p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes Citation Yu, Qiyao (Vivian) (2021) p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes. Senior thesis (Major), California Institute of Technology. doi:10.7907/ccbk-ht28. https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935 Abstract Let E be a CM elliptic curve over the rationals and p a good, ordinary prime. We show the implication: corank (Z p ) Sel (p ∞ ) (E/ Q ) = 1 ⟹ ord (s=1) L(s,E/ Q ) = 1 for the p ∞ -Selmer group Sel (p ∞ ) (E/ Q ) and the complex L-function L(s,E/ Q Q) in the case of p = 2, 3. Item Type: Thesis (Senior thesis (Major)) Subject Keywords: Number Theory Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Eric Temple Bell Undergraduate Mathematics Research Prize, 2021. Taussky-Todd Mathematics Prize Fund, 2020. Thesis Availability: Public (worldwide access) Research Advisor(s): Burungale, Ashay Thesis Committee: Burungale, Ashay (chair) Mantovan, Elena Ramakrishnan, Dinakar Defense Date: 4 May 2021 Record Number: CaltechTHESIS:06022021-000654935 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935 DOI: 10.7907/ccbk-ht28 ORCID: Author ORCID Yu, Qiyao (Vivian) 0000-0002-6820-4371 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 14225 Collection: CaltechTHESIS Deposited By: Qiyao Yu Deposited On: 02 Jun 2021 21:07 Last Modified: 03 Nov 2021 20:18 Thesis Files PDF (complete thesis) - Final Version See Usage Policy. 359kB Repository Staff Only: item control page

, 𝑝-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes Thesis by Qiyao Yu In Partial Fulfillment of the Requirements for the Degree of Bachelor of Science CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2021 Defended May 4, 2021 ii © 2021 Qiyao Yu ORCID: 0000-0002-6820-4371 All rights reserved iii ACKNOWLEDGEMENTS The study of this 𝑝-converse Theorem first started as a collaborative Summer Undergraduate Research Fellowship (SURF) project with Jacob Ressler under the guidance of Dr. Ashay Burungale in 2020. It was then completed during the course of this senior thesis at Caltech mentored by Dr. Burungale. This project is also partially supported by the Olga Taussky-Todd Prize. I would like to thank my mentor Dr. Ashay Burungale for his guidance, suggestions, and support during the tough period of the pandemic. I would also like to thank Prof. Dinakar Ramakrishnan and Prof. Elena Mantovan for serving on the thesis committee and providing extremely helpful advice. iv ABSTRACT Let 𝐸 be a CM elliptic curve over the rationals and 𝑝 a good, ordinary prime. We show the implication corankZ 𝑝 Sel 𝑝∞ (𝐸 /Q ) = 1 =⇒ ord𝑠=1 𝐿 (𝑠, 𝐸 /Q ) = 1 for the 𝑝 ∞ Selmer group Sel 𝑝∞ (𝐸 /Q ) and the complex 𝐿-function 𝐿(𝑠, 𝐸 /Q ) in the case of 𝑝 = 2, 3. v CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: Heegner Main Conjecture . . . . . . . . . . . . . . . . . . . . . . 2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Heegner Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter III: Heegner Main Conjecture: Verification and Modification . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Non-torsion Heegner Cohomological Class . . . . . . . . . . . . . . 3.3 Rank Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Equality of Characteristic Ideals . . . . . . . . . . . . . . . . . . . . Chapter IV: 𝑝-Converse Theorem . . . . . . . . . . . . . . . . . . . . . . . Chapter V: Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v 1 4 4 6 6 7 8 8 14 14 15 23 25 26 1 Chapter 1 INTRODUCTION Let 𝐸 be a rational elliptic curve. A fundamental arithmetic invariant of 𝐸 is the Mordell-Weil rank, which is defined to be the rank of the finitely generated abelian group 𝐸 (Q), i.e. rational points on the elliptic curve. The Mordell-Weil rank is typically expected to be 0 or 1, but currently there is no systematic approach to find generators of 𝐸 (Q). An equally mysterious object that governs the arithmetic of 𝐸 is the conjecturally finite Tate-Shafarevich group X(𝐸 /Q ). The 𝑝 ∞ -Selmer group Sel 𝑝∞ (𝐸 /Q ) encodes information about the arithmetic of 𝐸 via the fundamental exact sequence 𝜅 𝜌 0 → 𝐸 (Q) ⊗Z Q 𝑝 /Z 𝑝 → − Sel 𝑝∞ (𝐸 /Q ) → − X(𝐸 /Q ) [ 𝑝 ∞ ] → 0 (1.0.1) Here 𝜅 arises from the Kummer map and 𝜌 is induced by the map in the long exact sequence of cohomologies. An underlying object on the analytic side is the complex Hasse-Weil 𝐿-function 𝐿 (𝑠, 𝐸 /Q ) corresponding to the elliptic curve 𝐸 /Q with 𝑠 ∈ C. An important analytic invariant is the analytic rank given by the order of vanishing at 𝑠 = 1, which is denoted by ord𝑠=1 𝐿 (𝑠, 𝐸 /Q ). The BSD conjecture predicts a deep relation between the arithmetic and analytic sides. We include here one particular instance of the conjecture: Conjecture 1 (BSD conjecture). Let 𝐸 be an elliptic curve over the rationals. For 𝑟 = 0, 1, the following are equivalent: (1) rankZ 𝐸 (Q) = 𝑟 and X(𝐸 /Q ) is finite. (2) corankZ 𝑝 Sel 𝑝∞ (𝐸 /Q ) = 𝑟 for a prime 𝑝. (3) ord𝑠=1 𝐿 (𝑠, 𝐸 /Q ) = 𝑟. The implication (1) ⇒ (2) follows from the fundamental exact sequence 1.0.1. The implication (3) ⇒ (1) is a fundamental result on the BSD conjecture due to CoatesWiles, Gross-Zagier, Kolyvagin and Rubin. Thus, the implication (2) ⇒ (3) is 2 essentially a converse to this fundamental result, and we refer to it as the 𝑝-converse theorem. There has been significant progress made towards the 𝑝-converse theorem. When 𝑟 = 0 and 𝐸 is a CM elliptic curve, the 𝑝-converse was first established when ( 𝑝, |O𝐾× |) = 1 by Rubin in the early 90’s. Burungale-Tian then established the case for 𝑝 = 2, 3 in 2019. When 𝑟 = 0 and 𝐸 is non-CM, Skinner-Urban showed the 𝑝-converse for ordinary primes 𝑝 > 2 with certain assumptions on the Galois representation 𝐸 [ 𝑝]. Wan proved certain cases for supersingular primes 𝑝 > 2. For 𝑟 = 1 and 𝐸 a non-CM elliptic curve, Skinner and Zhang independently showed the result for good ordinary primes 𝑝 > 3 with certain assumptions on 𝐸 [ 𝑝]. For 𝑟 = 1 and 𝐸 a CM elliptic curve, Burungale-Tian (Burungale and Tian, 2020) and Burungale-Skinner-Tian have established the case for good ordinary and good supersingular prime 𝑝 > 3 respectively. The main result of the article is a 𝑝-converse in the case of CM elliptic curves. Theorem 1. Let 𝐸 be a CM elliptic curve over the rationals. Let 𝑝 ∈ {2, 3} be a good ordinary prime for the elliptic curve 𝐸 /Q . Then, corankZ 𝑝 Sel 𝑝∞ (𝐸 /Q ) = 1 =⇒ ord𝑠=1 𝐿 (𝑠, 𝐸 /Q ) = 1 In particular, whenever corankZ 𝑝 Sel 𝑝∞ (𝐸 /Q ) = 1, we know rankZ 𝐸 (Q) = 1 and X(𝐸 /Q ) is finite We restricted ourselves to 𝑝 ∈ {2, 3} as Burungale-Tian established the same statement in (Burungale and Tian, 2020) except that they assume 𝑝 > 3. Together with their result, the rank 1 𝑝-converse is established for all CM elliptic curves with good ordinary reduction at 𝑝. The approach to prove Theorem 1 closely follows that in (Burungale and Tian, 2020), with modifications to steps that used the assumption 𝑝 > 3. For completeness, we still include the essential components of the argument in (Burungale and Tian, 2020), with emphasis on the modified parts. In (Burungale and Tian, 2020), the proof first reduces the 𝑝-converse implication to an equivalent statement via an auxiliary Rankin-Selberg setup. With the generalized Gross-Zagier formula due to Yuan-Zhang-Zhang (Yuan, S.-W. Zhang, and W. Zhang, 2013), the equivalent statement is proven by showing a certain Heegner 3 point is non-torsion. The proof of the Heegner point being non-torsion relies on the Galois descent of a certain Heegner Main Conjecture, which is Iwasawa-theoretic in nature. When we include 𝑝 = 2, 3, the original approach goes through assuming the Heegner Main Conjecture (HMC). Thus, the major modifications take place in the reformulation and proof of the HMC when 𝑝 = 2, 3. Notations Here we introduce notations that will be used throughout unless specified otherwise. Fix an algebraic closure Q of Q. For a subfield 𝐹 ⊂ Q, we let 𝐺 𝐹 = Gal(Q/𝐹). Let Σ denote a set of places of 𝐹. Let 𝐹Σ ⊂ Q denote the maximal extension of 𝐹 unramified outside Σ and 𝐺 𝐹,Σ = Gal(𝐹Σ /𝐹). Let A𝐹 denote the adeles over 𝐹. For a finite subset 𝑆 of places in 𝐹, let A𝐹(𝑆) denote the adeles outside 𝑆 amd A𝐹,𝑆 denote the 𝑆-part of the adeles. For a Z-module 𝑀, let 𝑀Q := 𝑀 ⊗Z Q. 4 Chapter 2 HEEGNER MAIN CONJECTURE 2.1 Setup This section introduces the objects and hypotheses that will be used in the formulation of the Heegner Main Conjecture. We closely follow §2.1 of (Burungale and Tian, 2020). Let 𝑝 be a prime. Fix two embeddings 𝜄∞ : Q ↩→ C and 𝜄 𝑝 : Q ↩→ C 𝑝 . Let 𝑣 𝑝 be the 𝑝-adic valuation induced via 𝜄 𝑝 so that 𝑣 𝑝 ( 𝑝) = 1. Let 𝐾 be an imaginary quadratic field and O𝐾 its ring of integers. We consider 𝐾 to be a subfield of C through the embedding 𝜄∞ . Let 𝑐 be the complex conjugation on C that induces the unique non-trivial element 𝜏 of Gal(𝐾/Q) via 𝜄∞ . We assume that 𝑝 splits in 𝐾. Let 𝔭 denote the prime above 𝑝 in 𝐾 that was determined via the embedding 𝜄 𝑝 . For a positive integer 𝑚, let 𝐻𝑚 denote the ring class field of 𝐾 with conductor 𝑚 and O𝑚 = Z + 𝑚O𝐾 the corresponding order. Let 𝐻 denote the Hilbert class field. Let 𝐾∞ be the unique Z2𝑝 -extension of 𝐾 and let Γ𝐾 = Gal(𝐾∞ /𝐾). Note that 𝜏 − denote the anticyclotomic Z -extension of 𝐾 on which 𝜏 acts acts on Γ𝐾 . Let 𝐾∞ 𝑝 − − − − with by inversion. Denote Gal(𝐾∞ /𝐾) by Γ𝐾 . Let 𝐾𝑛 be the subextension of 𝐾∞ degree 𝑝 𝑛 over 𝐾. Fix a splitting1 Γ𝐾 = Γ−𝐾 × Γ+𝐾 . Self-dual pair Let 𝑔 ∈ 𝑆2 (Γ0 (𝑁𝑔 ), 𝜔) be a weight two elliptic newform with Neben-type 𝜔 and let 𝐸 𝑔 be the corresponding Hecke field. Let 𝐴 = 𝐴𝑔 be an abelian variety over Q associated to 𝑔 by Eichler-Shimura such that Ö 𝐿(𝑠, 𝐴/Q ) = 𝐿 (𝑠, 𝑔 𝜎 ) 𝜎:𝐸 𝑔 ↩→C Recall that the endomorphism ring for 𝐴 contains an order O𝑔 in the Hecke field and O𝑔 is generated over Z by the Hecke eigenvalues of 𝑔. Let 𝜒 be an arithmetic Hecke character over 𝐾. Let 𝐸 𝑔,𝜒 ⊂ C be the subfield  × generated over Q by the Hecke eigenvalues of 𝑔 and the image of 𝜒 on A𝐾(∞) . 1 When 𝑝 ≠ 2, there exists a canonical splitting. 5 Let ℘ be a prime above 𝑝 in 𝐸 𝑔,𝜒 induced via the embedding 𝜄 𝑝 . Let 𝐿 be the completion of 𝐸 𝑔,𝜒 at ℘ and let O be the integer ring of 𝐿. Let O𝑔,𝜒 ⊂ 𝐸 𝑔,𝜒 be generated over O𝑔 by the values of 𝜒. Let ℘0 be the prime of O𝑔,𝜒 given by ℘ ∩ O𝑔,𝜒 . Let O0 be the localization of O𝑔,𝜒 at ℘0 . We can see that O0 is a subring of O. Let 𝐿 0 be the fraction field of O0 . We define the integral and the rational Iwasawa algebras: Λ◦ = OJΓ−𝐾 K, Λ = Λ◦ ⊗O 𝐿. We define an involution 𝜄 on Λ by 𝜄(𝛾) = 𝛾 −1 for 𝛾 ∈ Γ−𝐾 . For any Λ-module 𝑀, we let 𝑀 𝜄 denote the module twisted by 𝜄. Now suppose that 𝜒 is of finite order and 𝜔 · 𝜒| A× = 1 (2.1.1) The Rankin-Selberg convolution 𝐿 (𝑠, 𝑔 × 𝜒) corresponding to the pair (𝑔, 𝜒) is self-dual with functional equation around 𝑠 = 1. Now we are ready to introduce an abelian variety associated to the pair (𝑔, 𝜒) using a variant of Eichler-Shimura construction. Definition 1. Let (𝑔, 𝜒) be as above. Let 𝐵 be the Serre tensor 𝐴 ⊗ 𝜒, an abelian variety defined over 𝐾 such that Ö 𝐿 (𝑠, 𝐵/𝐾 ) = 𝐿 (𝑠, 𝑔 𝜎 × 𝜒 𝜎 ) 𝜎:𝐸 𝑔 ↩→C Given 2.1.1, the dual abelian variety 𝐵∨ is isogeneous to 𝐵. Fix such an isogeny from now on. Selmer groups −. We introduce Selmer groups associated to 𝐵 over the anticyclotomic tower 𝐾∞ Define S(𝐵) := lim lim Sel 𝑝 𝑚 (𝐵/𝐾𝑛− ) ⊗O0 𝐿 ←−− ←−− 𝑛 𝑚 X(𝐵) := (lim lim Sel 𝑝 𝑚 (𝐵/𝐾𝑛− )) ∨ ⊗O0 𝐿 −−→ −−→ 𝑛 𝑚 Here (·) ∨ denotes taking the Pontryagin dual. Analogously, one can define S(𝐵∨ ) and X(𝐵∨ ) for the dual abelian variety 𝐵∨ . These Selmer groups carry a natural Λ-module structure through Galois action. 6 2.2 Heegner Points The Heegner Main Conjecture concerns Iwasawa theory of Heegner points on the abelian variety 𝐵 along the anticyclotomic tower. For the pair (𝑔, 𝜒), we always suppose the following generalized Heegner hypothesis: • 𝜔 · 𝜒| A× = 1. • 𝜖 (𝑔, 𝜒) = −1. Here 𝜖 (𝑔, 𝜒) is the global root number of the Rankin-Selberg convolution. With these hypothesis, Disegni has constructed in (Disegni, 2017) a norm compatible sequence of Heegner points
𝑃( 𝑓𝛼,𝑛 , 𝜒) 𝑛 ∈ lim 𝐵(𝐾𝑛− )Q ←−− 𝑛 At 𝑛 = 0, we have 𝑃𝑔,𝜒 denoting the Heegner point over 𝐾. The construction of the sequence of Heegner points is valid for 𝑝 = 2, 3. Details can be found in §2.2 of (Burungale and Tian, 2020).
Let 𝑇𝑝 𝐵 denote the Tate module. The image of 𝑃( 𝑓𝛼,𝑛 , 𝜒) 𝑛 under the Kummer map 𝐵(𝐾𝑛− ) ⊗O0 𝐿 → 𝐻 1 (𝐾𝑛− , 𝑇𝑝 𝐵) ⊗O0 𝐿 gives the Heegner cohomology class 𝜅 ∈ S(𝐵). 2.3 Formulation Now we are ready to introduce the Heegner Main Conjecture formulated for the CM modular forms. We want to note that such a conjecture is primarily due to Perrin-Riou. Recall that a Hecke eigenform 𝑔 is said to be CM if there exists an imaginary quadratic field 𝐾 and an arithmetic Hecke character 𝜓 over 𝐾 such that 𝑔 is the theta series 𝜃 (𝜓) associated to 𝜓. In this case, we say that 𝑔 has CM by 𝐾. Conjecture 2 (Heegner Main Conjecture (CM-case)). Let (𝑔, 𝜒) be a self-dual pair of a weight two CM modular form (level 𝑁𝑔 ) with CM by 𝐾 and a finite order Hecke character over an imaginary quadratic field 𝐾 with root number −1. Let 𝐵 the corresponding abelian variety over 𝐾. Suppose 𝑝 is an ordinary prime for the pair and 𝑝 - 𝑁𝑔 · cond𝑟 ( 𝜒). Then, 7 (1) 𝜅 ∈ S(𝐵) is Λ-non-torsion and rankΛ S(𝐵) = rankΛ X(𝐵) = 1. (2) charΛ S(𝐵)/(𝜅) · charΛ (S(𝐵)/(𝜅)) 𝜄 = charΛ X(𝐵)tor Here (·)𝑡𝑜𝑟 denotes the Λ-torsion submodule. Originally, the conjecture was formulated only for odd primes, but we have verified the objects involved, including the Heegner cohomology class 𝜅 and the anticyclotomic Selmer groups, are welldefined even when 𝑝 = 2. Although the even prime case is often avoided in Iwasawa-theoretic arguments, it can be included in this formulation as we are only considering a rational statement, where both the Iwasawa algebra Λ and the Selmer groups are tensored with an extension of Q 𝑝 . 2.4 Main Result In (Burungale and Tian, 2020), Conjecture 2 was proven under the assumption 𝑝 > 3. We generalize their approach and obtain the following result that includes 𝑝 = 2, 3. Theorem 2. Let 𝜓 be the Hecke character over 𝐾 with infinity type (1, 0) corresponding to the CM modular form 𝑔 and suppose 𝜓 𝜒 corresponds to a CM elliptic curve and has root number −1. Then, the CM-case of the Heegner Main Conjecture (Conjecture 2) holds. Compared to the formulation of Conjecture 2, the only additional assumption in Theorem 2 is the requirement of “𝜓 𝜒 corresponds to a CM elliptic curve.” The reason behind this assumption will be explained at the end of section 3.4. Moreover, as the reader will see in Chapter 4, the version of Heegner Main Conjecture proven by Theorem 2 suffices for the purpose of our 𝑝-converse theorem. 8 Chapter 3 HEEGNER MAIN CONJECTURE: VERIFICATION AND MODIFICATION In this section, we explain the main steps taken to generalize Burungale-Tian’s approach towards the Heegner Main Conjecture (HMC) and prove Theorem 2. We focus on resolving the key obstacles encountered in the case of 𝑝 = 2, 3 and include other relevant arguments for logical completeness. There are three natural pieces in the generalization. The first piece is the statement about the Heegner cohomological class 𝜅 being Λ-non-torsion. The second piece is concerned with the rank of Selmer groups S(𝐵) and X(𝐵) as Λ-modules. We will henceforth refer to it as the “rank statement.” The third piece is statement (2) in the HMC, namely, the rational equality of characteristic ideals concerning Selmer groups. We will refer to the third piece as the “equality of characteristic ideals.” 3.1 Preliminaries Before explaining the three pieces in detail, we first introduce some objects that will be useful later. Selmer Groups We introduce Selmer groups arising in the process of proving the HMC. We begin with the general definitions and move on to properties that will be used in the rank statement. Let 𝜓, 𝑔, 𝜒 be as in §2.3. Let 𝐵 be the abelian variety over 𝐾 associated to the pair (𝑔, 𝜒) in Definition 1. Since we are under the assumption that 𝑝 - cond( 𝜒), the abelian variety 𝐵 has ordinary reduction at primes above 𝑝. Let 𝑉𝑔 denote the 𝑝-adic Galois representation 𝜌𝑔 : 𝐺 Q → 𝐺 𝐿 2 (𝐿) associated to 𝑔. Let 𝐿( 𝜒) denote the one dimensional 𝐺 𝐾 -representation over 𝐿 associated with 𝜒. We can then define a 𝑝-adic Galois representation of 𝐺 𝐾 that is ordinary at 𝑝: 𝑉 := 𝑉𝑔 | 𝐺 𝐾 ⊗ 𝐿 𝐿( 𝜒) 9 For any Hecke character, we use a superscript ∗ to indicate that it has been precomposed with the non-trivial element 𝜏 of Gal(𝐾/Q). Define 𝜆 = 𝜓 𝜒. There exists an isomorphism of 𝐿 [𝐺 𝐾 ] modules: 𝑉  𝐿(𝜓 𝜒) ⊕ 𝐿(𝜓 ∗ 𝜒) = 𝐿 (𝜆) ⊕ 𝐿(𝜓 ∗ 𝜒) Choose a 𝐺 𝐾 -stable lattice 𝑇 ⊂ 𝑉 such that 𝑇  O (𝜓 𝜒) ⊕ O (𝜓 ∗ 𝜒) = O (𝜆) ⊕ O (𝜓 ∗ 𝜒) Define 𝑊 = 𝑉/𝑇, 𝑊 (𝜆) = 𝐿 (𝜆)/O (𝜆), 𝑊 (𝜓 ∗ 𝜒) = 𝐿 (𝜓 ∗ 𝜒)/O (𝜓 ∗ 𝜒). 𝐺 𝐿 2/Q Selmer Groups We want to define the Bloch-Kato Selmer groups on the 𝐺 𝐿 2/Q -side from the local Bloch-Kato Selmer groups at places 𝑣 of 𝐾. Let ? represent either 𝑉, 𝑇, or 𝑊. For 𝑣 - 𝑝, the local Bloch-Kato Selmer groups are defined as:   1 1 1 𝐻 𝑓 (𝐾𝑣 , 𝑉) := ker 𝐻 (𝐾𝑣 , 𝑉) → 𝐻 (𝐼𝑣 , 𝑉) For 𝑣 | 𝑝, the local Bloch-Kato Selmer groups are defined as:   𝐻 1𝑓 (𝐾𝑣 , 𝑉) := ker 𝐻 1 (𝐾𝑣 , 𝑉) → 𝐻 1 (𝐾𝑣 , 𝑉 − ) Here 𝑉 − denotes the maximal unramified quotient of 𝑉 | 𝐺 𝐾𝑣 . Define 𝐻 1𝑓 (𝐾𝑣 , 𝑇) and 𝐻 1𝑓 (𝐾𝑣 , 𝑊) as the preimage and image resp. of 𝐻 1𝑓 (𝐾𝑣 , 𝑉) under the natural maps 𝐻 1 (𝐾𝑣 , 𝑇) → 𝐻 1 (𝐾𝑣 , 𝑉) → 𝐻 1 (𝐾𝑣 , 𝑊). Using these local conditions, we define the Bloch-Kato Selmer group ) ( Ö 𝐻 1 (𝐾𝑣 , ?) 𝐻 1𝑓 (𝐾, ?) = ker 𝐻 1 (𝐾, ?) → 1 𝑣 𝐻 𝑓 (𝐾𝑣 , ?) We also want to define the Λ-adic Selmer groups using a similar approach. For 𝑣 - 𝑝, the Λ-adic local Bloch-Kato Selmer groups are defined as:   𝐻 1𝑓 (𝐾𝑣 , 𝑉 ⊗ 𝐿 Λ) := ker 𝐻 1 (𝐾𝑣 , 𝑉 ⊗ 𝐿 Λ) → 𝐻 1 (𝐼𝑣 , 𝑉 ⊗ 𝐿 Λ) 10 For 𝑣 | 𝑝, the local Bloch-Kato Selmer groups are defined as:   𝐻 1𝑓 (𝐾𝑣 , 𝑉 ⊗ 𝐿 Λ) := ker 𝐻 1 (𝐾𝑣 , 𝑉 ⊗ 𝐿 Λ) → 𝐻 1 (𝐾𝑣 , 𝑉 − ⊗ 𝐿 Λ) Here 𝑉 − denotes the maximal unramified quotient of 𝑉 | 𝐺 𝐾𝑣 . One can analogously define 𝐻 1𝑓 (𝐾𝑣 , ? ⊗ 𝐿 Λ). Definition 2. The Λ-adic Selmer group 𝑆(𝑉 ⊗ 𝐿 Λ) corresponding to the pair (𝑔, 𝜒) is given by ( ) Ö


𝑆(𝑉 ⊗ 𝐿 Λ) := ker 𝐻 1 𝐾, 𝑉 ⊗ 𝐿 0 Λ → 𝐻 1 𝐾𝑣 , 𝑉 ⊗ 𝐿 0 Λ /𝐻 1𝑓 𝐾𝑣 , 𝑉 ⊗ 𝐿 0 Λ . 𝑣 The discrete Selmer group Sel (𝐾, 𝑊 ⊗O Λ◦ ) is given by ) ( Ö 𝐻 1 (𝐾𝑣 , 𝑊 ⊗O Λ◦ ) /𝐻 1𝑓 (𝐾𝑣 , 𝑊 ⊗O Λ◦ ) . Sel (𝐾, 𝑊 ⊗O Λ◦ ) := ker 𝐻 1 (𝐾, 𝑊 ⊗O Λ◦ ) → 𝑣 Its Pontryagin dual is given by 𝑋 (𝑊) = homO (Sel (𝐾, 𝑊 ⊗O Λ◦ ) , 𝐿/O) Analogous objects can be defined for the dual Galois representations. Remark 1. The Λ-modules 𝑆(𝑉 ⊗ 𝐿 Λ) and 𝑋 (𝑊) 𝐿 := 𝑋 (𝑊) ⊗O 𝐿 the same as the Λ-modules S(𝐵) and X(𝐵). 𝐺 𝐿 1/𝐾 Selmer Groups We want to define the Bloch-Kato Selmer groups on the 𝐺 𝐿 1/𝐾 side from the local Bloch-Kato Selmer groups corresponding to Hecke characters. For a finite place 𝑣 of 𝐾, the local Bloch-Kato Selmer groups are   1 (𝐾 , 𝐿(𝜆)),  𝐻𝑢𝑟 𝑣     1 𝐻 𝑓 (𝐾𝑣 , 𝐿(𝜆)) = 0,      𝐻 1 (𝐾𝑣 , 𝐿(𝜆)),  𝑣-𝑝 𝑣 | 𝔭∗ 𝑣|𝔭 and   1 (𝐾 , 𝐿(𝜓 ∗ 𝜒)),  𝐻𝑢𝑟 𝑣     1 ∗ 1 𝐻 𝑓 (𝐾𝑣 , 𝐿(𝜓 𝜒)) = 𝐻 (𝐾𝑣 , 𝐿(𝜓 ∗ 𝜒)),      0,  𝑣-𝑝 𝑣 | 𝔭∗ 𝑣|𝔭 11 The unramified local Galoid cohomology is given by   1 1 1 𝐻𝑢𝑟 (𝐾𝑣 , ·) = ker 𝐻𝑢𝑟 (𝐾𝑣 , ·) → 𝐻𝑢𝑟 (𝐼𝑣 , ·) Let Σ be the set of primes of 𝐾 lying above 𝑝. Using these local conditions, we define the Selmer groups Ö © ª SelΣ (𝐾, 𝐿(·)) = ker ­𝐻 1 (𝐾, 𝐿(·)) → (3.1.1) 𝐻 1 (𝐾𝑣 , 𝐿(·))/𝐻 1𝑓 (𝐾𝑣 , 𝐿(·)) ® 𝑣-𝑝 « ¬ Ö © ª 𝐻 1𝑓 (𝐾, 𝐿(·)) = Sel(𝐾, 𝐿(·)) = ker ­SelΣ (𝐾, 𝐿(·)) → 𝐻 1 (𝐾𝑣 , 𝐿(·))/𝐻 1𝑓 (𝐾𝑣 , 𝐿(·)) ® 𝑣| 𝑝 « ¬ (3.1.2) Ö © ª 𝐻 1 (𝐾𝑣 , 𝐿(·)) ® SelΣ (𝐾, 𝐿(·)) = ker ­Sel(𝐾, 𝐿(·)) → 𝑣| 𝑝 « ¬ (3.1.3) We also want to introduce Λ-adic local conditions as in the 𝐺 𝐿 2/Q case and subsequently define the Λ-adic Selmer groups associated to the Hecke characters 𝜆 and 𝜓 ∗ 𝜒. Definition 3. Let · denote 𝜆 or 𝜓 ∗ 𝜒. The Λ-adic Selmer groups 𝑆(𝐿(·) ⊗ 𝐿 Λ) is given by ( ) Ö 𝑆(𝐿 (·) ⊗ 𝐿 Λ) := ker 𝐻 1 (𝐾, 𝐿(·) ⊗ 𝐿 Λ) → 𝐻 1 (𝐾𝑣 , 𝐿(·) ⊗ 𝐿 Λ) /𝐻 1𝑓 (𝐾𝑣 , 𝐿(·) ⊗ 𝐿 Λ) . 𝑣 We analogously define the discrete Selmer groups and their Pontryagin duals Sel (𝐾, 𝑊 (·) ⊗O Λ◦ ) , 𝑋 (·). Relating 𝐺 𝐿 2/Q and 𝐺 𝐿 1/𝐾 Selmer Groups Now we introduce some properties relating the Iwasawa-theoretic Selmer groups we have introduced above. Lemma 1 (Lemma 3.7 in (Burungale and Tian, 2020)). Let 𝜓 (resp. 𝜒) be a Hecke character over an imaginary quadratic field 𝐾 with infinity type (1, 0) (resp. finite order, unramified at 𝑝) and 𝜆 = 𝜓 𝜒. Let 𝑔 be the CM modular form associated to 𝜓 and 𝑉 the p-adic Galois representation of 𝐺 𝐾 corresponding to the pair (𝑔, 𝜒). Then we have an isomorphism 𝑆(𝑉 ⊗ 𝐿 Λ)  𝑆(𝐿 (𝜆) ⊗ 𝐿 Λ) ⊕ 𝑆(𝐿 (𝜓 ∗ 𝜒) ⊗ 𝐿 Λ) 12 of Λ-modules and isomorphisms Sel(𝐾, 𝑊 ⊗O Λ◦ )  Sel(𝐾, 𝑊 (𝜆) ⊗O Λ◦ ) ⊕ Sel(𝐾, 𝑊 (𝜓 ∗ 𝜒) ⊗O Λ◦ ) and 𝑋 (𝑊)  𝑋 (𝜆) ⊕ 𝑋 (𝜓 ∗ 𝜒) of Λ◦ -modules. Proof. The decomposition of Selmer groups follows directly from the definition and thus the proof in (Burungale and Tian, 2020) holds for 𝑝 = 2, 3.  𝑝-adic 𝐿-functions In this part, we introduce 𝑝-adic 𝐿-functions arising in the process of proving the HMC. It is worth noting that although 𝑝-adic 𝐿-functions do not appear in the formulation of the HMC, they seem to be inevitable in the proof. Just like the previous parts on Selmer groups, we will introduce underlying 𝑝-adic 𝐿-functions on both the 𝐺 𝐿 2/Q and 𝐺 𝐿 1/𝐾 sides. We will also relate them through decomposition. We assumed that the root number of the pair (𝑔, 𝜒) is −1. Thus, without loss of generality, we can suppose the root numbers: 1 𝜖 ( , 𝜓 ∗ 𝜒) = +1, 2 1 𝜖 ( , 𝜆) = −1. 2 (3.1.4) 𝐺 𝐿 2/Q -side 𝑝-adic 𝐿-function Definition 4 (Definition 3.12 in (Burungale and Tian, 2020)). Recall that 𝑔 is 𝑝-ordinary with 𝑈 𝑝 -eigenvalue 𝛼. Let 𝐿 𝑝 (𝑔 × 𝜒) ∈ OJΓ𝐾 K ⊗O 𝐿 be the two variable Rankin-Selberg 𝑝-adic 𝐿-function characterized by the interpolation property (for all sufficiently 𝑝-ramified finite order characters 𝜒0 : Γ𝐾 → C×𝑝 )
𝜒ˆ0 𝐿 𝑝 (𝑔 × 𝜒) = 𝑒 𝑝 𝑔 × ( 𝜒 𝜒0) −1 𝛼 𝑣(cond( 𝜒 0 ))
· 𝐿 ( 𝑝) (1, 𝑔 × ( 𝜒 𝜒0) −1 ) Ω𝑔 Here • Ω𝑔 := 𝐿 (1, 𝑎𝑑 (𝑔)), where 𝑎𝑑 (𝑔) denotes the adjoint. 13    
• 𝑒 𝑝 𝑔 × ( 𝜒 𝜒0) −1 = 𝜀 0, 𝜒𝔭 𝜒𝔭0 · 𝜀 0, 𝜒𝔭∗ 𝜒𝔭0 ∗ , where 𝜀 (0, ·) is the local 𝜀-factor. • 𝐿 ( 𝑝) (·) is the 𝐿-function with Euler factors at primes above 𝑝 removed. 𝐺 𝐿 1/𝐾 -side 𝑝-adic 𝐿-function Let 𝑊 denote a finite flat extension of the Witt ring 𝑊 (F), where F is an algebraic closure of F 𝑝 . Let 𝜋 : Γ♯𝐾  Γ𝐾 be a finite cover arising from a finite extension of 𝐾∞ contained in 𝐾 𝑎𝑏 (fixing a tame level prime to 𝑝). Let Σ be a 𝑝-adic CM type of 𝐾 corresponding to the embedding 𝜄∞ defined above. This is not to be confused with the set of places used in the definition of Selmer groups1. There exists an element 𝐿 Σ ∈ 𝑊JΓ♯𝐾 K uniquely characterized by an interpolation property. The domain of interpolation consists of arithmetic Hecke characters 𝜆0 : Γ♯𝐾 → C×𝑝 with infinity type (𝑘 + 𝜅, −𝜅) for 𝑘, 𝜅 ∈ Z such that • 𝑘 ≥ 1 and 𝜅 ≥ 0, or • 𝑘 ≤ 1 and 𝑘 + 𝜅 > 0. Here is the interpolation property. There exist 𝑝-adic CM periods ΩΣ,𝑝 ∈ C×𝑝 and complex CM periods ΩΣ,∞ ∈ C× such that for any character 𝜆0 in the interpolation domain, we have   𝐿 ( 𝑝) (0, (𝜆0) −1 ) 𝜋 𝜅 Γ(𝑘 + 𝜅) 𝐿 Σ (𝜆0) 0 −1 · [O𝐾× : Z× ] = 𝑒 𝑝 (𝜆 ) · · 𝑘+2𝜅 𝑘+2𝜅 (𝐼𝑚𝜃) ΩΣ,𝑝 ΩΣ,∞ Here Γ represents the usual Γ-function and 𝜃 ∈ 𝐾. For our purpose, we choose a certain tame level and consider the restriction of 𝐿 Σ to certain open subsets of Γ♯𝐾 . Definition 5 (Definition 3.13 in (Burungale and Tian, 2020)). Let 𝜆 0 be a 𝑝-adic Hecke character over 𝐾 with values in 𝐿. Let 𝐿 Σ (𝜆 0 ) ∈ 𝑊JΓ𝐾 K ⊗ 𝐿 be the Katz 𝑝-adic 𝐿-function given by 𝜒ˆ0 (𝐿 Σ (𝜆 0 )) := 𝐿 Σ (𝜆 0 𝜒0) The interpolation property of 𝐿 Σ (𝜆 0 ) follows from that of 𝐿 Σ . 1 Future use should be clear from context 14 Relating 𝐺 𝐿 2/Q and 𝐺 𝐿 1/𝐾 𝑝-adic 𝐿-function Lemma 2 (Lemma 3.14 in (Burungale and Tian, 2020)). Let the notation be as above. Let Σ∗ denote the conjugation CM type of Σ. There exists an equality up to × a constant in Q 𝑝 : 𝐿 𝑝 (𝑔 × 𝜒)  𝐿 Σ (𝜆) · 𝐿 Σ∗ (𝜓 ∗ 𝜒) 3.2 Non-torsion Heegner Cohomological Class This part of the verification is done in collaboration with Jacob Ressler. We have verified that the approach used by Burungale-Tian for this piece goes through even when 𝑝 = 2, 3. 3.3 Rank Statement This part of the modifications is done in collaboration with Jacob Ressler. Combining Lemma 1 and Remark 1, the Selmer groups S(𝐵) and X(𝐵) associated to an abelian variety 𝐵 can be decomposed into direct sums of Bloch-Kato Selmer groups associated to Hecke characters. The rank statement was then proven by examining the rank of the individual Selmer groups for the characters using Theorem 2.14 in (Arnold, 2007), which assume 𝑝 > 3. Thus, our main modification is to adapt the proof of Theorem 2.14 to 𝑝 = 2, 3. The original proof of Arnold relies on the non-triviality of an Euler system of elliptic units. The non-triviality was achieved by demonstrating that the Katz’s 𝑝-adic 𝐿-function associated to this Euler system is non-trivial. The restriction 𝑝 > 3 was originally imposed in the construction of the elliptic units, in order to avoid non-integral coefficient in front of the associated 𝑝-adic 𝐿-function. We fix this issue by adopting the canonical 12th-root of certain elliptic units introduced in (Oukhaba and Viguié, 2013). The 𝑝-adic 𝐿-function constructed from OukhabaViguié’s elliptic units is compatible with the argument in Arnold and is shown to be non-vanishing under desirable circumstances. Upon showing the non-triviality of Euler system, the rest of the argument for proving the statememts in Theorem 2.14 of (Arnold, 2007) follows from properties of Euler system in (Rubin, 2000) and does not assume 𝑝 > 3. To be precise, we have achieved the following: Theorem 3 (Generalization of Theorem 2.14 in (Arnold, 2007)). Let · denote a Hecke character satisfying conditions in §1.1 of (Arnold, 2007). − , ·) and 𝑍 (𝐾 − , ·) are torsion Λ◦ -modules and there is a divisibility of charac𝑋Σ (𝐾∞ ∞ 15 − , ·) | char 𝑍 (𝐾 − , ·). Moreover, SelΣ (𝐾 − , 𝑇 ∗ (·)) is torsionteristic ideals char 𝑋Σ (𝐾∞ ∞ ∞ ◦ free of rank 1 over Λ . − , ·) has rank 1 over Λ◦ and If the sign in the functional equation of · is −1, then 𝑋 (𝐾∞ − , 𝑇 ∗ (·)) = SelΣ (𝐾 − , 𝑇 ∗ (·)). Otherwise, 𝑋 (𝐾 − , ·) is a torsion Λ◦ -module. Sel(𝐾∞ ∞ ∞ Remark 2 (Arnold’s Notation). In the theorem above, we have kept some of the notations for Selmer groups in (Arnold, 2007). Here 𝑇 ∗ (·) is the Tate dual2 of 𝑇 (·) = O (·). The Iwasawa-theoretic Selmer group − , −) in defined to be a projective limit of the Bloch-Kato Selmer groups in SelΣ (𝐾∞ equation 3.1.1, where the limit is taken with respect to corestriction maps between − . Analogous definitions give Sel(𝐾 − , −) finite extensions of 𝐾 contained in 𝐾∞ ∞ − − and SelΣ (𝐾∞ , −). The divisible Selmer group 𝑋Σ (𝐾∞ , ·) is the Pontryagin dual of − , 𝑊 (·)). Analogous definition gives 𝑋 Σ (𝐾 − , ·) and 𝑋 (𝐾 − , ·). SelΣ (𝐾∞ ∞ ∞ These formulations of Iwasawa-theoretic Selmer groups are equivalent to what we have in Definition 3. − ) is defined as: The Λ◦ -module 𝑍 (𝐾∞ − − − 𝑍 (𝐾∞ , ·) := SelΣ (𝐾∞ , 𝑇 ∗ (·))/𝐶 (𝐾∞ , ·) − , ·) is the Λ◦ -submodule of SelΣ (𝐾 − , 𝑇 ∗ (·)) generated by {𝑐 (𝐾 − , ·) | Here 𝐶 (𝐾∞ 𝔞 ∞ ∞ (𝔞, 𝔣𝑝) = 1}. Note that 𝔣 is an ideal of 𝐾 related to the tame level we fixed in the definition of 𝐿 Σ and 𝔞 is an integral ideal of 𝐾. Most importantly, 𝑐 𝔞 denotes an Euler system constructed from our modified elliptic units. We will only use Arnold’s notations when specified. 3.4 Equality of Characteristic Ideals In Burungale-Tian, the proof for the equality of characteristic ideals was split into two sides: the 𝐺 𝐿 2/Q -side and the 𝐺 𝐿 1/𝐾 -side. We examine each side separately for parts that need modification. On the 𝐺 𝐿 2/Q -side, we desire the following: Theorem 4. There is an equality of ideals in Λ:  0  𝐿 𝑝 (𝑔 × 𝜒) = charΛ S(𝐵)/(𝜅) · charΛ (S(𝐵)/(𝜅)) 𝜄 · 𝑅(𝑔 × 𝜒) 2 We have flipped the notation for 𝑇 and 𝑇 ∗ compared to Arnold. (3.4.1) 16 Here 𝑅(𝑔 × 𝜒) is the anticyclotomic regulator associated to some Λ-adic height 0 pairing on 𝑆(𝑉 ⊗ 𝐿) and 𝐿 𝑝 (𝑔 × 𝜒) denotes the cyclotomic derivative of the twovariable Rankin-Selberg 𝑝-adic 𝐿-function 𝐿 𝑝 (𝑔 × 𝜒). Proof. The proof by Burungale-Tian relies on a Λ-adic Gross-Zagier formula established in Theorem C of (Disegni, 2017): 0 h𝜅, 𝜅 𝜄 i  𝐿 𝑝 (𝑔 × 𝜒) × Here, “” denotes equality up to some constant in Q 𝑝 . In particular, this equality holds as ideals in Λ arising from both sides. From our current understanding, the pairing and formula includes the case of 𝑝 = 2, 3 and there are no obstacles for adopting the same approach.  On the 𝐺 𝐿 1/𝐾 -side, we want the following identities: Theorem 5. There are equalities of ideals in Λ: (𝐿 Σ−∗ (𝜓 ∗ 𝜒)) = charΛ 𝑋 (𝜓 ∗ 𝜒) (𝐿 0Σ (𝜆)) = charΛ 𝑋 (𝜆)𝑡𝑜𝑟 · 𝑅(𝜆) (3.4.2) (3.4.3) Combining the two we get:
𝐿 0Σ (𝜆) · 𝐿 Σ−∗ (𝜓 ∗ 𝜒) = charΛ 𝑋 (𝜆)𝑡𝑜𝑟 · charΛ 𝑋 (𝜓 ∗ 𝜒) · 𝑅(𝜆) (3.4.4) Recall, 𝐿 Σ (𝜆) and 𝐿 Σ∗ (𝜓 ∗ 𝜒)) are two variable Katz’s 𝑝-adic 𝐿-function defined in §3.1. The superscript 0 denotes taking cyclotomic derivative and the superscript − denotes the projection to the anticyclotomic component by setting the cyclotomic variable to 0. The anticyclotomic regulator 𝑅(𝜆) is associated to height pairing on anticyclotomic Bloch-Kato Selmer groups. In Burungale-Tian, the two equalities were proven using anticyclotomic results in Theorem 2.14, Theorem 3.9, and Theorem 4.17 of (Arnold, 2007). Modifications of Arnold’s theorems and their proofs are needed as they assume 𝑝 > 3. The case of 𝜓 ∗ 𝜒 We will first discuss the equality of ideals involving 𝜓 ∗ 𝜒, whose root number satisfies 1 𝜖 ( , 𝜓 ∗ 𝜒) = +1. 2 17 The original proof of equation 3.4.2 by Arnold relies on Rubin’s Iwasawa Main Conjecture (IMC) established in (Rubin, 1991), which is an integral statement and assumes 𝑝 > 3 in an essential manner. Instead of trying to directly generalize Rubin’s result, the key modification in this paper is to use a rational IMC, which includes the case of 𝑝 = 2, 3. This rational IMC is a corollary of the main result by Johnson-Leung and Kings in (Johnson-Leung and Kings, 2011). Theorem 6 (Rational Iwasawa Main Conjecture). Suppose 𝑝 splits into two distinct primes in 𝐾, then there is an equality of ideals in Z 𝑝 JΓ𝐾 K ⊗Z 𝑝 Q 𝑝 :     char E ∞ /C ∞ ⊗Z 𝑝 Q 𝑝 = char 𝐴∞ ⊗Z 𝑝 Q 𝑝 and char(𝑋∞ ⊗Z 𝑝 Q 𝑝 ) = char(𝑈∞ /C ∞ ⊗Z 𝑝 Q 𝑝 ) Proof. This follows from the proof of Theorem 5.7 in (Johnson-Leung and Kings, 2011) and the relation to Rubin’s classical formulation of IMC given in §5.4 of (Johnson-Leung and Kings, 2011).  Here the notation for the Iwasawa modules are consistent with Rubin’s original notations in §4 of (Rubin, 1991). This result is rational in the sense that both the Iwasawa algebra and the modules are tensored with Q 𝑝 . In a sense, they are “locally inverted” at 𝑝. For the inclusion (𝐿 Σ−∗ (𝜓 ∗ 𝜒)) ⊃ charΛ 𝑋 (𝜓 ∗ 𝜒), we follow Arnold’s idea of proof but make use of the rational IMC instead of Rubin’s IMC. The rational IMC was first used to generalize a two variable main conjecture (Theorem 3.2 in (Arnold, 2007)) for the representation 𝑊 (𝜓 ∗ 𝜒). Arnold’s notations in Remark 2 are used in the following theorem. Theorem 7 (Generalization of Theorem 3.2 in Arnold, 2007). 𝑋 (𝐾∞ , 𝜓 ∗ 𝜒) is a torsion OJΓ𝐾 K-module. SelΣ (𝐾∞ , 𝑇 ∗ (𝜓 ∗ 𝜒)) has rank 1 over OJΓ𝐾 K, and there are equalities of ideals in OJΓ𝐾 K ⊗O 𝐿   char 𝑋 (𝐾∞ , 𝜓 ∗ 𝜒) ⊗O 𝐿 = char 𝐻 1 (𝐾∞,𝔭 , 𝑇 (𝜓 ∗ 𝜒))/loc𝔭 𝐶 (𝐾∞ , 𝜓 ∗ 𝜒) ⊗O 𝐿 and char 𝑋Σ (𝐾∞ , 𝜓 ∗ 𝜒) ⊗O 𝐿 = char 𝑍 (𝐾∞ , 𝜓 ∗ 𝜒) ⊗O 𝐿 18 Proof. Here we give a sketch of the proof to this theorem. Let 𝐿 ∞ be an abelian extension of 𝐾 which is finite over 𝐾∞ and splits both 𝑉 = 𝐿 (𝜓 ∗ 𝜒) and 𝑉 ∗ . By splitting over 𝑉 we mean that the character associated to the Galois representation is trivial when restricted to 𝐺 𝐿 ∞ . There is a decomposition Gal(𝐿 ∞ /𝐾)  Gal(𝐾∞ /𝐾) × Δ. When 𝑝 | [𝐿 ∞ : 𝐾∞ ], such a decomposition is not canonical. Similar to Rubin’s notation, for any subfield 𝐹 of 𝐿 ∞ , define 𝑀 Σ (𝐹), 𝑀 (𝐹), 𝑀Σ (𝐹) to be the maximal abelian 𝑝-extension of 𝐹 unramified outside of 𝑝, 𝔭, and everywhere respectively. When 𝐹 is finite over 𝐾, we let E (𝐹) denote the completion of the global units O𝐹× ⊗ O. If 𝐹/𝐾 is infinite, let E (𝐹) = lim E (𝐹 0). If 𝔞 ⊂ 𝐾 is an ←−− integral ideal prime to 𝔣𝑝, define U𝔞 (𝐹) ⊂ E (𝐹) to be the submodule generated by the elliptic unit Euler system. Define U (𝐹) to be the submodule generated by all such U𝔞 (𝐹) ⊂ E (𝐹). There is an O-module isomorphism   Σ 𝛼 : Hom Gal(𝑀 (𝐿 ∞ )/𝐿 ∞ ), Φ/O → SelΣ (𝐿 ∞ , 𝑊 ∗ ) The analogous statement is true for 𝑀 (𝐿 ∞ ) and 𝑀Σ (𝐿 ∞ ). There also exist isomorphisms 𝛽 : E (𝐿 ∞ ) ⊗ O → SelΣ (𝐿 ∞ , 𝑇) 𝛽 : U (𝐿 ∞ ) ⊗ O → 𝐶 (𝐿 ∞ ) By the assumption that 𝐿 ∞ splits 𝑉 ∗ , we can view 𝜓 ∗ 𝜒 as a character of Gal(𝐿 ∞ /𝐾)  Gal(𝐾∞ /𝐾) × Δ and write 𝜓 ∗ 𝜒 = 𝜅 ∗ 𝜈 ∗ . Here 𝜈 ∗ is a character of the finite group Δ. After descenting Selmer groups from 𝐿 ∞ to 𝐾∞ , we take Δ-invariant of the OJGal(𝐿 ∞ /𝐾)K-modules present in the isomorphisms 𝛼 and 𝛽. Applying Theorem 6 and twisting theorems in Ch.6. of (Rubin, 2000) gives the desired result.  This two variable main conjecture is then descended to the anticyclotomic tower to give the inclusion. The descent argument does not assume 𝑝 > 3. The reverse inclusion (𝐿 Σ−∗ (𝜓 ∗ 𝜒)) ⊂ charΛ 𝑋 (𝜓 ∗ 𝜒) follows from Theorem 3. The modification of Euler systems for 𝑝 = 2, 3 is needed here. 19 The case of 𝜆 Now we move on to the equality of ideals involving 𝜆 = 𝜓 𝜒, whose root number satisfies 1 𝜖 ( , 𝜆) = −1. 2 This equality is shown in Theorem 4.17 of (Arnold, 2007) under the assumption 𝑝 > 3. One essential component of Arnold’s proof is a 𝑝-adic height pairing on Selmer groups associated to Hecke characters. This height pairing, first introduced by Perrin-Riou in (Perrin-Riou, 1992), exists for Selmer groups associated to any suitablely nice 𝑝-adic Galois representation of a number field. The height pairing carries good properties such as having a bounded image, being Galois-equivariant and compatible with restrictions (corestrictions). Moreover, Rubin has proven a height formula when the pairing was evaluated on classes associated to the elliptic units. To summarize, we have the following: Theorem 8 (Theorem 4.7 in (Arnold, 2007)). Recall that for each finite extension 𝐹 of 𝐾, local Tate duality gives a pairing: h , i𝐹𝔭 : 𝐻 1 (𝐹𝔭 , 𝑇 ∗ ) × 𝐻 1 (𝐹𝔭 , 𝑇) → O − . Set h , i𝑛 := h , i𝐾𝑛,𝔭 For all 𝑛, there is a 𝑝-adic height pairing ℎ𝑛 : Sel(𝐾𝑛− , 𝑇 ∗ ) × Sel(𝐾𝑛− , 𝑇) → 𝐿 canonically determined by a chosen isomorphism3 𝜌 : Γ+𝐾 ≃ Z 𝑝 up to sign, satisfying: 1. (bounded image) There is an integer 𝑘, independent of 𝑛, such that the image of ℎ𝑛 lies in 𝑝 −𝑘 O. 2. (Galois-equivariance) For any 𝑎 ∈ Sel(𝐾𝑛− , 𝑇 ∗ ), 𝑏 ∈ Sel(𝐾𝑛− , 𝑇), and 𝜎 ∈ Gal(𝐾𝑛− /𝐾), we have ℎ𝑛 (𝑎 𝜎 , 𝑏 𝜎 ) = ℎ𝑛 (𝑎, 𝑏) − , 𝑇), we have 3. (compatibility) For 𝑎 𝑛 ∈ Sel(𝐾𝑛− , 𝑇 ∗ ) and 𝑏 𝑛+1 ∈ Sel(𝐾𝑛+1 ℎ𝑛 (𝑎 𝑛 , cor(𝑏 𝑛+1 )) = ℎ𝑛+1 (res(𝑎 𝑛 ), 𝑏 𝑛+1 ) 3 sending a chosen topological generator 𝛾 of Gal(𝐾 + /𝐾) to 1 ∈ Z 𝑝 ∞ 20 ˜ the 4. (height formula) If 𝑏 ∈ Sel(𝐾𝑛− , 𝑇), then (after extending scalars to 𝐿, completion of Q𝑢𝑟 𝑝 · 𝐿), ℎ𝑛 (𝑐 𝔞 (𝐾𝑛− ), 𝑏) = h𝛼𝑛 , loc𝔭 𝑏i𝑛 The construction of the pairing and the proof of height formula is essentially the same for 𝑝 = 2, 3. This height pairing was further generalized into an Iwasawatheoretic pairing and inherits many properties, including the height formula, from the original height pairing by definition. Definition 6 (Iwasawa-theoretic Height Pairing). Let Λ̃◦ := ÕJΓ−𝐾 K and Λ̃ := Λ̃◦ ⊗Õ ˜ where Õ the integer ring of 𝐿. ˜ We first define the Iwasawa-theoretic Tate pairing 𝐿,     1 − ∗ ˆ ˜ 1 − 𝜄ˆ ˜ h , i∞ : 𝐻 (𝐾∞,𝔭 , 𝑇 ) ⊗ 𝐿 ⊗Λ̃◦ 𝐻 (𝐾∞,𝔭 , 𝑇) ⊗ 𝐿 → Λ̃◦ by setting h𝑎 ∞ , 𝑏 ∞ i∞ = lim ←−− Õ h𝑎 𝑛𝜎 , 𝑏 𝑛 i𝑛 𝜎 −1 𝑛 𝜎∈Gal(𝐾 𝑛− /𝐾) We then define the Iwasawa-theoretic 𝑝-adic height pairing

− − ℎ∞ : Sel(𝐾∞ , 𝑇 ∗ ) ⊗ˆ 𝐿˜ ⊗Λ̃◦ Sel(𝐾∞ , 𝑇) 𝜄 ⊗ˆ 𝐿˜ → Λ̃◦ by setting ℎ∞ (𝑎 ∞ , 𝑏 ∞ ) = lim ←−− Õ ℎ𝑛 (𝑎 𝑛𝜎 , 𝑏 𝑛 )𝜎 −1 𝑛 𝜎∈Gal(𝐾 𝑛− /𝐾) − , 𝑇), there is a height Remark 3 (Iwasawa Height Formula). For every 𝑏 ∞ ∈ Sel(𝐾∞ formula − ℎ∞ (𝑐 𝔞 (𝐾∞ ), 𝑏 ∞ ) = h𝛼∞ , loc𝔭 𝑏 ∞ i∞ − , 𝑇 ∗) ⊗ ˆ 𝐿˜ defined in Lemma 4.6 of (Arnold, 2007). for 𝛼∞ ∈ 𝐻 1 (𝐾∞,𝔭 Another important piece in Arnold’s argument towards equation 3.4.3 is the duality result about characteristic ideals of the divisible Selmer groups 𝑋 (·) (see Proposition 4.1 and Proposition 4.2 of (Arnold, 2007)). The duality result does not assume 𝑝 > 3. The Iwasawa height formula, combined with the duality result and results derived from the rational IMC, gives the equality of the root number −1 case. To summarize, in the root number −1 case, apart from employing the rational IMC, no additional modifications are needed for 𝑝 = 2, 3. 21 We are now ready to combine results from the 𝐺 𝐿 2/Q -side and the 𝐺 𝐿 1/𝐾 -side, namely, equation 3.4.1 and 3.4.4, to prove (2) of Conjecture 2 under the assumptions in Theorem 2. The following strategy closely follows that discussed in BurungaleTian but uses our generalized results above. Proof of Equality of Characteristic Ideals. The LHS of equation 3.4.1 and 3.4.4 are equal due to a corollary of Lemma 2, which says: 0 𝐿 𝑝 (𝑔 × 𝜒) = 𝐿 0Σ (𝜆) · 𝐿 Σ−∗ (𝜓 ∗ 𝜒) × up to some constant in Q 𝑝 . Thus, combining equation 3.4.1 and 3.4.4 gives the following equality in Λ: charΛ S(𝐵)/(𝜅) · charΛ (S(𝐵)/(𝜅)) 𝜄 · 𝑅(𝑔 × 𝜒) = charΛ 𝑋 (𝜆)𝑡𝑜𝑟 · charΛ 𝑋 (𝜓 ∗ 𝜒) · 𝑅(𝜆) (3.4.5) From the decomposition of the Selmer group X(𝐵) given by Lemma 1 and Remark 1, we have charΛ X(𝐵)tor = charΛ 𝑋 (𝜆)𝑡𝑜𝑟 · charΛ 𝑋 (𝜓 ∗ 𝜒)𝑡𝑜𝑟 When proving the rank statement, 𝑋 (𝜓 ∗ 𝜒) was determined to be a torsion Λ-module. Thus, the equality becomes charΛ X(𝐵)tor = charΛ 𝑋 (𝜆)𝑡𝑜𝑟 · charΛ 𝑋 (𝜓 ∗ 𝜒) (3.4.6) Employing again the decomposition of Selmer groups and the rank statement, the definition of Λ-adic height pairing on Selmer groups gives the following relation between regulators: 𝑅(𝑔 × 𝜒) = 𝑅(𝜆) (3.4.7) With equation 3.4.5, 3.4.6, and 3.4.7, the equality of characteristic ideals will be complete if we can show that the regulator 𝑅(𝜆) does not vanish in the case of 𝑝 = 2, 3. When 𝑝 > 3, Rubin has demonstrated 𝑅(𝜆) ≠ 0 in the appendix of (Agboola and Howard, 2006), when 𝜆 is associated to a CM elliptic curve. Let 𝐹 be a finite extension of 𝐾 contained in 𝐾∞ . Rubin first defined the following 22 more generalized versions of the 𝑝-adic height pairing ℎ𝑛 : ℎ 𝐹 : Sel(𝐹, 𝑇) ⊗ Sel(𝐹, 𝑇 ∗ ) → Gal(𝐾∞ /𝐹) ⊗ Q 𝑝 + ℎ 𝐹,𝑐𝑦𝑐𝑙 : Sel(𝐹, 𝑇) ⊗ Sel(𝐹, 𝑇 ∗ ) → Gal(𝐹𝐾∞ /𝐹) ⊗ Q 𝑝 − ℎ 𝐹,𝑎𝑛𝑡𝑖 : Sel(𝐹, 𝑇) ⊗ Sel(𝐹, 𝑇 ∗ ) → Gal(𝐹𝐾∞ /𝐹) ⊗ Q 𝑝 ℎ 𝐹,𝔭 : Sel(𝐹, 𝑇) ⊗ Sel(𝐹, 𝑇 ∗ ) → Gal(𝐹 𝐿 ∞ /𝐹) ⊗ Q 𝑝 Here 𝐿 ∞ is the unique Z 𝑝 -extension of 𝐾 unramified outside 𝔭. The first piece in the proof of non-vanishing is Theorem A.1 in (Agboola and Howard, 2006), which states that on a CM elliptic curve, the 𝔭-adic height ℎ 𝐹,𝔭 of a point of infinite order is nonzero. This result is due to Bertrand and does not use the assumption 𝑝 > 3. The second piece is showing that there are 𝐾𝑛− -rational points of infinite order which are universal norms in the Selmer group Sel(𝐾𝑛− , 𝑇) and Sel(𝐾𝑛− , 𝑇 ∗ ). This part uses results about the rank of Sel(𝐾𝑛− , 𝑇) and 𝑋 (𝜆) as Λ-modules when 𝜆 is of root number −1. They assume 𝑝 > 3. Luckily, we have already generalized these statements in Theorem 3 to include 𝑝 = 2, 3 using a modified Euler system. The third piece is showing that if 𝑅(𝜆) is zero, then the 𝑝-adic height pairing ℎ𝑛 is zero. This piece does not use the assumption 𝑝 > 3. Combining these three pieces above, we have demonstrated that when 𝑝 = 2, 3, the regulator 𝑅(𝜆) does not vanish for 𝜆 associated to a CM elliptic curve. When 𝜆 is more general, for example, associated to a CM abelian variety, we have not yet investigated the possibility of proving the non-vanishing of 𝑅(𝜆) for 𝑝 = 2, 3. This is the reason for imposing the additional assumption in Theorem 2 compared to Conjecture 2 proven in Burungale-Tian. Collecting the steps above, we have arrived at the equality of characteristic ideals and Theorem 2 is complete.  23 Chapter 4 𝑝-CONVERSE THEOREM In this section, we briefly outline how Theorem 2 helps to prove the CM rank 1 𝑝-converse theorem, namely Theorem 1, when 𝑝 = 2, 3. We would like to reemphasize that the main modification for 𝑝 = 2, 3 takes place in Theorem 2. With Theorem 2 established, the steps discussed below are only a restatement of that in Burungale-Tian and not original to this project. The purpose of restating their arguments is to highlight the fact that the modified version of HMC, namely Theorem 2, still suffices to prove the 𝑝-converse. Recall that 𝐸 be a CM elliptic curve over an imaginary quadratic field 𝐾 and let 𝜆 be the algebraic Hecke character associated to 𝐸. According to Rohrlich in (Rohrlich, 1984), there exists a finite order Hecke character 𝜒 unramified at 𝑝, such that 𝜒 (4.0.1) ord𝑠=1 𝐿 (𝑠, 𝜆∗ · ∗ ) = 0 𝜒 When corankZ 𝑝 Sel 𝑝∞ (𝐸 /Q ) = 1, the Parity Conjecture by Nekovár̆ shows that the root number of 𝜆 is −1. Let 𝑔 be the CM modular form associated to 𝜆 𝜒−1 , then the Rankin-Selberg 𝐿-function associated to the pair (𝑔, 𝜒) decomposes in the following form: 𝜒 𝐿 (𝑠, 𝑔 × 𝜒) = 𝐿(𝑠, 𝜆)𝐿 (𝑠, 𝜆∗ · ∗ ) 𝜒 Thus, by the choice of the twist 𝜒, showing the 𝑝-converse is equivalent to showing corankZ 𝑝 Sel 𝑝∞ (𝐸 /Q ) = 1 =⇒ ord𝑠=1 𝐿 (𝑠, 𝑔 × 𝜒) = 1 The properties of 𝜆 and 𝜒 ensure that the generalized Heegner hypothesis is satisfied for the pair (𝑔, 𝜒). Thus, there exists a Heegner point 𝑃𝑔,𝜒 ∈ 𝐵(𝐾) ⊗Z Q with 𝐵/𝐾 a CM abelian variety associated to the pair (𝑔, 𝜒). According to the generalized Gross-Zagier formula by Yuan-Zhang-Zhang (Yuan, S.-W. Zhang, and W. Zhang, 2013), there is an equivalence: ord𝑠=1 𝐿 (𝑠, 𝑔 × 𝜒) = 1 ⇔ 𝑃𝑔,𝜒 ≠ 0 24 Due to the work of Kolyvagin/Rubin, 4.0.1 implies corankO℘ Sel℘∞ (𝜆∗ · 𝜒 )=0 𝜒∗ Combining all of the above, the 𝑝-converse theorem becomes equivalent to the implication corankO℘ Sel℘∞ (𝐵/𝐾 ) = 1 =⇒ 𝑃𝑔,𝜒 ≠ 0 This implication is proven from the Galois decent of Theorem 2, as 𝑃𝑔,𝜒 is at the lowest level of the norm-compatible system of Heegner points. The use of Theorem 2 is justified as the pair (𝑔, 𝜒) satisfies the requirement of Conjecture 2 by construction and the Hecke character 𝜆 with root number −1 is associated to the CM elliptic curve 𝐸 by definition. 25 Chapter 5 CONCLUSION AND FUTURE WORK The specific CM case of the Heegner Main Conjecture discussed in the proof of our CM rank 1 𝑝-converse theorem has some independent interest, especially the part about anticyclotomic main conjectures. 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