K3 surfaces are absolute Hodge, the Galois action of Gal(𝐿/𝐿) leaves 𝑁 𝑆(𝑋) ⊗ Q𝑙 invariant. It follows that 𝜌 𝑋,𝑙 factors through 𝐺𝑂 (𝑇𝑙 , 𝜓𝑙 ) and the identity component 𝐺 ◦𝑋,𝑙 commutes with 𝐹 ⊗ Q𝑙 . The discussion above implies the following: Corollary 5.0.4. With notations as above, the identity component 𝐺 ◦𝑋,𝑙 can be identified with the centralizer of 𝐹 ⊗ Q𝑙 in 𝐺𝑂 (𝑇𝑙 , 𝜓𝑙 ). Let us consider the orthogonal similitude 𝜒 which is a 1-dimensional representation of 𝐺 𝑋,𝑙 . For each prime 𝔭 ∈ 𝑆 above a rational prime 𝑝, the Frobenius Fr𝔭 ∈ Gal(𝐿/𝐿) is well defined up to conjugacy. By abuse of notation, let us also use Fr𝔭 for its image under 𝜌 𝑋,𝑙 . Recall that 𝑑 = [𝐹 : 𝐾] where 𝐾 is the subfield of 𝐹 over which 𝑝 splits completely. The trace on the algebraic representation ∧𝑑 𝐻 2 (𝑋 𝐿 , Q𝑙 ) ⊗ 𝜒−𝑑 gives a well-defined invariant 𝑎 𝔭 := Tr(Fr𝔭 | ∧𝑑 𝐻 2 (𝑋 𝐿 , Q𝑙 ) ⊗ 𝜒−𝑑 ). The following proposition is similar to [3, Proposition 3.11]. Proposition 5.0.5. With notations as above, 𝔭 is not 𝜇-ordinary if and only if 𝑎 𝔭 is
an integer in [−𝐶, 𝐶] where 𝐶 = 22 𝑑 . Proof. The characteristic polynomial of Fr𝔭 acting on 𝐻 2 (𝑋 𝐿 , Q𝑙 ) has the form 𝑃(𝑇) := 𝑇 22 + 𝑐 1𝑇 21 + · · · + 𝑐 21𝑇 + 𝑐 22 . 47 It is known from Deligne’s works on the Weil’s conjecture for K3 surfaces [5, 6] that the action of Fr𝔭 is semi-simple and the 𝑐𝑖 are integers, independent of 𝑙. Let us split 𝑃(𝑇) over Q into 𝑃(𝑇) = (𝑇 − 𝛼1 ) . . . (𝑇 − 𝛼22 ). From the Weil’s conjecture, the 𝛼𝑖 are algebraic integers in O𝐿 with Archimedean absolute value 𝑝 and 𝔮-adic units for any prime 𝔮 of 𝐿 away from 𝑝. It is also known that the 𝑝-adic valuations of 𝛼𝑖 are encoded in the slopes with 2 (𝑋 /𝑊) (see [1, Chapter 8]). The multiplicities of the Newton polygon of 𝐻𝑐𝑟𝑖𝑠 𝔭 𝑑 2 trace of Fr𝔭 on ∧ 𝐻 (𝑋 𝐿 , Q𝑙 ) is given by ∑︁ Π𝑖∈𝐼 𝛼𝑖 𝑏 := Tr(Fr𝔭 | ∧𝑑 𝐻 2 (𝑋 𝐿 , Q𝑙 )) = 𝐼 ⊂{1,...,22},#𝐼=𝑑 which is an integer as 𝑏 can be written as a polynomial with integer coefficients in the 𝑐𝑖 . Since 𝔭 is not 𝜇-ordinary, from Proposition 5.0.2, the smallest slope of 2 (𝑋 /𝑊) is greater than 1 − 1 . Thus, the sum of any 𝑑 slopes of 𝐻 2 (𝑋 /𝑊) 𝐻𝑐𝑟𝑖𝑠 𝔭 𝔭 𝑐𝑟𝑖𝑠 𝑑 is greater than 𝑑 − 1, i.e., the 𝑝-valuation of Π𝑖∈𝐼 𝛼𝑖 is greater than 𝑑 − 1. Hence, the 𝑝-valuation of 𝑏 must be at least 𝑑 or 𝑝 𝑑 divides 𝑏. In addition, as each 𝛼𝑖 has absolute value 𝑝, we deduce that     𝑑 22 𝑑 22 −𝑝 ≤𝑏≤𝑝 . 𝑑 𝑑 The result then follows from the fact that Fr𝔭 acts on 𝜒 via multiplication by 𝑝. □ The next discussion follows [3, Section 4.1]. Let Γ denote the image of Gal(𝐿/𝐿) under 𝜌 𝑋,𝑙 . We have the following commutative diagram of groups 𝐺 ◦𝑋,𝑙 ≃ G 𝐹 Gal(𝐿/𝐿) ∋ Fr𝔭 𝜌 𝑋,𝑙 𝐺 𝑋,𝑙 ≃ Γ Gal(𝐹 𝐿/𝐿) ∋ 𝜎 𝜋0 𝜋0 (𝐺 𝑋,𝑙 ) 𝜄 Gal(𝐹/Q) 𝜅 𝜌 𝑋,𝑙 ≃ Gal(𝐿 𝑐𝑜𝑛𝑛 /𝐿). Our assumptions ensure that 𝜅 is a surjection and 𝜄 is an injection between abelian groups. 48 Let 𝜎 ∈ Gal(𝐹 𝐿/𝐿) be the element that corresponds to Fr𝔭 ∈ Gal(𝐿/𝐿) restricted to 𝐹 𝐿. Let 𝑆 𝜎 ⊂ 𝑆 be the set of primes of 𝐿 whose Frobeni restrict to 𝜎. The subfield 𝐾 of 𝐹 over which 𝑝 splits completely is the same as 𝐹 ⟨𝜄(𝜎)⟩ . Let 𝐺 (𝜎) 𝑋,𝑙 be the connected component of 𝐺 𝑋,𝑙 corresponding to 𝜅(𝜎). We define the following conjugacy-invariant algebraic function that only depends on 𝜎. 𝑏 𝜎 := Tr(− | ∧𝑑 𝐻 2 (𝑋, Q𝑙 ) ⊗ 𝜒−𝑑 ) : 𝐺 (𝜎) 𝑋,𝑙 → Q𝑙 . The next result is similar to [3, Proposition 4.9] and [28, Theorem 1]. Proposition 5.0.6. The density of the set of non 𝜇-ordinary prime is equal to the number of 𝜎 ∈ Gal(𝐹 𝐿/𝐿) for which 𝑏 𝜎 is a constant, divided by [𝐹 𝐿 : 𝐿]. Proof. Let us fix 𝜎 ∈ 𝜎 ∈ Gal(𝐹 𝐿/𝐿). Since Γ is a closed and compact subgroup of 𝐺 𝐿(𝐻 2 (𝑋 𝐿 , Q𝑙 )), it is an 𝑙-adic analytic subgroup and admits a normalized Haar measure of 1. Let 𝜎Γ denote the image of the coset 𝜎Gal(𝐿/𝐹 𝐿) ⊂ Gal(𝐿/𝐿) 1 under 𝜌 𝑋,𝑙 so that 𝜎Γ has measure [𝐹 𝐿:𝐿] . For each integer 𝑛 ∈ [−𝐶, 𝐶] where 𝐶 is as in Proposition 5.0.5, let 𝑇𝑛 be the subset of 𝐺 (𝜎) 𝑋,𝑙 where 𝑏 𝜎 is equal to 𝑛. Then 𝑇𝑛 is a conjugacy-invariant closed subset of 𝐺 𝑋,𝑙 as 𝜋0 (𝐺 𝑋,𝑙 ) is abelian and 𝑏 𝜎 is algebraic. Let 𝑍 be the union of all 𝑇𝑛 which is again conjugacy-invariant and closed. Let 𝑍 ′ := 𝑍 ∩ 𝜎Γ. The boundary of 𝑍 ′ has measure 0 since it is an 𝑙-adic analytic subset of Γ. From Proposition 5.0.5, any non 𝜇-ordinary prime 𝔭 ∈ 𝑆 𝜎 has Fr𝔭 in 𝑍 ′. By Chebotarev’s density and [29, Corollary 6.10], the density of these primes is the same as the Haar measure of 𝑍 ′. Since 𝜎Γ is Zariski dense in 𝐺 (𝜎) 𝑋,𝑙 , Serre’s results [29, Proposition 5.12 and 5.2.1.2] imply that the measure of 𝑍 ′ is 1 if 𝑍 contains 𝐺 (𝜎) 𝑋,𝑙 and 0 otherwise. (𝜎) Next, we prove that 𝑍 contains 𝐺 (𝜎) 𝑋,𝑙 if and only if 𝑏 𝜎 is a constant on 𝐺 𝑋,𝑙 . As (𝜎) 𝐺 (𝜎) 𝑋,𝑙 is connected and 𝑍 is the preimage of finitely many points, if 𝑍 contains 𝐺 𝑋,𝑙 , then 𝑏 𝜎 is a constant. On the other hand, if 𝑏 𝜎 is a constant 𝑐 on 𝐺 (𝜎) 𝑋,𝑙 , it suffices to prove that 𝑐 is an integer in [−𝐶, 𝐶]. By Chebotarev’s density, there exist infinitely many primes in 𝑆 𝜎 above distinct rational primes 𝑝 and 𝑝′ whose Frobeni lie in 𝜎Γ. At these primes 𝑐 is equal to 𝑏 𝑝 /𝑝 𝑑 and 𝑏 𝑝 ′ /𝑝′ 𝑑 , respectively, where 𝑏 𝑝 and 𝑏 𝑝 ′ are integers. We deduce that 𝑐 is a rational number whose denominator is a divisor of both 𝑝 and 𝑝′, hence an integer. Because −𝑝 𝑑 𝐶 ≤ 𝑏 𝑝 ≤ 𝑝 𝑑 𝐶, we have 𝑐 ∈ [−𝐶, 𝐶]. The proposition follows from summing over all 𝜎 ∈ Gal(𝐹 𝐿/𝐿). □ 49 Because the density in Proposition 5.0.6 is preserved under matrix scaling and restriction to the kernel of the orthogonal similitude, we may replace 𝐺 ◦𝑋,𝑙 with the group of 𝐹-linear special isometries 𝑆𝑂 𝐹 (𝑇𝑙 , 𝜓𝑙 ). For abbreviation, let us write G for the 𝑙-adic algebraic group 𝑆𝑂 (𝑇𝑙 , 𝜓𝑙 ) and G 𝐹 for 𝑆𝑂 𝐹 (𝑇𝑙 , 𝜓𝑙 ) as a subgroup. The next discussion follows [3, Section 4.2, 4.3] in order to pick a basis to realize G 𝐹 in block diagonal form inside 𝑆𝑂 (𝑇𝑙 , 𝜓𝑙 ) after a sufficiently large extension of Q𝑙 . The trace 𝑏 𝜎 remains the same after passing to extensions of Q𝑙 . If 𝐹 is CM field, we assume that 𝐹 has degree 2𝑚 and 𝑇 (𝑋) has rank 2𝑚𝑛. Let 𝐹 0 be the maximal totally real subfield of 𝐹 of degree 𝑚. We have the following decomposition into character spaces as an 𝐹 0 ⊗ Q𝑙 -module Ê 𝑇𝑙 ⊗ Q𝑙 = (𝑇𝑙𝜏 , 𝜓𝑙𝜏 ), 𝜏∈Gal(𝐹 0 /Q) where each 𝑇𝑙𝜏 has dimension 2𝑛. From Hodge symmetry, the bilinear symmetric ∗ form 𝜓𝑙𝜏 on 𝑇𝑙𝜎 ⊕ 𝑇𝑙𝜎 has the form ! 0 𝐽𝜏 𝐽𝜏 0 for some diagonal matrix 𝐽𝜏 . So there is a further decomposition Ê ∗ (𝑇𝑙𝜎 ⊕ 𝑇𝑙𝜎 , 𝜓𝑙𝜏 ). 𝑇𝑙 ⊗ Q𝑙 = (5.1) (𝜎,𝜎 ∗ )∈Gal(𝐹/Q) 𝜎| 𝐹0 =𝜏 If 𝐹 is totally real, we assume that 𝐹 has degree 𝑚 and 𝑇𝑙 has rank 2𝑛 over 𝐹 ⊗ Q𝑙 . In this case, complex conjugation is the identity map on Gal(𝐹/Q) and we have the following decomposition into character spaces as an 𝐹 ⊗ Q𝑙 -module: Ê 𝑇𝑙 ⊗ Q𝑙 = (𝑇𝑙𝜏 , 𝜓𝑙𝜏 ), (5.2) 𝜏∈Gal(𝐹/Q) where again 𝑇𝑙𝜏 has dimension 2𝑛 and 𝜓𝑙𝜏 is a symmetric form. The next lemma is an adaptation of [3, Lemma 4.12]. Lemma 5.0.1. With notations as above: 1. If 𝐹 is CM, we have 𝑚 Ö 𝑖=1 𝑆𝐿 𝑛 ≃ G 𝐹 ↩→ G 50 given by (𝑀1 , . . . , 𝑀𝑚 ) ↦→ diag(𝑀1 , 𝑀1∗ , . . . , 𝑀𝑚 , 𝑀𝑚∗ ), 𝑀𝑖⊤ 𝐽𝜏𝑖 for all 𝜏𝑖 ∈ Gal(𝐹0 /Q). where 𝑀𝑖∗ := 𝐽𝜏−1 𝑖 2. If 𝐹 is totally real, we have 𝑚 Ö 𝑆𝑂 2𝑛 ≃ G 𝐹 ↩→ G 𝑖=1 given by (𝑀1 , . . . , 𝑀𝑚 ) ↦→ diag(𝑀1 , . . . , 𝑀𝑚 ), where 𝑀𝑖⊤ 𝜓𝑙𝜏𝑖 𝑀𝑖 = 𝜓𝑙𝜏𝑖 for all 𝜏𝑖 ∈ Gal(𝐹/Q). Proof. Over Q𝑙 , the action of 𝐹 ⊗ Q𝑙 can be diagonalized with respect to the eigenspaces 𝑇𝑙𝜎 of dimension 𝑛 for 𝜎 ∈ Gal(𝐹/Q). 1. If 𝐹 is CM, a matrix 𝐴 ∈ G that commutes with such diagonal matrix must be of block diagonal form, say 𝐴 = diag(𝑀1 , 𝑀1′ , . . . , 𝑀𝑚 , 𝑀𝑚′ ), where (𝑀𝑖 , 𝑀𝑖′) corresponds to conjugate pair (𝜏𝑖 , 𝜏𝑖∗ ). From (5.1), the orthogonal condition 𝐴⊤ 𝜓𝑙 𝐴 = 𝜓𝑙 is satisfied if and only if 𝑀𝑖′ = 𝑀𝑖∗ where 𝑀𝑖∗ := 𝐽𝜏−1 𝑀𝑖⊤ 𝐽𝜏𝑖 for all 𝜏𝑖 . 𝑖 2. The totally real case follows similarly from (5.2). □ Let 𝑇 be a maximal torus of G 𝐹 given by diag(𝑇1 , 𝑇1−1 , . . . , 𝑇𝑚 , 𝑇𝑚−1 ) where each 𝑇𝑖 is diagonal of size 𝑛. Because 𝑇 has rank 𝑛𝑚, it is also a maximal torus of G. The Weyl group 𝑊G := 𝑁 G (𝑇)/𝑇 of G can be realized as a subgroup of the symmetric group 𝔖2𝑚𝑛 . We consider the set of permutations of {±1, ±2, . . . , ±𝑚𝑛} where the index set {±(𝑖𝑛 − 𝑛 + 1), . . . , ±𝑖𝑛} corresponds to the pair (𝑇𝑖 , 𝑇𝑖−1 ). Then 𝑊G consists of permutations 𝜋 for which 𝜋(𝑖) + 𝜋(−𝑖) = 0 and such that there is an even number of 1 ≤ 𝑖 ≤ 𝑚𝑛 for which 𝜋(𝑖) > 0. We consider two subgroups of 𝑊G : 𝐻 := 𝑁 G (G 𝐹 ) ∩ 𝑁 G (𝑇)/𝑇, 51 and the Weyl group of G 𝐹 : 𝑊 𝐹 := 𝑁 G 𝐹 (𝑇)/𝑇 . If 𝐹 is CM, the group 𝑊 𝐹 can be identified with the product of symmetric groups Î𝑚 𝑖=1 𝔖𝑛 and embedded in 𝑊G via (𝜋1 , . . . , 𝜋 𝑚 ) ↦→ (𝜋1 , 𝜋1 , . . . , 𝜋 𝑚 , 𝜋 𝑚 ). If 𝐹 is totally real, the group 𝑊 𝐹 can be identified with the product of orthogonal Î𝑚 Weyl groups 𝑖=1 𝑊𝑆𝑂 2𝑛 via (𝜋1 , . . . , 𝜋𝑚 ) ↦→ (𝜋1 , . . . , 𝜋𝑚 ). The next proposition is similar to [3, Proposition 4.14]. Proposition 5.0.7. We have a natural isomorphism 𝑁 G (G 𝐹 )/G 𝐹 ≃ 𝐻/𝑊 𝐹 . Proof. We define a map from 𝑁 G (G 𝐹 ) to 𝐻/𝑊 𝐹 whose kernel is G 𝐹 . Let 𝑔 ∈ 𝑁 G (G 𝐹 ) and consider 𝑔𝑇 𝑔 −1 which is a maximal torus of G 𝐹 . Since maximal tori are conjugates, there exists 𝑔0 ∈ G 𝐹 up to 𝑁 G 𝐹 (𝑇) such that 𝑔𝑇 𝑔 −1 = 𝑔0𝑇 𝑔0−1 . It follows that 𝑔0−1 𝑔 is in 𝑁 G (G 𝐹 ) ∩ 𝑁 G (𝑇) and the map 𝑔 ↦→ 𝑔0−1 𝑔𝑊 𝐹 is well defined. The kernel of this map consists of 𝑔 such that 𝑔0−1 𝑔 ∈ 𝑁 G 𝐹 (𝑇) or 𝑔 ∈ G𝐹 . □ Let G𝑋 := G ∩ 𝐺 𝑋,𝑙 so that 𝜋0 (G𝑋 ) = 𝜋 𝑜 (𝐺 𝑋,𝑙 ). Because this group is abelian, G𝑋 is contained in 𝑁 G (G 𝐹 ). The previous proposition yields a well-defined injective homomorphism 𝜙 : 𝜋0 (G𝑋 ) ↩→ 𝑁 G (G 𝐹 )/G 𝐹 ≃ 𝐻/𝑊 𝐹 . The next lemma is an adaptation of [3, Lemma 4.15]. Lemma 5.0.2. With notations as above, there is an isomorphism 𝜖 : 𝐻/𝑊 𝐹 ≃ 𝑆 ⋊ 𝔖𝑚 , where 𝑆 ⊆ {±1} 𝑚 is the subgroup with an even number of minuses. Proof. Because 𝐻 normalizes G 𝐹 which is in block diagonal form, 𝐻 consists of block permutation matrices of block size 𝑛 × 𝑛 in the CM case and 2𝑛 × 2𝑛 in the totally real case. Since 𝑊 𝐹 consists of products of permutations within 𝑛 × 𝑛 blocks 52 in the CM case and within 2𝑛 × 2𝑛 blocks in the real case, the quotient coset 𝐻/𝑊 𝐹 is specified by a permutation of the 𝑚 blocks together with an interchange within a block. The result then follows from the fact that 𝐻 as a subgroup of the Weyl group 𝑊𝑆𝑂 2𝑚𝑛 has an even number of 1 ≤ 𝑖 ≤ 𝑚𝑛 such that 𝜋(𝑖) < 0. □ We adapt the proof of the main theorem [3, Theorem 4.20] to our setting. Recall that for a 𝜎 ∈ Gal(𝐹 𝐿/𝐿), its imagine in 𝜋0 (G𝑋 ) is 𝜌(𝜅(𝜎)) and while its image in Gal(𝐹/Q) is 𝜄(𝜎). In addition, 𝐾 = 𝐹 ⟨𝜎⟩ and 𝑑 is equal to the order of 𝜎 in Gal(𝐹/Q). It follows that the order of 𝜌(𝜅(𝜎)) in 𝜋0 (G𝑋 ) is a divisor 𝑒 of 𝑑. Next we pick a coset representative 𝐵𝜎 in 𝑁 G (G 𝐹 ) of G 𝐹 such that 𝐵𝜎 G 𝐹 = G𝑋(𝜎) . Let 𝑞 : 𝑁 G (G 𝐹 ) → 𝐻/𝑊 𝐹 denote the quotient from Proposition 5.0.7. Then we need to pick 𝐵 such that 𝜖 (𝑞(𝐵𝜎 )) = 𝜙(𝜌(𝜅(𝜎))). From the discussion above, we have a decomposition 𝐵𝜎 = 𝑃𝜎 𝑇𝜎 ∈ 𝑁 G (G 𝐹 ) where 𝑃𝜎 is the block permutation matrix that corresponds to 𝜙(𝜌(𝜅(𝜎))) and 𝑇𝜎 is a diagonal matrix in 𝑇. Then as a permutation 𝑃𝜎 has order 𝑒 that divides 𝑑. Proposition 5.0.8. With notations as above, for any 𝜎 ∈ Gal(𝐹 𝐿/𝐿), the function 𝑏 𝜎 = Tr(− | ∧𝑑 𝐻 2 (𝑋, Q𝑙 ) ⊗ 𝜒−𝑑 ) : 𝐵𝜎 G 𝐹 → Q𝑙 is not a constant function. Proof. It suffices to show that Tr(𝑃𝜎 𝑇𝜎 𝐴 | ∧𝑑 𝐻 2 (𝑋, Q𝑙 ) ⊗ 𝜒−𝑑 ) is not a constant when 𝐴 varies in G 𝐹 . Assume that 𝐹 is CM. Let 𝑇𝜎 = (𝑇1 , 𝑇1−1 , . . . , 𝑇𝑚 , 𝑇𝑚−1 ), ∗ ), and set 𝐴 = ( 𝐴1 , 𝐴1∗ , . . . , 𝐴𝑚 , 𝐴𝑚 𝐶 := 𝑇𝜎 𝐴 = (𝐶1 , 𝐸 1 , . . . , 𝐶𝑚 , 𝐸 𝑚 ). 53 Let Ω denote the set of cycle types 𝜏 in 𝔖𝑑 , #𝜏 denote the number of permutations Î in 𝔖𝑑 with cycle type 𝜏, and define Tr𝜏 (𝑀) := 𝑙∈𝜏 Tr(𝑀 𝑙 ). The permutation form of the exponential formula ([31], Chap 7) yields 1 ∑︁ Tr(𝑃𝜎 𝐶 | ∧𝑑 𝐻 2 (𝑋, Q𝑙 )) = #𝜏 Tr𝜏 (𝑃𝜎 𝐶). 𝑑! 𝜏∈Ω Because 𝑃𝜎 is a block permutation matrix of order 𝑒 and 𝐶 is block diagonal, for 𝑙 divisible by 𝑒, the matrix (𝑃𝜎 𝐶) 𝑙 has block diagonal form, i.e., it has nonvanishing trace. Taking 𝐴 to be the identity matrix 𝐼 or a matrix of the form ( 𝐴1 , 𝐴1∗ , . . . , 𝐼𝑚 , 𝐼𝑚 ), one sees that 𝑏 𝜎 is not a constant. The totally real case is similar except that we work with block permutation matrices of block size 2𝑛. □ Propositions 5.0.8 and 5.0.11 imply the following theorem: Theorem 5.0.9. Let 𝑋 be a K3 surface over a number field 𝐿 such that E (𝑋) is an abelian field 𝐹 and 𝐿 conn ⊂ 𝐹 𝐿. If 𝐹 is totally real then we assume further that 𝑇 (𝑋) has even rank as an E (𝑋)-module. Then the set of primes of 𝐿 of 𝜇-ordinary reduction has density 1. 1 . Corollary 5.0.10. The set of primes of 𝐿 with ordinary reduction has density [𝐹 𝐿:𝐿] Proof. Because the 𝜇-ordinary polygon is ordinary if and only if 𝑝 is totally split in 𝐹/Q, the set of ordinary primes of 𝐿 is the set of primes 𝔭 such that 𝜄(𝜎) is the 1 by Chebotarev’s density. identity element in Gal(𝐹/Q). This set has density [𝐹 𝐿:𝐿] □ Proposition 5.0.11. If X(𝐺, 𝑎, 𝑏, 𝑁) corresponds to a family of K3 surfaces, then a generic surface 𝑋 in the family satisfies the conditions of Theorem 5.0.12. Proof. Let 𝑋 be a generic point that corresponds to the cyclic cover 𝐶𝛼 . This means 𝑋 is defined over 𝐿 = Q(𝛼1 , . . . , 𝛼𝑁 ) and 𝐹 = 𝐸 𝑚 the 𝑚-th cyclotomic field. From the definition of 𝐿 𝑐𝑜𝑛𝑛 , it suffices to show that the image of Gal(𝐹 𝐿) under 𝜌 𝑋,𝑙 is contained in 𝐺 ◦𝑋,𝑙 = G 𝐹 , i.e., Gal(𝐹 𝐿) commutes with the 𝐹-action. This follows from the fact that the 𝐹-action is multiplication to the coordinates 𝑦 and 𝑧 in the defining equations of 𝐶𝛼 and 𝐷. □ 54 An analogue of Elkies’s supersingular prime theorem Theorem 5.0.12. Let 𝑋𝛼 be a K3 surface in the family (5, (1, 1, 1, 2), (1, 1, 3), 4). In particular, 𝑋𝛼 arises from the pair of cyclic covers 𝐶𝛼 : 𝑦 5 = 𝑥(𝑥 − 1)(𝑥 − 𝛼) and 𝐷 : 𝑧5 = 𝑡 (𝑡 − 1). 2 3 −𝛼+1) Assume in addition that 𝑗 := (𝛼 ∈ Q ∩ [0, 27 4 ] and the reduction of 𝐶𝛼 at 5 is 𝛼2 (𝛼−1) 2 singular. Then 𝑋𝛼 can be defined over Q and there exists infinitely many primes at which the reduction of 𝑋𝛼 has basic Newton polygon. Proof. In [17], the authors show that 𝐶𝛼 is defined over Q and there exist infinitely many basic rational primes for 𝐶𝛼 . From Corollary 3.0.4, 𝑋𝛼 is defined over Q. From Corollary 4.0.4, 𝑋𝛼 has basic polygon at 𝑝 if and only if 𝐶𝛼 has basic polygon at 𝑝. Hence 𝑋𝛼 has infinitely rational primes of basic reduction. □ Corollary 5.0.13. With notations as above, there exist infinitely many primes 𝑝 at which the Picard rank of 𝑋𝛼 jumps up. Proof. The family of K3 surfaces has 𝑚 = 5 and 𝑁 = 4 so the Picard rank in characteristic zero is 22 − 𝜙(𝑚)(𝑁 − 2) = 14. At basic primes congruent to 2, 3, 4 mod 5, the basic polygon coincides with the supersingular polygon. Because the Picard rank is 22 at a supersingular prime (Tate’s conjecture), it follows that there exist infinitely many primes at which the Picard rank of 𝑋𝛼 jumps up by 8. □ Remark 5.0.3. This phenomenon is recently proven in general for any K3 surface over a number field in [30]. Our examples confirm their result and further calculate the explicit jump. Bibliography [1] Pierre Berthelot and Arthur Ogus. Notes on Crystalline Cohomology. (MN-21). Princeton University Press, 1978. URL http://www.jstor.org/stable/ j.ctt130hk6f. [2] Fedor A. Bogomolov and Yuri G. Zarhin. Ordinary reduction of 𝐾3 surfaces. Cent. Eur. J. Math., 7(2):206–213, 2009. ISSN 1895-1074,1644-3616. doi: 10.2478/s11533-009-0013-8. URL https://doi.org/10.2478/s11533009-0013-8. 55 [3] Victoria Cantoral-Farfan, Wanlin Li, Elena Mantovan, Rachel Pries, and Yunqing Tang. 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Institut für Experimentelle Mathematik Essen: Preprint. IEM, 1998. URL https://books.google. com/books?id=sDewmAEACAAJ. 58 [36] Liang Xiao and Xinwen Zhu. Cycles on Shimura varieties via geometric Satake, 2017. Preprint available on arXiv. [37] Yu.G. Zarhin. Hodge groups of k3 surfaces. Journal für die reine und angewandte Mathematik, 341:193–220, 1983. URL http://eudml.org/doc/ 152536. 59 APPENDIX With the help of a computer program, we can search for all K3 type monodromy data with 𝑚 ≤ 100. The following table lists the families with 𝑚 ≤ 22. Note that there are no examples when 23 ≤ 𝑚 ≤ 100. The two last columns list all admissible heights and Artin invariants that occur for our families at all the primes. 𝑚 𝑁 𝑎 𝑏 𝐾2 ℎ 𝜎0 3 3 (2, 2, 2) (1, 1, 1) 0 1, ∞ 1 4 3 (2, 3, 3) (1, 1, 2) 0 1, ∞ 1 5 3 (3, 3, 4) (1, 1, 3) 0 1, ∞ 1, 2 6 3 (3, 4, 5) (1, 1, 4) 0 1, ∞ 1 6 3 (2, 5, 5) (1, 2, 3) 0 1, ∞ 1 7 3 (4, 5, 5) (1, 1, 5) 0 1, 3, ∞ 1, 3 7 3 (4, 4, 6) (1, 2, 4) 0 1, 3, ∞ 1, 3 7 3 (3, 5, 6) (1, 3, 3) 0 1, 3, ∞ 1, 3 7 3 (2, 6, 6) (2, 2, 3) 0 1, 3, ∞ 1, 3 8 3 (4, 5, 7) (1, 2, 5) 0 1, 2, ∞ 1 8 3 (5, 5, 6) (1, 2, 5) 0 1, 2, ∞ 1 8 3 (3, 6, 7) (1, 3, 4) 0 1, 2, ∞ 1 8 3 (3, 6, 7) (2, 3, 3) 0 1, 2, ∞ 1 9 3 (5, 6, 7) (1, 1, 7) 0 1, 3, ∞ 1, 3 9 3 (5, 5, 8) (1, 3, 5) 0 1, 3, ∞ 1, 3 9 3 (5, 6, 7) (1, 3, 5) 0 1, 3, ∞ 1, 3 9 3 (4, 6, 8) (1, 4, 4) 0 1, 3, ∞ 1, 3 9 3 (2, 8, 8) (2, 3, 4) 0 1, 3, ∞ 1, 3 9 3 (4, 6, 8) (2, 3, 4) 0 1, 3, ∞ 1, 3 10 3 (5, 7, 8) (1, 1, 8) 0 1, ∞ 1, 2 10 3 (5, 7, 8) (1, 2, 7) 0 1, ∞ 1, 2 10 3 (6, 7, 7) (1, 2, 7) 0 1, ∞ 1, 2 10 3 (5, 6, 9) (1, 3, 6) 0 1, ∞ 1, 2 10 3 (4, 7, 9) (1, 4, 5) 0 1, ∞ 1, 2 10 3 (5, 7, 8) (1, 4, 5) 0 1, ∞ 1, 2 10 3 (6, 7, 7) (1, 4, 5) 0 1, ∞ 1, 2 10 3 (2, 9, 9) (2, 3, 5) 0 1, ∞ 1, 2 10 3 (3, 8, 9) (2, 3, 5) 0 1, ∞ 1, 2 10 3 (5, 6, 9) (2, 3, 5) 0 1, ∞ 1, 2 60 10 3 (3, 8, 9) (3, 3, 4) 0 1, ∞ 1, 2 10 3 (5, 6, 9) (3, 3, 4) 0 1, ∞ 1, 2 11 3 (6, 8, 8) (1, 2, 8) 0 1, 5, ∞ 1, 5 11 3 (6, 8, 8) (1, 2, 8) 0 1, 5, ∞ 1, 5 11 3 (6, 7, 9) (1, 3, 7) 0 1, 5, ∞ 1, 5 11 3 (5, 8, 9) (1, 5, 5) 0 1, 5, ∞ 1 11 3 (6, 6, 10) (2, 3, 6) 0 1, 5, ∞ 1 11 3 (6, 6, 10) (2, 3, 6) 0 1, 5, ∞ 1 11 3 (4, 8, 10) (2, 4, 5) 0 1, 5, ∞ 1 11 3 (6, 6, 10) (2, 3, 6) 0 1, 5, ∞ 1 11 3 (3, 9, 10) (3, 3, 5) 0 1, 5, ∞ 1 12 3 (7, 8, 9) (1, 2, 9) 0 1, 2, ∞ 1 12 3 (6, 7, 11) (1, 4, 7) 0 1, 2, ∞ 1 12 3 (7, 7, 10) (1, 4, 7) 0 1, 2, ∞ 1 12 3 (7, 8, 9) (1, 4, 7) 0 1, 2, ∞ 1 12 3 (5, 8, 11) (1, 5, 6) 0 1, 2, ∞ 1 12 3 (5, 9, 10) (1, 5, 6) 0 1, 2, ∞ 1 12 3 (6, 7, 11) (2, 3, 7) 0 1, 2, ∞ 1 12 3 (7, 7, 10) (2, 3, 7) 0 1, 2, ∞ 1 12 3 (7, 7, 10) (2, 3, 7) 0 1, 2, ∞ 1 12 3 (7, 8, 9) (2, 3, 7) 0 1, 2, ∞ 1 12 3 (5, 8, 11) (2, 5, 5) 0 1, 2, ∞ 1 12 3 (5, 9, 10) (2, 5, 5) 0 1, 2, ∞ 1 12 3 (3, 10, 11) (3, 4, 5) 0 1, 2, ∞ 1 12 3 (5, 8, 11) (2, 5, 5) 0 1, 2, ∞ 1 12 3 (5, 8, 11) (3, 4, 5) 0 1, 2, ∞ 1 12 3 (5, 9, 10) (3, 4, 5) 0 1, 2, ∞ 1 13 3 (8, 9, 9) (1, 3, 9) 0 1, 3, ∞ 1, 2, 3 13 3 (8, 8, 10) (1, 4, 8) 0 1, 3, ∞ 1, 2, 3 13 3 (7, 9, 10) (1, 5, 7) 0 1, 3, ∞ 1, 2, 3 13 3 (8, 9, 9) (1, 3, 9) 0 1, 3, ∞ 1, 2, 3 13 3 (6, 8, 12) (3, 4, 6) 0 1, 3, ∞ 1, 2, 3 13 3 (5, 9, 12) (3, 5, 5) 0 1, 3, ∞ 1, 2, 3 13 3 (4, 10, 12) (4, 4, 5) 0 1, 3, ∞ 1, 2, 3 14 3 (7, 10, 11) (1, 2, 11) 0 1, 3, ∞ 1, 3 14 3 (7, 9, 12) (1, 4, 9) 0 1, 3, ∞ 1, 3 14 3 (7, 10, 11) (1, 5, 8) 0 1, 3, ∞ 1, 3 14 3 (7, 9, 12) (2, 3, 9) 0 1, 3, ∞ 1, 3 61 14 3 (9, 9, 10) (2, 3, 9) 0 1, 3, ∞ 1, 3 14 3 (5, 10, 13) (2, 5, 7) 0 1, 3, ∞ 1, 3 14 3 (5, 11, 12) (2, 5, 7) 0 1, 3, ∞ 1, 3 14 3 (6, 11, 11) (2, 5, 7) 0 1, 3, ∞ 1, 3 14 3 (7, 10, 11) (2, 5, 7) 0 1, 3, ∞ 1, 3 14 3 (7, 9, 12) (3, 3, 8) 0 1, 3, ∞ 1, 3 14 3 (3, 12, 13) (3, 4, 7) 0 1, 3, ∞ 1, 3 14 3 (6, 9, 13) (3, 4, 7) 0 1, 3, ∞ 1, 3 14 3 (7, 9, 12) (3, 4, 7) 0 1, 3, ∞ 1, 3 14 3 (9, 9, 10) (3, 4, 7) 0 1, 3, ∞ 1, 3 14 3 (5, 11, 12) (4, 5, 5) 0 1, 3, ∞ 1, 3 14 3 (7, 10, 11) (4, 5, 5) 0 1, 3, ∞ 1, 3 15 3 (8, 10, 12) (1, 2, 12) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 11, 11) (1, 3, 11) 0 1, 2, 4, ∞ 1, 2 15 3 (9, 10, 11) (1, 3, 11) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 10, 12) (1, 4, 10) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 9, 13) (1, 5, 9) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 10, 12) (1, 6, 8) 0 1, 2, 4, ∞ 1, 2 15 3 (7, 10, 13) (1, 7, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (7, 11, 12) (1, 7, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 10, 12) (2, 4, 9) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 8, 14) (2, 5, 8) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 10, 12) (2, 5, 8) 0 1, 2, 4, ∞ 1, 2 15 3 (6, 10, 14) (2, 6, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 8, 14) (3, 4, 8) 0 1, 2, 4, ∞ 1, 2 15 3 (8, 10, 12) (3, 4, 8) 0 1, 2, 4, ∞ 1, 2 15 3 (3, 13, 14) (3, 5, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (5, 11, 14) (3, 5, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (6, 11, 13) (3, 5, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (7, 9, 14) (3, 5, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (7, 10, 13) (3, 5, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (7, 11, 12) (3, 5, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (4, 12, 14) (4, 4, 7) 0 1, 2, 4, ∞ 1, 2 15 3 (4, 12, 14) (4, 5, 6) 0 1, 2, 4, ∞ 1, 2 16 3 (8, 11, 13) (1, 2, 13) 0 1, 2, 4, ∞ 1, 2 16 3 (8, 11, 13) (1, 4, 11) 0 1, 2, 4, ∞ 1, 2 16 3 (10, 11, 11) (1, 4, 11) 0 1, 2, 4, ∞ 1, 2 16 3 (9, 11, 12) (1, 6, 9) 0 1, 2, 4, ∞ 1, 2 62 16 3 (7, 11, 14) (1, 7, 8) 0 1, 2, 4, ∞ 1, 2 16 3 (7, 12, 13) (1, 7, 8) 0 1, 2, 4, ∞ 1, 2 16 3 (9, 11, 12) (2, 3, 11) 0 1, 2, 4, ∞ 1, 2 16 3 (8, 9, 15) (2, 5, 9) 0 1, 2, 4, ∞ 1, 2 16 3 (7, 12, 13) (2, 7, 7) 0 1, 2, 4, ∞ 1, 2 16 3 (8, 9, 15) (3, 4, 9) 0 1, 2, 4, ∞ 1, 2 16 3 (9, 9, 14) (3, 4, 9) 0 1, 2, 4, ∞ 1, 2 16 3 (3, 14, 15) (3, 5, 8) 0 1, 2, 4, ∞ 1, 2 16 3 (5, 12, 15) (3, 5, 8) 0 1, 2, 4, ∞ 1, 2 16 3 (5, 13, 14) (4, 5, 7) 0 1, 2, 4, ∞ 1, 2 16 3 (7, 10, 15) (4, 5, 7) 0 1, 2, 4, ∞ 1, 2 16 3 (5, 12, 15) (5, 5, 6) 0 1, 2, 4, ∞ 1, 2 17 3 (10, 12, 12) (2, 3, 12) 0 1, ∞ 1, 2, 4, 8 17 3 (10, 10, 14) (2, 5, 10) 0 1, ∞ 1, 2, 4, 8 17 3 (8, 12, 14) (2, 7, 8) 0 1, ∞ 1, 2, 4, 8 17 3 (9, 10, 15) (3, 5, 9) 0 1, ∞ 1, 2, 4, 8 17 3 (7, 12, 15) (3, 7, 7) 0 1, ∞ 1, 2, 4, 8 17 3 (5, 14, 15) (5, 5, 7) 0 1, ∞ 1, 2, 4, 8 18 3 (9, 13, 14) (1, 3, 14) 0 1, 3, ∞ 1, 3 18 3 (11, 12, 13) (1, 4, 13) 0 1, 3, ∞ 1, 3 18 3 (9, 11, 16) (1, 6, 11) 0 1, 3, ∞ 1, 3 18 3 (11, 11, 14) (1, 6, 11) 0 1, 3, ∞ 1, 3 18 3 (8, 13, 15) (1, 8, 9) 0 1, 3, ∞ 1, 3 18 3 (11, 12, 13) (1, 8, 9) 0 1, 3, ∞ 1, 3 18 3 (9, 13, 14) (2, 3, 13) 0 1, 3, ∞ 1, 3 18 3 (10, 13, 13) (2, 3, 13) 0 1, 3, ∞ 1, 3 18 3 (10, 11, 15) (2, 5, 11) 0 1, 3, ∞ 1, 3 18 3 (7, 12, 17) (2, 7, 9) 0 1, 3, ∞ 1, 3 18 3 (7, 14, 15) (2, 7, 9) 0 1, 3, ∞ 1, 3 18 3 (9, 11, 16) (3, 4, 11) 0 1, 3, ∞ 1, 3 18 3 (11, 12, 13) (3, 4, 11) 0 1, 3, ∞ 1, 3 18 3 (9, 10, 17) (3, 5, 10) 0 1, 3, ∞ 1, 3 18 3 (7, 13, 16) (3, 7, 8) 0 1, 3, ∞ 1, 3 18 3 (9, 13, 14) (3, 7, 8) 0 1, 3, ∞ 1, 3 18 3 (4, 15, 17) (4, 5, 9) 0 1, 3, ∞ 1, 3 18 3 (10, 11, 15) (4, 5, 9) 0 1, 3, ∞ 1, 3 18 3 (7, 12, 17) (4, 7, 7) 0 1, 3, ∞ 1, 3 18 3 (5, 15, 16) (5, 5, 8) 0 1, 3, ∞ 1, 3 63 18 3 (5, 14, 17) (5, 6, 7) 0 1, 3, ∞ 1, 3 18 3 (7, 14, 15) (5, 6, 7) 0 1, 3, ∞ 1, 3 18 3 (9, 10, 17) (5, 6, 7) 0 1, 3, ∞ 1, 3 19 3 (12, 12, 14) (3, 4, 12) 0 1, 3, 9, ∞ 1, 3, 9 19 3 (11, 12, 15) (3, 5, 11) 0 1, 3, 9, ∞ 1, 3, 9 19 3 (9, 14, 15) (3, 7, 9) 0 1, 3, 9, ∞ 1, 3, 9 19 3 (10, 12, 16) (4, 5, 10) 0 1, 3, 9, ∞ 1, 3, 9 19 3 (8, 14, 16) (4, 7, 8) 0 1, 3, 9, ∞ 1, 3, 9 19 3 (7, 15, 16) (5, 7, 7) 0 1, 3, 9, ∞ 1, 3, 9 20 3 (12, 13, 15) (1, 6, 13) 0 1, 2, 4 ∞ 1, 2 20 3 (11, 14, 15) (1, 8, 11) 0 1, 2, 4 ∞ 1, 2 20 3 (9, 15, 16) (1, 9, 10) 0 1, 2, 4 ∞ 1, 2 20 3 (10, 13, 17) (2, 5, 13) 0 1, 2, 4 ∞ 1, 2 20 3 (11, 13, 16) (2, 5, 13) 0 1, 2, 4 ∞ 1, 2 20 3 (13, 13, 14) (2, 5, 13) 0 1, 2, 4 ∞ 1, 2 20 3 (11, 14, 15) (2, 7, 11) 0 1, 2, 4 ∞ 1, 2 20 3 (9, 15, 16) (2, 9, 9) 0 1, 2, 4 ∞ 1, 2 20 3 (12, 13, 15) (3, 4, 13) 0 1, 2, 4 ∞ 1, 2 20 3 (12, 13, 15) (3, 6, 11) 0 1, 2, 4 ∞ 1, 2 20 3 (7, 15, 18) (3, 7, 10) 0 1, 2, 4 ∞ 1, 2 20 3 (9, 15, 16) (3, 7, 10) 0 1, 2, 4 ∞ 1, 2 20 3 (9, 15, 16) (3, 8, 9) 0 1, 2, 4 ∞ 1, 2 20 3 (10, 11, 19) (4, 5, 11) 0 1, 2, 4 ∞ 1, 2 20 3 (10, 13, 17) (4, 5, 11) 0 1, 2, 4 ∞ 1, 2 20 3 (11, 11, 18) (4, 5, 11) 0 1, 2, 4 ∞ 1, 2 20 3 (11, 12, 17) (4, 5, 11) 0 1, 2, 4 ∞ 1, 2 20 3 (7, 15, 18) (4, 7, 9) 0 1, 2, 4 ∞ 1, 2 20 3 (9, 12, 19) (5, 6, 9) 0 1, 2, 4 ∞ 1, 2 20 3 (9, 13, 18) (5, 6, 9) 0 1, 2, 4 ∞ 1, 2 20 3 (7, 14, 19) (5, 7, 8) 0 1, 2, 4 ∞ 1, 2 20 3 (7, 16, 17) (5, 7, 8) 0 1, 2, 4 ∞ 1, 2 20 3 (9, 14, 17) (5, 7, 8) 0 1, 2, 4 ∞ 1, 2 20 3 (7, 15, 18) (6, 7, 7) 0 1, 2, 4 ∞ 1, 2 3 4 (1, 1, 2, 2) (1, 1, 1) 0 1, 2, ∞ 1 4 4 (1, 1, 3, 3) (1, 1, 2) 0 1, 2, ∞ 1 4 4 (1, 2, 2, 3) (1, 1, 2) 0 1, 2, ∞ 1 5 4 (1, 3, 3, 3) (1, 1, 3) 0 1, 2, 4, ∞ 1, 2 5 4 (2, 2, 2, 4) (1, 2, 2) 0 1, 2, 4, ∞ 1, 2 64 6 4 (1, 1, 5, 5) (1, 2, 3) 0 1, 2, ∞ 1 6 4 (1, 2, 4, 5) (1, 2, 3) 0 1, 2, ∞ 1 6 4 (1, 3, 3, 5) (1, 1, 4) 0 1, 2, ∞ 1 6 4 (1, 3, 3, 5) (1, 2, 3) 0 1, 2, ∞ 1 6 4 (1, 3, 4, 4) (1, 1, 4) 0 1, 2, ∞ 1 6 4 (1, 3, 4, 4) (1, 2, 3) 0 1, 2, ∞ 1 6 4 (2, 2, 3, 5) (1, 2, 3) 0 1, 2, ∞ 1 6 4 (2, 3, 3, 4) (1, 1, 4) 0 1, 2, ∞ 1 6 4 (2, 3, 3, 4) (1, 2, 3) 0 1, 2, ∞ 1 7 4 (2, 4, 4, 4) (1, 2, 4) 0 1, 2, 3, 6, ∞ 1,3 7 4 (3, 3, 3, 5) (1, 3, 3) 0 1, 2, 3, 6, ∞ 1,3 8 4 (1, 3, 6, 6) (1, 3, 4) 0 1, 2, 4, ∞ 1 8 4 (1, 5, 5, 5) (1, 2, 5) 0 1, 2, 4, ∞ 1 8 4 (2, 4, 5, 5) (1, 2, 5) 0 1, 2, 4, ∞ 1 8 4 (3, 3, 3, 7) (1, 3, 4) 0 1, 2, 4, ∞ 1 8 4 (3, 3, 3, 7) (2, 3, 3) 0 1, 2, 4, ∞ 1 8 4 (3, 3, 4, 6) (1, 3, 4) 0 1, 2, 4, ∞ 1 8 4 (3, 3, 4, 6) (2, 3, 3) 0 1, 2, 4, ∞ 1 9 4 (1, 5, 5, 7) (1, 3, 5) 0 1, 2, 3, 6, ∞ 1, 3 9 4 (2, 4, 4, 8) (2, 3, 4) 0 1, 2, 3, 6, ∞ 1, 3 9 4 (3, 5, 5, 5) (1, 3, 5) 0 1, 2, 3, 6, ∞ 1, 3 9 4 (4, 4, 4, 6) (1, 4, 4) 0 1, 2, 3, 6, ∞ 1, 3 9 4 (4, 4, 4, 6) (2, 3, 4) 0 1, 2, 3, 6, ∞ 1, 3 10 4 (1, 4, 7, 8) (1, 4, 5) 0 1, 2, 4, ∞ 1, 2 10 4 (1, 5, 7, 7) (1, 2, 7) 0 1, 2, 4, ∞ 1, 2 10 4 (1, 5, 7, 7) (1, 4, 5) 0 1, 2, 4, ∞ 1, 2 10 4 (2, 3, 6, 9) (2, 3, 5) 0 1, 2, 4, ∞ 1, 2 10 4 (2, 4, 7, 7) (1, 4, 5) 0 1, 2, 4, ∞ 1, 2 10 4 (3, 3, 5, 9) (2, 3, 5) 0 1, 2, 4, ∞ 1, 2 10 4 (3, 3, 5, 9) (3, 3, 4) 0 1, 2, 4, ∞ 1, 2 10 4 (3, 3, 6, 8) (2, 3, 5) 0 1, 2, 4, ∞ 1, 2 10 4 (3, 5, 6, 6) (1, 3, 6) 0 1, 2, 4, ∞ 1, 2 10 4 (3, 5, 6, 6) (2, 3, 5) 0 1, 2, 4, ∞ 1, 2 10 4 (4, 4, 5, 7) (1, 4, 5) 0 1, 2, 4, ∞ 1, 2 12 4 (1, 5, 8, 10) (1, 5, 6) 0 1, 2, 4, ∞ 1 12 4 (1, 7, 7, 9) (1, 4, 7) 0 1, 2, 4, ∞ 1 12 4 (2, 6, 7, 9) (1, 4, 7) 1 1, 2, 4, ∞ 1 12 4 (2, 7, 7, 8) (2, 3, 7) 0 1, 2, 4, ∞ 1 65 12 4 (3, 5, 5, 11) (3, 4, 5) 0 1, 2, 4, ∞ 1 12 4 (3, 5, 6, 10) (3, 4, 5) 0 1, 2, 4, ∞ 1 12 4 (3, 5, 8, 8) (1, 5, 6) 1 1, 2, 4, ∞ 1 12 4 (3, 7, 7, 7) (1, 4, 7) 0 1, 2, 4, ∞ 1 12 4 (3, 7, 7, 7) (2, 3, 7) 0 1, 2, 4, ∞ 1 12 4 (4, 5, 5, 10) (1, 5, 6) 0 1, 2, 4, ∞ 1 12 4 (4, 5, 5, 10) (3, 4, 5) 0 1, 2, 4, ∞ 1 12 4 (4, 6, 7, 7) (1, 4, 7) 0 1, 2, 4, ∞ 1 12 4 (4, 6, 7, 7) (2, 3, 7) 0 1, 2, 4, ∞ 1 12 4 (5, 5, 5, 9) (1, 5, 6) 0 1, 2, 4, ∞ 1 12 4 (5, 5, 5, 9) (2, 5, 5) 0 1, 2, 4, ∞ 1 12 4 (5, 5, 5, 9) (3, 4, 5) 0 1, 2, 4, ∞ 1 12 4 (5, 5, 6, 8) (1, 5, 6) 0 1, 2, 4, ∞ 1 12 4 (5, 5, 6, 8) (2, 5, 5) 0 1, 2, 4, ∞ 1 12 4 (5, 5, 6, 8) (3, 4, 5) 0 1, 2, 4, ∞ 1 14 4 (1, 5, 11, 11) (2, 5, 7) 2 1, 2, 3, 6, ∞ 14 4 (1, 9, 9, 9) (3, 4, 7) 3 1, 2, 3, 6, ∞ 14 4 (2, 5, 10, 11) (2, 5, 7) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (3, 3, 9, 13) (3, 4, 7) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (3, 4, 9, 12) (3, 4, 7) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (3, 7, 9, 9) (2, 3, 9) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (3, 7, 9, 9) (3, 4, 7) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (4, 6, 9, 9) (3, 4, 7) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (5, 5, 5, 13) (2, 5, 7) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (5, 5, 7, 11) (2, 5, 7) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (5, 5, 7, 11) (4, 5, 5) 0 1, 2, 3, 6, ∞ 1, 3 14 4 (5, 5, 8, 10) (2, 5, 7) 0 1, 2, 3, 6, ∞ 1, 3 15 4 (2, 8, 8, 12) (2, 5, 8) 0 1, 2, 4, 8, ∞ 1, 2 15 4 (3, 7, 7, 13) (3, 5, 7) 0 1, 2, 4, 8, ∞ 1, 2 15 4 (4, 8, 8, 10) (3, 4, 8) 0 1, 2, 4, 8, ∞ 1, 2 15 4 (5, 7, 7, 11) (3, 5, 7) 0 1, 2, 4, 8, ∞ 1, 2 15 4 (6, 8, 8, 8) (2, 5, 8) 0 1, 2, 4, 8, ∞ 1, 2 15 4 (7, 7, 7, 9) (3, 5, 7) 0 1, 2, 4, 8, ∞ 1, 2 16 4 (7, 7, 7, 11) (1, 7, 8) 0 1, 2, 4, 8, ∞ 1, 2 18 4 (4, 5, 12, 15) (4, 5, 9) 0 1, 2, 3, 6, ∞ 1, 3 18 4 (5, 5, 11, 15) (4, 5, 9) 0 1, 2, 3, 6, ∞ 1, 3 18 4 (7, 7, 7, 15) (2, 7, 9) 0 1, 2, 3, 6, ∞ 1, 3 18 4 (7, 7, 10, 12) (2, 7, 9) 0 1, 2, 3, 6, ∞ 1, 3 66 22 4 (7, 7, 13, 17) (4, 7, 11) 0 1, 2, 5, 10, ∞ 1, 5 22 4 (9, 9, 9, 17) (2, 9, 11) 0 1, 2, 5, 10, ∞ 1, 5 3 5 (1, 1, 1, 1, 2) (1, 1, 1) 0 1, 2, 3, ∞ 1, 2 4 5 (1, 1, 1, 2, 3) (1, 1, 2) 0 1, 2, 3, ∞ 1, 2 4 5 (1, 1, 2, 2, 2) (1, 1, 2) 0 1, 2, 3, ∞ 1, 2 5 5 (2, 2, 2, 2, 2) (1, 2, 2) 0 1 - 4, ∞ 1, 2, 4 6 5 (1, 1, 1, 4, 5) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 6 5 (1, 1, 2, 3, 5) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 6 5 (1, 1, 2, 4, 4) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 6 5 (1, 1, 3, 3, 4) (1, 1, 4) 0 1, 2, 3, ∞ 1, 2 6 5 (1, 1, 3, 3, 4) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 6 5 (1, 2, 2, 2, 5) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 6 5 (1, 2, 2, 3, 4) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 6 5 (1, 2, 3, 3, 3) (1, 1, 4) 1 1, 2, 3, ∞ 1, 2 6 5 (1, 2, 3, 3, 3) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 6 5 (2, 2, 2, 3, 3) (1, 2, 3) 0 1, 2, 3, ∞ 1, 2 8 5 (1, 3, 3, 3, 6) (1, 3, 4) 0 1 - 4, 6, ∞ 1, 2 8 5 (3, 3, 3, 3, 4) (1, 3, 4) 0 1 - 4, 6, ∞ 1, 2 8 5 (3, 3, 3, 3, 4) (2, 3, 3) 0 1 - 4, 6, ∞ 1, 2 9 5 (2, 4, 4, 4, 4) (2, 3, 4) 0 1, 2, 3, 6, 9, ∞ 1, 2, 3, 6 10 5 (1, 1, 4, 7, 7) (1, 4, 5) 0 1 - 4, ∞ 1, 2, 4 10 5 (1, 4, 4, 4, 7) (1, 4, 5) 0 1 - 4, ∞ 1, 2, 4 10 5 (2, 3, 3, 3, 9) (2, 3, 5) 0 1 - 4, ∞ 1, 2, 4 10 5 (2, 3, 3, 6, 6) (2, 3, 5) 0 1 - 4, ∞ 1, 2, 4 10 5 (3, 3, 3, 3, 8) (2, 3, 5) 0 1 - 4, ∞ 1, 2, 4 10 5 (3, 3, 3, 5, 6) (2, 3, 5) 0 1 - 4, ∞ 1, 2, 4 12 5 (1, 5, 5, 5, 8) (1, 5, 6) 0 1 - 4, 6, ∞ 1, 2 12 5 (3, 5, 5, 5, 6) (3, 4, 5) 0 1 - 4, 6, ∞ 1, 2 12 5 (4, 5, 5, 5, 5) (1, 5, 6) 0 1 - 4, 6, ∞ 1, 2 12 5 (4, 5, 5, 5, 5) (3, 4, 5) 0 1 - 4, 6, ∞ 1, 2 14 5 (2, 5, 5, 5, 11) (2, 5, 7) 0 1, 2, 3, 6, 9, ∞ 1, 2, 3, 6 14 5 (3, 3, 4, 9, 9) (3, 4, 7) 0 1, 2, 3, 6, 9, ∞ 1, 2, 3, 6 14 5 (5, 5, 5, 5, 8) (2, 5, 7) 0 1, 2, 3, 6, 9, ∞ 1, 2, 3, 6 3 6 (1, 1, 1, 1, 1, 1) (1, 1, 1) 0 1 - 4, ∞ 1, 2 4 6 (1, 1, 1, 1, 1, 3) (1, 1, 2) 0 1 - 4, ∞ 1, 2 4 6 (1, 1, 1, 1, 2, 2) (1, 1, 2) 0 1 - 4, ∞ 1, 2 6 6 (1, 1, 1, 1, 3, 5) (1, 2, 3) 0 1 - 4, ∞ 1, 2 6 6 (1, 1, 1, 1, 4, 4) (1, 2, 3) 0 1 - 4, ∞ 1, 2 67 6 6 (1, 1, 1, 2, 2, 5) (1, 2, 3) 0 1 - 4, ∞ 1, 2 6 6 (1, 1, 1, 2, 3, 4) (1, 2, 3) 0 1 - 4, ∞ 1, 2 6 6 (1, 1, 1, 3, 3, 3) (1, 1, 4) 1 1 - 4, ∞ 6 6 (1, 1, 1, 3, 3, 3) (1, 2, 3) 0 1 - 4, ∞ 1, 2 6 6 (1, 1, 2, 2, 2, 4) (1, 2, 3) 0 1 - 4, ∞ 1, 2 6 6 (1, 1, 2, 2, 3, 3) (1, 2, 3) 0 1 - 4, ∞ 1, 2 6 6 (1, 2, 2, 2, 2, 3) (1, 2, 3) 0 1 - 4, ∞ 1, 2 8 6 (1, 3, 3, 3, 3, 3) (1, 3, 4) 0 1 - 4, 6, 8, ∞ 1, 2 10 6 (2, 3, 3, 3, 3, 6) (2, 3, 5) 0 1 - 4, 8, ∞ 1, 2, 4 10 6 (3, 3, 3, 3, 3, 5) (2, 3, 5) 0 1 - 4, 8, ∞ 1, 2, 4 4 7 (1, 1, 1, 1, 1, 1, 2) (1, 1, 2) 0 1 - 5, ∞ 1, 2, 3 6 7 (1, 1, 1, 1, 1, 2, 5) (1, 2, 3) 0 1 - 5, ∞ 1, 2, 3 6 7 (1, 1, 1, 1, 1, 3, 4) (1, 2, 3) 0 1 - 5, ∞ 1, 2, 3 6 7 (1, 1, 1, 1, 2, 2, 4) (1, 2, 3) 0 1 - 5, ∞ 1, 2, 3 6 7 (1, 1, 1, 1, 2, 3, 3) (1, 2, 3) 0 1 - 5, ∞ 1, 2, 3 6 7 (1, 1, 1, 2, 2, 2, 3) (1, 2, 3) 0 1 - 5, ∞ 1, 2, 3 6 7 (1, 1, 2, 2, 2, 2, 2) (1, 2, 3) 0 1 - 5, ∞ 1, 2, 3 10 7 (2, 3, 3, 3, 3, 3, 3) (2, 3, 5) 0 1 - 5, 8, ∞ 1 - 4, 6 4 8 (1, 1, 1, 1, 1, 1, 1, 1) (1, 1, 2) 0 1 - 6, ∞ 1, 2, 3 6 8 (1, 1, 1, 1, 1, 1, 1, 5) (1, 2, 3) 0 1 - 6, ∞ 1, 2, 3 6 8 (1, 1, 1, 1, 1, 1, 2, 4) (1, 2, 3) 0 1 - 6, ∞ 1, 2, 3 6 8 (1, 1, 1, 1, 1, 2, 2, 3) (1, 2, 3) 0 1 - 6, ∞ 1, 2, 3 6 8 (1, 1, 1, 1, 1, 1, 3, 3) (1, 2, 3) 0 1 - 6, ∞ 1, 2, 3 6 8 (1, 1, 1, 1, 2, 2, 2, 2) (1, 2, 3) 0 1 - 6, ∞ 1, 2, 3 6 9 (1, 1, 1, 1, 1, 1, 1, 1, 4) (1, 2, 3) 0 1 - 7, ∞ 1-4 6 9 (1, 1, 1, 1, 1, 1, 1, 2, 3) (1, 2, 3) 0 1 - 7, ∞ 1-4 6 9 (1, 1, 1, 1, 1, 1, 2, 2, 2) (1, 2, 3) 0 1 - 7, ∞ 1-4 6 10 (1, 1, 1, 1, 1, 1, 1, 1, 1, 3) (1, 2, 3) 0 1 - 8, ∞ 1-4 6 10 (1, 1, 1, 1, 1, 1, 1, 1, 2, 2) (1, 2, 3) 0 1 - 8, ∞ 1-4 6 11 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2) (1, 2, 3) 0 1 - 9, ∞ 1-5 6 12 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (1, 2, 3) 0 1 - 10, ∞ 1-5 Table 5.1: Special families of k3-type surfaces