On the Equivariant Tamagawa Number Conjecture Citation Navilarekallu, Tejaswi (2006) On the Equivariant Tamagawa Number Conjecture. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7HZ0-F068. https://resolver.caltech.edu/CaltechETD:etd-05242006-225912 Abstract For a finite Galois extension K/Q of number fields with Galois group G and a motive M = M' ⊗ h⁰(Spec(K))(0) with coefficients in Q[G], the equivariant Tamagawa number conjecture relates the special value L*(M,0) of the motivic L-function to an element of K₀(Z[G];R) constructed via complexes associated to M. The conjecture for nonabelian groups G is very much unexplored. In this thesis, we will develop some techniques to verify the conjecture for Artin motives and motives attached to elliptic curves. In particular, we consider motives h⁰(Spec(K))(0) for an A₄-extension K/Q and, h¹ (E x Spec(L))(1) for an S₃-extension L/Q and an elliptic curve E/Q. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Equivariant Tamagawa number Conjecture; modular symbols; period isomorphism; Tate sequences Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Flach, Matthias Thesis Committee: Flach, Matthias (chair) Ramakrishnan, Dinakar Wambach, Eric Dimitrov, Mladen Defense Date: 8 May 2006 Record Number: CaltechETD:etd-05242006-225912 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05242006-225912 DOI: 10.7907/7HZ0-F068 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2017 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 31 May 2006 Last Modified: 08 Nov 2022 19:02 Thesis Files Preview PDF - Final Version See Usage Policy. 525kB Repository Staff Only: item control page


 
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S ∗ S S    S S ∗ S b χ∈G b χ∈G  ∗ S  × b χ∈G δb : ζ(R[G])× → K0 (Z[G], R), 608l_8 K (Z[G], R) ;<9*`6087_8lba^V`; ‡8 K uv_mNefx  wEo  „ † 9;iZfuq5 ^V`898:rTe08lZ0w:8:92mNZ08w ^VZkw:mNZ09`_e0wl` ^VZ8lb<8ljk8lZ?` RΩ(K/k) ∈ K (Z[G], R)  ! 8 b<8l` T Ω(K/k) = RΩ(K/k) − δ(L  57608 b (0)) 8:rTef; N^V_;a^VZT`*5 ^Vj ^Vu?^ 7^qZ?efj= 8l_2w:mNZ 8:wl`ef_8/oRmN_ `608jkmN`; ‡8 M = h (< x8:w (K))  ;i`6k^VZz^Nwl`;<mNZ ?= > `60898ljz;<9;ijzxfb<8n^Vbiu‡8l=f_I^ A = Q[G]† 9`I^V`8:9/`6O^V` T Ω(K/k) = 0 2 57608 N^VZf;<96f;iZfumVo T Ω(K/k) ;iZ K (M, R) | 608l_8 M ;<9^]j ^%G ;ij ^Vb Z mN_{f8l_w:mNZT`I^V;iZf;iZfu Z[G] | ;<98:r?ef; v^Vb<8lZ?``m `608 9`_mNZfu < `I^V_ zw:mNZ 8:wl`ef_8n^N9#9`I^V`8:{z;iZ  . 1  b<9mf|v`608 N^VZf;<96f;iZfumVo T Ω(K/k) ;iZ K (Z[G]) ;<9B8:r?ef; v^Vb<8lZ?` ` m`608#w:8lZT`_I^VbGw:mNZ 8:wl`ef_8:9 mVo  . ~  d 57608#8:rTef; N^V_;a^VZT`dw:mNZ 8:wl`ef_8^Vb<9m7_8:w:m ‡8l_9*98 ‡8l_I^Vb‡_8Eg0Z08ljk8lZ?`9 mVo*`608 < `I^V_ w:mNZ 8:wl`ef_84{}e08`m y 6f;iZ?=fef_u0|  _mv99 |fpef=f;iZ |^VZ0{ mN`608l_9  wEo  €  †  c0mN_Z0mNZO^V=8lbi;a^VZ8} `8lZ09;<mNZ09 K/k |}`608 N^VZf;<96f;iZfumVo T Ω(K/k) ;<9 GZ0m Z)oRmN_7w:8l_`I^V;iZ {};i608:{}_I^Vb 8} `8lZ09;<mNZ09  wEo 0= † ^VZ0{]osmN_n^VZ ;iZ}g0Zf;i`8Ko ^Vjz;ibi>]mVoBr?eO^V`8l_Zf;<mNZƒ8} `8lZ09;<mNZ09  wEo   †   Z08mVo2`608 ‡8l> ;<{f8 ^N9e098:{;iZ   ;<9`6O^V``608qj ^Vx   0 0 ∗ S 0     0 k    0    0           K0 (Z[G], R) → K0 (M, R) × K0 (Z[G  ], R) × K0 (Z[G]) ;<9 ;iZ 8:wl`; ‡8oRmN_ G = Q |#`608 rTeO^V`8l_Zf;<mNZ uv_mNefx  608l_8 M ;<9 ^ j ^%};ij ^Vb Z mN_{f8l_ w:mNZ?`I^V;iZf;iZfu †  m 8 ‡8l_ |0`6f;<9o ^V;ib<97`mz60mNb<{osmN_ G = A  9I8:8 xfx8lZ0{};  †  Z[G] Y[Z `6f;<9z`608:9;<9 | 8 96O^Vbib_8:9`_;<wl`z`m ^ xO^V_`;<wlefba^V_ 8} `8lZ09;<mNZ K/Q ;i`6 K^Vb (K/Q) ≃ A | ^VZ0{ {f8 ‡8lb<mNx9ImNjk8`8:wI6fZf;<r?e08:97osmN_/w:mNjzxfef`;iZfu RΩ(K/Q)  ! 8e098`608n;<{f8 ^N97mVo y 6f;iZ?=fef_u   . ~  † `m]w:mNZ09`_e0wl``608k5d^V`8)98:rTe08lZ0w:8 ^VZ0{ 8} xfbi;<wl;i`bi> w:mNjzxfef`8k`608k{f8l`8l_jz;iZO^VZT`9 ^V_;<9;iZfu]oR_mNj `6f;<9 98:rTe08lZ0w:8q`mk8 v^VbieO^V`8 RΩ(K/Q) | ^VZ0{ ‡8l_;hoR> `608 N^VZf;<96f;iZfu)mVo T Ω(K/Q)   Z084w:mNefb<{]xmv99;i=fbi>  8    4   4     / 8}`8lZ0{ `608:98`8:w6fZf;<rTe08:9`m ‡8l_;hoQ> `608Kw:mNZ 8:wl`ef_8qosmN_^VZ ;iZ}g0Zf;i`8qo ^Vjz;ibi> mVo A  8}`8lZ09;<mNZ09  4    -7 h1(E )(1) K [Y Zk`f6 ;<9/w ^N9I8 /8w:mNZ09;<{f8l_^VZ)8lbibi;ixf`;<wwlef_ ‡8 E {f8Eg0Z08:{ m ‡8l_ k ^VZ0{ b<mTm k^V`/`608=O^N98w6O^VZfu‡8nwlef_ ‡8 z 57608l_8 ;<9^ ZO^V`ef_I^Vb/^Nwl`;<mNZ mVo G mNZ E |3^VZ0{ /8zb<8l` A = Q[G]  E := E × Spec(K) 57608)^N9I9mTwl;a^V`8:{ jkmN`; T;<w Ls oQefZ0wl`;<mNZ ;<9 L(E , s) = (L(E ⊗ χ, s)) | 6f;<wI6 `I^ ‡8:9 N^Vbie08:9;iZ Q  7 8:w ^Ve098zmVoB`608 w:8lZ?`8l_KmVo/9>Tjzjk8l`_>‡| `608jkmv9`;iZT`8l_8:9`;iZfu]9x8:wl;a^Vb v^Vbie08 C ≃ ζ(C[G]) mVo*`6f;<9 Ls oRefZ0wl`;<mNZ];<9 
      K K Spec(k)   K   b χ∈G  b χ∈G L∗ (EK , 1) = (L∗ (E ⊗ χ, 1))χ∈Gb ∈ ζ(R[G])× , 608l_8 /8 _;i`8 L `m{f8lZ0mN`8`6087b<8 ^N{};iZfuKw:mT8  wl;<8lZT`#;iZz`6085d^ >Tb<mN_/8GxO^VZ09;<mNZkmVo `608 LsoRefZ0wl;i`mNZ ^V` s = 1 \*^V_`qmVo*`6088:rTef; N^V_;a^VZT`nw:mNZ 8:wl`ef_8wlba^V;ijk9n`6O^V` δ(L ;<9^VZ b (E , 1)) | 6f;<w6 8lb<8ljk8lZ?`mVo K (Z[G]; R) |O;<9;iZ o ^Nwl`^VZ 8lb<8ljk8lZ?`mVo K (Z[G]; Q) 7 576f;<9o ^Nwl`;<9 GZ0m Z ;iZ 98 ‡8l_I^Vb w ^N98:9{}e08K`m < 6f;ij4ef_I^ ^VZ0{ mN`608l_9  wEo 0/%/ |  1 †  nm K|osmN_n^ _I^V`;<mNZO^Vb xf_;ijk8 l b<8l` T := lim E(Q̄)/l ^VZ0{ b<8l` V := T ⊗ Q  8l` S =8^ 0g Zf;i`898l`#mVoxf_;ijk8:9Bw:mNZT`I^V;iZf;iZfu`608;iZ}g0Zf;i`8←xf−_;ijk8:9 |‡xf_;ijk8:9/m ‡8l_ l ^VZ0{ xf_;ijk8:9/mVo =O^N{z_8:{}e0wl`;<mNZ  Z08qw ^VZ`608lZ]{f8Eg0Z08K^x 8l_ os8:wl` Z [G] w:mNjzxfb<8 RΓ (O , T )  > ;a^`608608l;iuv6?`xO^V;i_;iZfu0|0x 8l_;<mG{ ;<9mNjkmN_xf6f;<9j ^VZ0{]w:mNjzxO^V_;<9mNZ ;<9ImNjkmN_xf6f;<9j /84w ^VZ ^N99mGwl;a^V`8`m `6f;<9w:mNjzxfb<8 ^VZ]8lb<8ljk8lZ?`    ∗     ∗  0    K   0 
n l l n l l Zl l  c l K,Sl l  RΩ(E, Zl [G]) ∈ K0 (Zl [G]; Ql ). 576088:rTef; N^V_;a^VZT`Bw:mNZ 8:wl`ef_8wlba^V;ijk92`6O^V`#`608 l xO^V_`#mVo δ(L b (E , 1)) 8:rTeO^Vb<9 RΩ(E, Z [G]) 2576f;<9 :w mNZ 8:wl`ef_8]u‡8lZ08l_I^Vbi; 8:9)`608 _8lxfef`8:{ w:mNZ 8:wl`ef_8 mVo /;i_wI6 ^VZ0{ < ;iZfZ08l_`mNZ  tn>‡8l_  98:8  ~v~ osmN_ f{ 8l`I^V;ib<9 †  Y[Z`6f;<9 `608:9;<9 | 8uv; ‡8^ZO^V; ‡8^Vxfxf_m‡^Nw6z`m ‡8l_;hoQ>K`608w:mNZ 8:wl`ef_8oRmN_d`608/jkmN`; ‡8 h (E )(1)  ! 8qe098KjkmT{}efba^V_n9>Tj= mNb<9`m w:mNjzxfef`849x 8:wl;a^Vb v^Vbie08:9mVo Ls oRefZ0wl`;<mNZ09 ;i`6 ^V= 8lbi;a^VZ ` ;<9`9 |^VZ0{ 0e 98k^VZO^Vbi>G`;<wzjk8l`60mG{f9K`mu‡8l`KZTefjk8l_;<w ^Vb v^Vbie08:94mVo7`608z9I^Vjk8v| ;i`6 Z0mNZO^V=8lbi;a^VZ ` ;<9`9  Z0{f8l_ 9mNjk86T>TxmN`608:98:9 8 ;ibib e098`608:9I8 L v^Vbie08:9`m uv; ‡8qZ?efjk8l_;<w ^Vb 8 T;<{f8lZ0w:84`m `608Kw:mNZ 8:wl`ef_8# 57608 `608:9;<9 ;<9zmN_u?^VZf; 8:{ ^N9osmNbib<m 9 Y Z y 6O^Vxf`8l_ ~ /8=f_;<8 0 > {f8Eg0Z08`608 `8l_jk9 ;iZ ‡mNb ‡8:{ ^VZ0{ƒ9`I^V`8`6088:rTef; v^V_;a^VZ?`qw:mNZ 8:wl`ef_8#4 Y Z y 6O^Vxf`8l_7/ 8_8l;iZT`8l_xf_8l`K`608w:mNZ 8:wl`ef_8osmN_ _`;iZ kj mN`; ‡8:9 |N^VZ0{quv; ‡8B^7{f8l`I^V;ib<8:{{f8:9wl_;ixf`;<mNZ4mVo0^7jk8l`60mT{K`m ‡8l_;hoQ>n`608#w:mNZ 8:wl`ef_82;iZK`6082Z0mNZO^V=8lbi;a^VZ  ∗ K l                 K      1      1 98l``;iZfu /‚ e0wI6 mVo `6f;<9 y 6O^Vxf`8l_;<9xO^V_`mVo  ~:E  Bcfef_`608l_ | 8^Vxfxfbi>`6f;<9jk8l`60mG{ ^VZ0{ xf_m ‡8q`608 :w mNZ 8:wl`ef_8q;iZ ^ xO^V_`;<wlefba^V_w ^N98#B Y Z y 6O^Vxf`8l_ 1 8Kw:mNZ09;<{f8l_`608qjkmN`; ‡8:9n^V_;<9;iZfu oR_mNj(`608q=O^N98 w 6O^VZfu‡8mVo 8lbibi;ixf`;<wnwlef_ ‡8:9 ! 8uv; ‡8`608{f8l`I^V;ib<97mVo w:mNjzxfef`;iZfu L N^Vbie08:9/e09;iZfujkmT{}efba^V_79>Gj4=mNb<9 | ^VZ0{ e09;iZfu ^VZO^Vbi>G`;<w4jk8l`60mG{f9 ! 8^Vxfxfbi>]`608:984`m uv; ‡84Z?efjk8l_;<w ^Vb28 T;<{f8lZ0w:8:9n`m `608w:mNZ 8:wl`ef_8  ^Vu?^V;iZ ;iZ^Z0mNZO^V=8lbi;a^VZ 98l``;iZfu † * 576f;<9 y 6O^Vxf`8l_B;<9BxO^V_`/mVo `608^V_`;<wlb<8  ~ „ * Y Zk`608^Vxfx 8lZ0{};  | 8 vu ; ‡8k^)=f_;<8Eo7^Nw:w:mNefZT`4mVoB`608 6f;<9`mN_> mVo v^V_;<mNe09_8:9efbi`9K^VZ0{ƒw:mNZ 8:wl`ef_8:9q_8lba^V`8:{ƒ`m `608z9x8:wl;a^Vb v^Vbie08:9mVo Ls oQefZ0wl`;<mNZ09`6O^V`8 ‡8lZ?`eO^Vbibi> b<8:{`mz`608qoRmN_jefba^V`;<mNZ mVo*`608q8:r?ef; v^V_;a^VZ?`nw:mNZ 8:wl`ef_8#               
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c0Nm _n^ uv_mNefx G | 84{f8lZ0mN`8q=T> Gb `608K98l`mVo#^Vbib ;i__8:{}e0wl;i=fb<84w:mNjzxfb<8]w6O^V_I^Nwl`8l_9nmVo G  • c0mN_n^ Z?efj= 8l_gO8lb<{ k | /8K{f8lZ0mN`8K=?> G `608 q^Vb<mN;<9uv_mNefx mVo k̄/k  • c0mN_]^ g0Zf;i`8 xfba^Nw:8 l ;iZ^ ZTefj4=8l_gO8lb<{ k | 8ƒ{f8lZ0mN`8 =T> F r `608 c0_mN= 8lZf;ie09 8lb<8ljk8lZT` • ^V``I^NwI608:{]`m l ;iZ G  98l`mVo2^Vbib3;iZ}g0Zf;i`8qxfba^Nw:8:9  mVo*`608qefZ0{f8l_bi>G;iZfukZ?efj= 8l_gO8lb<{ †  • S c0mN_q^ Z?efj4=8l_ngO8lb<{ k | /8{f8lZ0mN`8=?> O `6084_;iZfumVo#;iZ?`8lu‡8l_9;iZ k  cfef_`608l_ |osmN_^zg0Zf;i`8 • 9I8l` S mVo xf_;ijk8:9 | O {f8lZ0mN`8:9`608_;iZfu mVo S ;iZT`8lu‡8l_9 c0mN_n^VZ?>_;iZfu R | /8Kb<8l` ζ(R) {f8lZ0mN`8q`608Kw:8lZT`8l_mVo R  • c0mN_n^VZ R jkmT{}efb<8 M | /8Kb<8l` M `mz= 8`608 R {}eO^Vb Hom (M, R)  • n ; ‡8lZ ^_;iZfuk60mNjkmNjkmN_xf6f;<9j φ : R → S ^VZ0{]^VZ R jkmT{}efb<8 M | /8Kb<8l` M := M ⊗ S  • c0mN_n^ _;iZfu R | /8qb<8l` R {f8lZ0mN`8q`608KmNxfxmv9;i`8_;iZfu  •    k  l k ∞   k k,S   ∗  R   S op R =
     


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  c0mN_^_;iZfu R  ;i`6efZf;i`8lb<8ljk8lZ?` † | 8b<8l` PMod(R) {f8lZ0mN`8`608w ^V`8lu‡mN_>)mVo g0Zf;i`8lbi>)u‡8lZ08l_I^V`8:{ fx _m 8:wl`; ‡8 R jkmG{}efb<8:9 ! 84{f8Eg0Z08 K (R) `mk=8q`608^V= 8lbi;a^VZ uv_mNefx u‡8lZ08l_I^V`8:{ =?> `6089>Tj= mNb<9 osmN_8 ^Nw6 P ∈ PMod(R) | ;i`6]_8lba^V`;<mNZ09 [P ] + [P ] = [P ] oRmN_8 ‡8l_> 960mN_`80^Nwl`98:rTe08lZ0w:8 [P ]    0   1 3 2 0 → P1 → P2 → P3 → 0. 57608uv_mNefx K (R) ;<9B{f8Eg0Z08:{z`m= 87`608^V= 8lbi;a^VZf; ^V`;<mNZ mVo`608{};i_8:wl`#bi;ijz;i` lim GL (R)  #rTef;  ^Vb<8lZ?`bi>‡|OmNZ08nw ^VZ {f8Eg0Z08 K (R) `m= 8`608q^V= 8lbi;a^VZ uv_mNefxu‡8lZ08l_I^V`8:{ =T>)9>Gj4−→=mNb<9 [P, α] osmN_8 ‡8l_> ^VZ0{ α ∈ Aut (P ) | ;i`6]_8lba^V`;<mNZ09 P ∈ PMod(R)  1 n n  1  R [P1 , α1 ] + [P2 , α2 ] = [P3 , α3 ], osmN_8 ‡8l_>w:mNjzjef`I^V`; ‡8{};a^Vuv_I^Vj mVo*960mN_`8f^Nwl`98:rTe08lZ0w:8:9   0 → (P1 , α1 ) → (P2 , α2 ) → (P3 , α3 ) → 0,  wEo  .,E  †  nmN`8#`6O^V`2^_;iZfun60mNjkmNjkmN_xf6f;<9j φ : R → S ;iZ0{}e0w:8:9d60mNjkmNjkmN_xf6f;ijk9 φ : K (R) → K (S) | so mN_ i = 0, 1   Z08w ^VZk{f8Eg0Z08^n_8lba^V`; ‡8 K  uv_mNefx K (R, φ) |T^VZ0{kjkmN_xf6f;<9jk9 K (S) → K (R, φ)  ∗ i i i  0 1 0 € ^VZ0{ K (R, φ) → K (R) 9e0w6 `6O^V` 0 0 K1 (R) → K1 (S) → K0 (R, φ) → K0 (R) → K0 (S) ;<9480^Nwl` Y Z `8l_jk9mVou‡8lZ08l_I^V`mN_9^VZ0{ _8lba^V`;<mNZ09 |d`608kuv_mNefx K (R, φ) ;<9qu‡8lZ08l_I^V`8:{ =?>ƒ`_;ixfb<8:9 | 608l_8 M, N ∈ PMod(R) ^VZ0{ λ : M ⊗ S → N ⊗ S ;<9^VZ ;<9mNjkmN_xf6f;<9j mVo S  [M, N ; λ] jkmG{}efb<8:9 |?^VZ0{ ;i`6z_8lba^V`;<mNZ092uv; ‡8lZk=T>960mN_`#80 ^Nwl`#98:rTe08lZ0w:8:9  98:8  .,E  §1   oRmN_2{f8l`I^V;ib<9 †  ! 608lZ `608jkmN_xf6f;<9j φ ;<9#8 G;<{f8lZ?` /8 _;i`8 K (R; S) oRmN_B`6087uv_mNefx K (R, φ)  b<9mf|NosmN_B^VZ R jkmT{}efb<8 | 8qb<8l` M := M ⊗ S  M nm K|db<8l` F =8 ^ gO8lb<{ ^VZ0{ b<8l` R =8 ^ w:8lZ?`_I^Vb9;ijzxfb<8)^Vbiu‡8l=f_I^ m ‡8l_ F  c ;  ^VZ 8G `8lZ09;<mNZ 9e0w6 `6O^V` R := R ⊗ F ≃ M (F ) ^VZ0{)^VZ ;iZ0{f8:w:mNjzxmv9I^V=fb<8;<{f8ljzxmN`8lZ?` e ∈ R  8l` V F /F =8n^Kg0Zf;i`8lbi> u‡8lZ08l_I^V`8:{ R jkmG{}efb<8^VZ0{ b<8l` φ ∈ End (V )  ! 8{f8Eg0Z08`608_8:{}e0w:8:{_I^VZ rr (V ) mVo V ^N9 0  R  R     0 0  S  R   0 0 F 0 0 n 0
 R R rrR (V ) := dimF 0 (e(V ⊗F F 0 )) ^VZ0{ `608_8:{}e0w:8:{ {f8l`8l_jz;iZO^VZ?` detred (φ) mVo φ ^N9 R detredR (φ) = det0 (φ ⊗ 1|e(V ⊗F F 0 )). 57608q_8:{}e0w:8:{ _I^VZ ;iZ0{}e0w:8:9n^jkmN_xf6f;<9j F  rrR : K0 (R) → Z. 57608 detred (φ) ;<92^VZz8lb<8ljk8lZ?` F | 6f;<wI6 ;<9d;iZ0{f8lx8lZ0{f8lZ?`/mVoO`608/wI60mN;<w:8:9#mVo F ^VZ0{ e dY[`d;iZ0{}e0w:8:9 ^jkmN_xf6f;<9j]|w ^Vbib<8:{ `608q_8:{}e0w:8:{ Z0mN_j j ^Vx | R
0 nrR : K1 (R) → F ×  wEo  .,E  †    ×        " 8l` R =8#^VZ?>K^V_=f;i`_I^V_>q_;iZfu  9 = 8EosmN_8v| b<8l` PMod(R) {f8lZ0mN`8#`608#w ^V`8lu‡mN_>KmVoGg0Zf;i`8lbi>u‡8lZ08l_I^V`8:{ xf_ m 8:wl`; ‡8 R jkmG{}efb<8:9KY[Z . „ |tn8lbi;iuvZ08 6O^N9w:mNZ09`_e0wl`8:{ ^ w ^V`8lu‡mN_> V (R) mVo G;i_`eO^Vb2mN= 8:wl`9
  E ^VZ0{ ^efZf; ‡8l_9I^Vbd{f8l`8l_jz;iZO^VZ?`oQefZ0wl`mN_  [ ] : PMod(R) → V (R) 9I^V`;<9 oR>G;iZfukw:8l_`I^V;iZ w:mNZ0{};i`;<mNZ09n^N9;iZ  „ #576084^V=m ‡8oRefZ0wl`mN_ZO^V`ef_I^Vbibi> 8G`8lZ0{f9`m ^oQefZ0wl`mN_  [ ] : D p (R) → V (R) 608l_8 D (R) ;<9/`608nw ^V`8lu‡mN_>)mVo x8l_ oR8:wl`7w:mNjzxfb<8}8:97mVo R jkmG{}efb<8# w:mNjzxfb<8 C mVo R jkmG{}efb<8 <; 9w ^Vbib<8:{
3;ho `608l_8K;<9^ w6O^V;iZ  p  •  C • → C1• ← C2• · · · → P • mVo r?eO^N9;  ;<9ImNjkmN_xf6f;<9jk9 | 608l_8 P ;<9B^n=mNefZ0{f8:{kw:mNjzxfb<8kmVog0Zf;i`8lbi> u‡8lZ08l_I^V`8:{kxf_m 8:wl`; ‡8jkmG{  fe b<8:9 Y[`7oRmNbib<m 9oR_mNj(`608qxf_mGmVo*mVo*`608K8};<9`I^VZ0w:8qmVo V (R) ;iZ  . „3`6O^V``608l_84^V_8K;<9mNjkmN_xf6f;<9jk9   •  ∼ Ki (R) −→ πi (V (R)) osmN_ i = 0, 1 608l_8 K (R) {f8lZ0mN`8:9B`608^Vbiu‡8l=f_I^V;<w K  uv_mNefx)^N99ImTwl;a^V`8:{ `m R  98:8 0/ . fosmN_#{f8l`I^V;ib<9 † | ^VZ0{ π (V (R)) ;<9 `608 uv_mNefx mVo4;<9ImNjkmN_xf6f;<9j wlba^N998:9mVomN= 8:wl`9 mVo V (R) ^VZ0{ π (V (R)) =  i 0  ) 1 n; ‡8lZ ^g0Zf;i`8lbi>u‡8lZ08l_I^V`8:{)9ef=f_;iZfu S mVo Q |?^VZ S  mN_{f8l_ A mVo ^g0Zf;i`8{};ijk8lZ09;<mNZO^Vb Q[^Vbiu‡8l=f_I^ ;<9^zg0Zf;i`8lbi> u‡8lZ08l_I^V`8:{ S  jkmT{}efb<8 9e0wI6 `6O^V` A ⊗ Q = A nc0mN_K^VZ?> 9e0w6 S  mN_{f8l_ | ^VZ0{ ^VZT> A gO8lb<{ 8 }`8lZ09;<mNZ F mVo Q |0mNZ08q6O^N9^zZ0mN`;<mNZ mVo2_8lba^V`; ‡8 G;i_`eO^VbdmN= 8:wl`9 V (A; F ) ^VZ0{ V (A ; Q ) | 608l_8 A = A ⊗ Z  9I8:8  „ oRmN_{f8l`I^V;ib<9 † B57608l_84^V_8;<9mNjkmN_xf6f;<9jk9 AutV (R) (1V (R)  Z   p  p Z p ∼ ^VZ0{ π0 (V (A; F )) −→ K0 (A; F ) ∼ π0 (V (Ap ; Qp )) −→ K0 (Ap ; Qp ), 6f;<w6ƒ^V_8qw:mNjzxO^V`;i=fb<8 ;i`6]`6084‚ ^ >‡mN_ > ;<8l`mN_;<9n98:rTe08lZ0w:8:9  wEo2\#_mNx ~ 0=;iZ  „ †    #$"* p „ 8 98ljz;<9;izj xfb<z8 _;ifZ u z57608lZ ^VZ R jkmG{}efb<8 P ;<9qxf_m 8:wl`; ‡8z;ho7^VZ0{ .  38l` R =8k^:w mNjzj4ef`I^V`; ‡) mNZfbi> ;ho;iz` ;<9 b<mGw ^Vbibi> Ro _8:8 ^V` V^ bib`608fx _;ijk8;<{f8 ^Vb<9zmVo R  Y[Z `6f;<9zw ^N9I8v|#`608{f8l`8l_jz;iZO^VZ?` oRefZ0wl` mN_w ^VZ]=8q{fE8 g0Z08:{ b<mGw ^Vbiib >=?> 
   DetRp (Pp ) = ∧rankRp (Pp ) Pp , rankRp (Pp ) osmN_78 ‡8l_>kxf_;ijk8 p ∈ Spec(R)  mN`8`6O^V`7`6?e097{f8Eg0Z08:{ Det (P ) ;<97^4uv_I^N{f8:{)bi;iZ08=fefZ0{}b<8# 38l` P(R) {f8lZ0mN`8`608nw ^V`8lu‡mN_>)mVo uv_I^N{f8:{ bi;iZ08=fefZ0{}b<8:9m ‡8l_ R  Y[Z `608n^V=m ‡8nZ0mN`I^V`;<mNZ |G;ho ;<9 ^g0Zf;i`8  ^V`*w:mNjzj4ef`I^V`; ‡8 Z[ ^Vbiu‡8l=f_I^`608lZ`608l_8/;<9*^ZO^V`ef_I^Vb}8:r?ef; v^Vb<8lZ0w:8/mVoOw ^V`8lu‡mN_;<8:9 A   R 
∼ P(A) → V (A)     ~   ‡8lZ ;ho R ;<9nZ0mN`qw:mNjzj4ef`I^V`; ‡8v| =fef`;<998ljz;<9;ijzxfb<8v| mNZ08 w ^VZ w:mNZ09`_e0wl`q`608{f8l`8l_jz;iZO^VZ?` oRefZ0wl`mN_q;iZ ^ 9;ijz;iba^V_os^N96f;<mNZ =T> b<mGm G;iZfu ^V``608 ;iZ0{f8:w:mNjzxmv9I^V=fb<8 ;<{f8ljzxmN`8lZ?`9 ! 8uv; ‡8 `6f;<9w:mNZ09`_e0wl`;<mNZ]=8lb<m 9;iZ0w:8q;i`;<9 ‡8l>`mz9mNjk8KmVo2mNef_w:mNjzxfef`I^V`;<mNZ09  > ! 8:{f{f8l_=fef_ZF 9B{f8:w:mNjzxmv9;i`;<mNZ /8w ^VZ ^N9I9efjk87`6O^V` R ;<9#^nw:8lZT`_I^VbO9;ijzxfb<8^Vbiu‡8l=f_I^qm ‡8l_ ^gO8lb<{ F  < m R ≃ M (D) osmN_29mNjk87{}; T;<9;<mNZ _;iZfu D ;i`6kw:8lZT`8l_ F d c0ef_`608l_ |‡=?>4g G ;iZfu4^VZ 80 ^Nwl`q‚ mN_;i`I^)8:rTef; N^Vb<8lZ0w:8 PMod(M (D)) → PMod(D) | 8j ^:> ^N99efjk8`6O^V` R = D  c ;  ^4gO8lb<{8} `8lZ09;<mNZ F /F 9e0w6`6O^V` D ⊗ F ≃ M (F )  38l` e = 8^VZ;iZ0{f8:w:mNjzx mv9I^V=fb<8 ;<{f8ljzxmN`8lZ?`zmVo M (F ) ^VZ0{ b<8l` e , . . . , e =8)^VZ mN_{f8l_8:{ F  =O^N9;<9 mVo eM (F )  38l` V =8)^g0Zf;i`8lbi> u‡8lZ08l_I^V`8:{ xf_m 8:wl`; ‡8  ^VZ0{ 608lZ0w:8koR_8:8 † D jkmT{}efb<8# 8l` {v , . . . , v } = 8 ^  =O^N9;<94mVo V  < 8l` b := ∧e v  576f;<9K;<94^VZ F  =O^N9;<9mVo Det (e(V ⊗ F ))  < ;iZ0w:8)^VZ?> D wI6O^VZfu‡8K;iZ `608q=O^N9;<9 {v } j4efbi`;ixfbi;<8:9 b =T>]^VZ]8lb<8ljk8lZ?`mVo F |}`608 F  9xO^Nw:849xO^VZfZ08:{ =?> b >G;<8lb<{f9^ 8lbib  {f8Eg0Z08:{]uv_I^N{f8:{ F  bi;iZ08q=fefZ0{}b<8## 576f;<9{f8Eg0Z08:9`608K{f8l`8l_jz;iZO^VZ?`oQefZ0wl`mN_           n   n 0 F 0 d 1 0 d 0
0 d  d
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8l` X = 84^ 9jkmTmN`6 xf_m 8:wl`; ‡8 v^V_;<8l` >]{f8Eg0Z08:{ m ‡8l_q^zZ?efj= 8l_ngO8lb<{ k c0mN_;iZT`8lu‡8l_9 n, r ;i`6 |}b<8l` M = 8`608qxfef_8  y 60m † jkmN`; ‡8 h (X)(r) # 57608q_8 ^Vbi; ^V`;<mNZ09nmVo M ^V_8 n≥0 |V^/g0bi`8l_8:{ k 9xO^Nw:8 ;i`6K;i`9 ZO^V`ef_I^Vb?{f8:wl_8 ^N9;iZfug0bi`8l_I^V`;<mNZ {F H (X/k)} • H (M ) := H (X/k) 96f;hoQ`8:{ =T> r |O^ w:mNjzxO^V`;i=fb<849>}9`8lj mVo l[ ^N{};<wK_8lxf_8:98lZ?`I^V`;<mNZ09nmVo G • H (M ) := H (X × k̄, Q (r))

L 

    n  n dR dR  i n dR  l n et k l k  i∈Z  |GosmN_8 ^Nw6 σ ∈ Hom(k, C) ^ Q mG{}u‡8q9`_e0wl`ef_8Km ‡8l_ R mN_ C ^Nw:w:mN_{};iZfuk` m 608l`06 8l_ v(σ) ;<9_8 ^Vb mN_w:mNjzxfb<8  l8 ` A = 8)^g0fZ ;i`8lbi> u‡l8 Z08l_I^V`8:{ 98ljz;<9;ijzxfb<8 Q[ ^Vbiu‡8l=f_I^]`6O^V`z^Nwl`94mNZ M  Y[`K`608l_8EosmN_8 ^Nwl`9 ZOV^ `ef_I^Vbibi>ƒmNZ V^ bib `608 _8 ^Vbi; ^V`;<mNZ094mVo M  ! 8 96O^Vbib#{f8lZ0mN`8 `608jkmN`; ‡8z=?> M `m ;iZ0{};<w ^V`8 `608 ^Nwl`;<mNZ mVo A #Oc mN_^g0Zf;i`8qxfba^Nw:8 v mVo k mVo*_8:9;<{}e08KwI6O^V_I^Nwl`8l_;<9`;<w p |0^VZ0{ ^xf_;ijk8 l 6= p {f8Eg0Z08   • Hσ (M ) := H n (σX(C), (2πi)r Q) 
 A Iv Lv (MA , T ) := detredAl (1 − Fr−1 v |Hl (MA ) ) ∈ ζ(Al )[T ]. w:mNjzxO^V`;i=f;ibi;i`[>w:mNZ 8:wl`ef_8 mVo5d^V`8 ;ijzxfbi;<8:9]`6O^V` `6f;<9]8lb<8ljk8lZT`]= 8lb<mNZfu‡9]`m ζ(A)[T ] ^VZ0{ ;<9 ;iZ0f{ 8lx8lZ0{f8lZ?`mVo*`608KwI60mN;<w:84mVo l  ! 8q96O^Vbib3608lZ0w:8EoRmN_`6 ^N9I9efjk8q`6f;<9 ! 8Kw ^VZ Z0m {f8Eg0Z08  L(MA , s) = Y Lv (MA , N v −s ). v c0mN_^VZT> g0Zf;i`8K98l`mVo*xf_;ijk8:9 S  w:mNZ?`I^V;iZf;iZfu)`608q;iZ}g0Zf;i`8Kxf_;ijk8:9 † |098l` LS (MA , s) = Y Lv (MA , N v −s ). v∈S / 57608)^V=m ‡8)xf_mG{}e0wl`94w:mNZ ‡8l_u‡8 ;iZ `608k6O^Vbhoxfba^VZ08 Re(s) >> 0  ef_;iZ?`8l_8:9`;<9K`m _8lba^V`8k`608 9 x8:wl;a^Vb N^Vbie08:9mVo `608:9I8 LsoQefZ0wl`;<mNZ09`mk^Vbiu‡8l=f_I^V;<wKxf_mNx8l_`;<8:9mVo M      A #G '  .  38l` K/k =8zg0Zf;i`8 K^Vb<mN;<98}`8lZ09;<mNZ mVo7ZTefj4=8l_gO8lb<{f9 |2^VZ0{ b<8l` G =8z;i`9 q^Vb<mN;<94uv_mNefx
57608lZ |}`608l_8n;<9^ZO^V`ef_I^Vb ^Nwl`;<mNZmVo A = Q[G] mNZ `608njkmN`; ‡8 M = h (Spec(K))(j) #57608 _ 8lba^V`; ‡8 LsoQefZ0wl`;<mNZ L (M , s) ^V``I^Nw608:{ƒ`m `6f;<9njkmN`; ‡8 ;<9n`608`efxfb<8 (L (χ, j + s)) mVo _`;iZ Ls oQefZ0wl`;<mNZ09q^V``I^NwI608:{ƒ`mk`608;i__8:{}e0wl;i=fb<8w:mNjzxfb<8ƒ w6O^V_I^Nwl`8l_9qmVo G  576f;<9n`efxfb<8 I` ^ ‡8:9 N^Vbie08:9;iZ    S  0 A S b χ∈G  Y b χ∈G C ≃ ζ(C[G]) ≃ ζ(A ⊗Q C). ~  38l` E = 8^VZ 8lbibi;ixf`;<wwlef_ ‡8{f8Eg0Z08:{ m ‡8l_ k | ^VZ0{]b<8l` K/k =8^g0Zf;i`8 q ^Vb<mN;<98G `8lZ09;<mNZ mVo ZTefj4=8l_#gO8lb<{f9 ;i`6 K^Vb<mN;<92uv_mNefx G *576087uv_mNefx G ^VZ0{ 608lZ0w:87`608^Vbiu‡8l=f_I^ A = Q[G] ^Nwl`9 mNZ `608jkmN`; ‡8 M = h (E × Spec(K))(1) / 57608 Ls oQefZ0wl`;<mNZ L(M , s) ^V``I^Nw608:{ `m `6f;<9 jkmN`; ‡8 ;<9`608 `efxfb<8 (L(E ⊗ χ, 1 + s)) mVo ^N99I8&! 8l;ib Ls oQefZ0wl`;<mNZ09 ` ;<9`8:{ =?> ;i__8:{}e0wl;i=fb<8w:mNjzxfb<8] w6O^V_I^Nwl`8l_9nmVo G  
   1 A Spec(k)   b χ∈G  .  ( O- -
  ϑ ∞ 0c mN_ v ∈ S |}`608Kt8lbi;iuvZ08Kw:mN60mNjkmNb<mNuv> ;<9`608qw:mN60mNjkmNb<mNuv>]mVo `608Kw:mNjzxfb<8

  ∞ α v Hv (M ) ⊗Q R −→ (Hv (M ) ⊗Q k̄v )Gv /F 0 ;iZ0{}e0w:8:{ =T> `608K;iZ0wlbie09;<mNZ H (M ) ⊗ R ,→ H (M ) ⊗ k̄ 257608Kw:mNjzxO^V_;<9mNZ ;<9mNjkmN_xf6f;<9j v v Q v Q ∼ Hv (M ) ⊗Q k̄v −→ HdR (M ) ⊗k,v k̄v ;<9 G  8:r?ef; v^V_;a^VZ?`^VZ0{ 608lZ0w:8q`6084^V=m ‡8Kw:mNjzxfb<8]w ^VZ]=8 _;i``8lZ ^N9    v α M ⊕v∈S∞ (Hv (M ) ⊗Q R)Gv −→ ⊕v∈S∞ HdR (M ) ⊗k,v kv /F 0 . c0mN_B`6087jkmN`; ‡8 M = h (X)(r) mNZ08w ^VZk{f8Eg0Z08         H (k, M ) ^VZ0{ ;i`9*g0Zf;i`8 xO^V_` 9 H (k, M )  98:8  „ osmN_{f8l`I^V;ib<9 †    n  i i f 67-$89%,+ "  (          
          AR      ! r∗  B 0 → H 0 (k, M ) ⊗Q R −→ ker αM −→ Hf1 (k, M ∗ (1)) ⊗Q R)∗ ∗ r δ B −→ Hf1 (k, M ) ⊗Q R −→ coker αM −→ (H 0 (k, M ∗ (1)) ⊗Q R)∗ → 0 "    #      $      /) 0    1 2      '43 ! 8K9I8l`  % &'       () rB #  +*  , -  .&   δ Ξ(M ) := [Hf0 (k, M )]  [Hf1 (k, M )]−1  [Hf1 (k, M ∗ (1))∗ ] [Hf0 (k, M ∗ (1))∗ ]−1  v∈S∞ [Hv (M )Gv ]−1  [HdR (M )/F 0 ] 608l_8 [ ] : D (A) → V (A) ;<9`608q{f8l`8l_jz;iZO^VZ?`oRefZ0wl`mN_^N9{f8Eg0Z08:{ ;iZ]`608xf_8 G;<mNe099I8:wl`;<mNZ 576084^V=m ‡84w:mNZ 8:wl`ef_8:{ 8f^Nwl`98:rTe08lZ0w:8:9uv; ‡8:9^VZ];<9mNjkmN_xf6f;<9j    p  ϑ∞ : Ξ(M ) ⊗A AR ≃ 1VAR .  .%. #$" < ;iZ0w:8 H 0 (M ) ;<94{f8Eg0Z08:{ ^N9K`608 ;ij ^Vu‡8 mVo`608 wl>}wlb<8 wlba^N9I9Kj ^Vx ;i`;<9 GZ0m Z `m]=8 g0Zf;i`8 {};ijk8lZ09;<mNZO^Vb&K57608 ‡8:wl`mN_q9xO^Nw:8 H (M ) ;<9q^Vb<9m 8Gx8:wl`8:{ `m =8Kg0Zf;i`8 {};ijk8lZ09;<mNZO^Vb |=fef` `6f;<9*;<9*Z0mN` TZ0m Zz;iZ u‡8lZ08l_I^Vb&2 Y ` ;<9 TZ0m Zz;iZ `608w ^N98:9 /8^V_87;iZ?`8l_8:9`8:{ ;iZ  ^VZ0{ ;iZz9mNjk8`_; G;a^Vb w ^N98:9 †  c0mN_ M = h (Spec(K))(r) | mNZ0846O^N9 H (M ) ≃ K (O ) ⊗ Q | 6f;<w6 ;<9 TZ0m Z `m =8Kg0Zf;i`8z{};ijk8lZ09;<mNZO^Vb2=T> ^ 57608:mN_8lj mVo
ef;ibib<8lZ  wEo 0/ .  † K cfef_`608l_ | oRmN_q^ 9jkmTmN`6ƒxf_m 8:wl`; ‡8 wlef_ ‡8 X/k ^VZ0{ M = h (X)(1) | mNZ086O^N9 H (M ) = Pic (X)(k) ⊗ Q | 6f;<wI6 ;<9 GZ0m Z `m =8 0g Zf;i`8K{};ijk8lZ09;<mNZO^Vbd=?>‚ mN_{f8lbib &! 8l;ib 57608:mN_8lj  f      1 f  0  1 f    1 2r−1 K   Z   0 1 f   Z #G '  . cOmN_ M = h0 (Spec(K))(0) | K/k ^g0Zf;i`8 K^Vb<mN;<98}`8lZ09;<mNZ mVo*Z?efj= 8l_gO8lb<{f9 ;i`6 q^Vb<mN;<9 uv_mNefx G |0mNZ08q6O^N9 Ξ(M ) = [US∞ ⊗Z Q]  [XS∞ ⊗Z Q]−1 608l_8osmN_K^ g0Zf;i`8z98l` S mVoxf_;ijk8:9 | U {f8lZ0mN`8:9K`608uv_mNefx mVo S  efZf;i`9q;iZ K | ^VZ0{ X = # 57608j ^Vx α ;<9`608_8luvefba^V`mN_j ^Vx ker(⊕ Z → Z)  S v∈S S M US∞ ⊗Z R → XS∞ ⊗Z R, 6f;<wI6 ;<9{f8Eg0Z08:{]=?>  u 7→ − X log |u|v · v. 57608K;<9mNjkmN_xf6f;<9j ϑ : Ξ(M ) ⊗ R[G] ≃ 1 ;<97;iZ0{}e0w:8:{]=T>)`608qj ^Vx α  ~  cOmN_ M = h (E )(1) | 608l_8 E ;<9`608q=O^N98KwI6O^VZfu‡84mVo#^VZ 8lbibi;ixf`;<w4wlef_ ‡8 E {f8Eg0Z08:{ m ‡8l_ |0mNZ086O^N9 k ∞ 1 v∈S VR[G] M  K   K Ξ(M ) = [E(K) ⊗Z Q]  [E(K)∗ ⊗Z Q]  [H 0 (EK , Ω1 )∗ ]  [H1 (EK (C), Q)+ ] 608l_8 ∗ {f8lZ0mN`8:9/`608{}eO^Vb^VZ0{ + {f8lZ0mN`8:9/`6089ef=fjkmG{}efb<8g f8:{k=?>z`608nw:mNjzxfb<8 w:mNZ efu?^% ` ;<mNZ2Y[Zz`6f;<9#w ^N98v|‡`608;<9mNjkmN_xf6f;<9j ϑ ;<9#mN=f`I^V;iZ08:{ T;a^n`608  l_mNZ  5 ^V`8 † 608l;iuv6?`/xO^V;i_;iZfu ^VZ0{]`608qx 8l_;<mG{;<9mNjkmN_xf6f;<9j    ∞  .~  ( O- -
  ϑ l 8l` A = 8^VZ R jkmG{}efb<8;iZ A  Z A 9ef=fjkmG{}efb<8 T mVo V ;<99I^V;<{`m = 8q^VZ A ba^V``;<w:8q;iZ V ;ho ;i`7;<9 =mN`6 g0Zf;i`8lbi> u‡8lZ08l_I^V`8:{ ^VZ0{oRefbib  `6O^V`;<9 | T ⊗ A = V †  Z A 9`_e0wl`ef_8 T mNZ M ;<9^498l` {T : v ∈ S } 608l_8v|‡osmN_/8 ^NwI6 v ∈ S , T ;<9^VZ A ba^V``;<w:8 i; Z H (M ) ^VZ0{ oRmN_8 ^Nw6ƒxf_;ijk8 l ∈ Spec(R) `608;ij ^Vu‡8 T mVo T ⊗ Z efZ0{f8l_`608 w:mNjzxO^V_;<9mNZ <; 9mNjkmN_xf6f;<9j H (M )⊗ Q ≃ H (M ) ;<93= mN`6;iZ0{f8lx 8lZ0{f8lZT`*mVo v ^VZ0{ G  9`I^V=fb<8# Z A 9`_e0wl`ef_8 ;<9xf_m 8:wl`; ‡8q;ho28 ^Nw6 T ;<9^xf_m 8:wl`; ‡8 A jkmG{}efb<8# 8l` T =8^xf_m 8:wl`; ‡8 A 9`_e0wl`ef_8KmNZ M 2 cOmN_^VZ?> xf_;ijk8 l b<8l` S = S ∪ {xf_;ijk8:9^V=m ‡8qb }  c0mN_^VZT> w:mNZ?`;iZ?e0mNe09 G  jkmG{}efb<8 N b<8l`

 
A  v ∞ ∞ v v l v Q l v l Z l k   v 
 l S RΓ(Ok,Sl , N ) := C • (GSl , N ), RΓc (Ok,Sl , N ) := Cone (RΓ(Ok,Sl , N ) → ⊕v∈Sl C • (Gv , N )) [−1]. Z08q6O^N9^ w:mNZ 8:wl`ef_I^Vb3;<9ImNjkmN_xf6f;<9j ∼ ϑl : Al ⊗A Ξ(M ) −→ [RΓc (Ok,Sl , Vl )] ≃ Al ⊗Al [RΓc (Ok,Sl , Tl )]. ! 87_8Eos8l_2`6087_8 ^N{f8l_#`m  „ < 8:wl`;<mNZ/osmN_2`608{f8l`I^V;ib<9/^V=mNef`2`608w:mNjzxfb<8 RΓ d57608;<9ImNjkmN_xf6f;<9j {f8Eg0Z08:9^VZ 8lb<8ljk8lZ?` ([RΓ (O , T )], Ξ(M ); ϑ ) ;iZ V (A ) × V (A) 757608K;<9mNjkmN_xf6f;<9j ϑ c wlba^N997mVod`6f;<978lb<8ljk8lZT`;<9;iZ0{f8lx 8lZ0{f8lZT`mVo3`608w60mN;<w:8qmVo3`60898l` S mN_7`608nwI60mN;<w:8qmVod`608nxf_m 8:wl`; ‡8 9`_e0wl`ef_8 T |0^VZ0{]`608l_8EoRmN_8 8Kw ^VZ {f8Eg0Z08qefZ0{f8l_w:8l_`I^V;iZ w:mN608l_8lZ0w:8q6T>TxmN`608:9;<9  wEo  „ † c l K,Sl l l l V (Al )   (Ξ(M ), ϑ∞ ) := Y [RΓc (OK,Sl , Tl )], Ξ(M ), Y p p ϑp ; ϑ ∞ ! . ! 8 96O^Vbib{f8lZ0mN`8 `608 ;<9ImNjkmN_xf6f;<9j wlba^N9I9 mVok`6f;<9ƒ8lb<8ljk8lZ?`ƒ;iZ π (V (A; R)) ≃ K (A; R) =T>  RΩ(M, A)  0 0 #G '  . 9 =8EoRmN_8v|/mNef_ g0_9`k8f^Vjzxfb<8];<9 M = h (Spec(K))(0) | 608l_8 K/k ;<9 ^ g0Zf;i`8 q^Vb<mN;<9 8} `8lZ09;<mNZ4mVofZTefj4=8l_ gO8lb<{f9 Y Zq`6f;<9dw ^N98 U ⊂ U ⊗Q ^VZ0{ X ⊂ X ⊗Q ^V_8 q^Vb<mN;<9 9`I^V=fb<8  jkmG{}efb<8:9 |G=fef``608l>)^V_8Z0mN`xf_m 8:wl`; ‡8# m /8 ‡8l_ |}`6085 ^V`898:r?e08lZ0w:8:97^N99mGwl;a^V`8:{)`m Z[G] `608n8} `8lZ09;<mNZ K/k xf_m T;<{f8:9^4w ^VZ0mNZf;<w ^Vb xf_m 8:wl`; ‡8 Z[G] 9`_e0wl`ef_8ne09;iZfu 6f;<wI6 mNZ08w ^VZ osmN_j4efba^V`84`608Kw:mNZ 8:wl`ef_8# ! 8 ;ibib398:8`6f;<9;iZ {f8l`I^V;ib ;iZ `608qZ08} `w6O^Vxf`8l_  0 S    S  S   S   ./ ~  ef_n98:w:mNZ0{]80^Vjzxfb<8q;<9 M = h (E )(1) | 608l_8 E ;<9=O^N9I8Kw6O^VZfu‡8mVo2^VZ 8lbibi;ixf`;<wwlef_ ‡8 {f8Eg0Z08:{ m ‡8l_ k d Y[Z `6f;<92w ^N98 H(σE (C), Z) ⊂ H(σE (C), Q) {f8Eg0Z08:9B^xf_m 8:wl`; ‡8 Z[G] 9`_e0wl`ef_8# Y Z y 6O^Vxf`8l_ 01 | /8 ;ibib w:mNZ09;<{f8l_n`6084w:mNZ 8:wl`ef_8KosmN_`6f;<9xO^V_`;<wlefba^V_xf_m 8:wl`; ‡8  9`_e0wl`ef_8# Z[G]  1  K K   K  K   /
 1 -
/.021  4 y Nm Z09;<f{ 8l_` 608Kb<mNZfu 8f^Nwl`9I8:r?e08lZ0w:8q;iZ K  `608:mN_>

··· / K1 (AR ) δ / K0 (A; R) / ··· / K0 (A) nr  ζ(AR )× Y[Z  „ | ef_Z09^VZ0{]c ba^Nw6 6O^ ‡84w:mNZ09`_e0wl`8:{ ^zw ^VZ0mNZf;<w ^Vb3j ^Vx  ~ . † δb : ζ(AR )× −→ K0 (A; R) 9e0wI6]`6O^V` δb ◦ nr = δ  9 8K9I^ ;iZ]`608 < 8:wl`;<mNZ ~  ~ |0`608qb<8 ^N{};iZfu w:mG8  wl;<8lZ?`mVo*`608 LsoRefZ0wl`;<mNZ ^V``I^Nw608:{ƒ`m `608jkmN`; ‡8 M bi;<8:9n;iZ ζ(A ) q57608 8:r?ef; v^V_;a^VZ?`Kw:mNZ 8:wl`ef_8_8lba^V`8:9`608;ij ^Vu‡8mVo ` 6f;<9b<8 ^N{};iZfu w:mG8  wl;<8lZ?`efZ0{f8l_`608qj ^Vx δb `m `608KmN= 8:wl` RΩ(M, A) ∈ K (A; R) w:mNZ09`_e0wl`8:{];iZ ` 608qxf_8 G;<mNe0998:wl`;<mNZ    R  × 0  67-$89%,+ "  2 + "   '#$ 4       1       1     3  1     3    .   ' L  '      L(M, s) ords=0 L(M, s)    -         (   ,    !&       " ' s=0 3 Spec(ζ(AC ))  1  ords=0 L(M, s) = rrA (Hf1 (K, M ∗ (1))∗ ) − rrA (Hf0 (K, M ∗ (1))∗ ) "   rr    (   A  3 !       "#  %     (       3 3  L∗ (M, 0) := lims→0 s−ords=0 L(M,s) L(M, s) b ∗ (M, 0)) ∈ K0 (A; R) L(M, A) := δ(L .1 -
 T Ω(M, A) := L(M, A) + RΩ(M, A).  ,) 3 
! T Ω(M, A) ∈ K0 (A; Q)  .     " T Ω(M, A) = 0 3 3 #$"* . 57608z^V=m ‡8 w:mNZ 8:wl`ef_8;<9^ u‡8lZ08l_I^Vbi; ^V`;<mNZ `m `608jkmN`; ‡8:9 ;i`6 Z0mNZ0w:mNjzj4ef`I^V`; ‡8kw:mG8   wl;<8lZT`9mVo2w:mNZ 8:wl`ef_8qoRmN_jefba^V`8:{]=?> c0mNZT`I^V;iZ08^VZ0{]\ 8l__;iZ  p;<mNe | ^VZ0{ D ^V`m  wEo  ~ / | ~ = †  ~ 576084w:mNZ 8:wl`ef_8mVo b<mTwI6 ^VZ0{ D ^V`mz;<9`608K9x8:wl;a^Vb w ^N98qmVo `6f;<9w:mNZ 8:wl`ef_8 ;i`6 A = Z  Y[Z`608Z08G `` /mkwI6O^Vxf`8l_9 /8q96O^Vbib w:mNZ09;<{f8l_` mzxO^V_`;<wlefba^V_jkmN`; ‡8:9^VZ0{uv; ‡8K^_8l;iZT`8l_xf_8 `I^V` ;<mNZ mVo*`608Kw:mNZ 8:wl`ef_8#       .=
   

 
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        8l` K/k =8 ^ g0Zf;i`8 q^Vb<mN;<98G`8lZ09;<mNZ mVoZ?efj= 8l_gO8lb<{f9 ;i`6 q^Vb<mN;<94uv_mNefx G  ef_;iZ?`8l_8:9` ;<9 ;iZ `608 jkmN`; ‡8 M = h (Spec(K))(0) | 6f;<w6 /8 _8lu?^V_{ ^N9z^ jkmN`; ‡8 {f8Eg0Z08:{ m ‡8l_ k ;i`6 w:mG8  wl;<8lZ?`9;iZ A = Q[G]  9#Z0mN`8:{ ;iZk`608 y 6O^Vxf`8l_ ~ |?`608 Ls oQefZ0wl`;<mNZ^N99mGwl;a^V`8:{ ;i`6 `6f;<9BjkmN`; ‡8;<9^q`efxfb<8mVo _`;iZ s oRefZ0wl`;<mNZ09B 5dmk=8xf_8:wl;<98v|0b<8l` S = 84^g0Zf;i`898l`mVo*xf_;ijk8:9;iZ k |O^VZ0{]b<8l` S := S = 8K`608K98l` L mVo3xf_;ijk8:97;iZ K bi>G;iZfuk^V= m ‡8`608nxf_;ijk8:97;iZ S * Y o ρ : G → GL(V ) ;<9^4w:mNjzxfb<8 _8lxf_8:98lZT`I^V`;<mNZ w:mN__8:9xmNZ0{};iZfu `mk^ wI6O^V_I^Nwl`8l_ χ ∈ Gb `608lZ 
  0       K k  k LS (χ, s) := Y p∈Sk det(1 − FrP N p−s |V IP )−1 ;<9`608 _`;iZ LsoRefZ0wl`;<mNZ  _8lba^V`; ‡8`m S † ^V``I^Nw608:{ `m `6084w6O^V_I^Nwl`8l_ χ  8l_8 P ;<9n^VZƒ^V_=f;i`_I^V_> fx _;ijk8q;iZ K bi>G;iZfu ^V= m ‡8 p ^VZ0{ I ;<9`608K;iZ08l_`;a^zuv_mNefx mVo P B57608qjkmN`; G;<w LsoQefZ0wl`;<mNZ];<9`608lZ vu ; ‡8lZ]=T>     P  LS (M, s) = (LS (χ, s))χ∈Gb . 576f;<9 LsoRefZ0wl`;<mNZ 6O^N9^4jk8l_mNjkmN_xf6f;<w4w:mNZ?`;iZ?eO^V`;<mNZ `m`608 60mNb<8 s xfba^VZ08K^VZ0{ `608l_8EoRmN_8 8w ^VZ :w mNZ09;<{f8l_;i`9q5d^ >Tb<mN_K9I8l_;<8:9n8}xO^VZ09;<mNZ ^V` s = 0 qcOmN_q`608_8:9`nmVoB`6f;<9qwI6O^Vxf`8l_ /8z96O^Vbib*9efxf_8:99 ` 608qjkmN`; ‡8 M oQ_mNj mNef_Z0mN`I^V`;<mNZ ^VZ0{ _;i`8 L (0) osmN_`608qb<8 ^N{};iZfu w:mG8  wl;<8lZ?`      ∗ S L∗S (M, 0) = (L∗S (χ, 0))χ∈Gb ∈ Y b χ∈G R ≃ ζ(R[G])× . nm K|0mNZ `608K^V_;i`6fjk8l`;<w49;<{f8 8q6O^ ‡8    .€  Ξ(M ) = [Hf0 (k, M )]  [Hf1 (k, M ∗ (1))∗ ]  v∈S∞ [Hv (M )Gv ]−1 . 57608KmN`608l_`8l_jk9`6O^V`^Vxfx 8 ^V_;iZ]`608q{f8Eg0Zf;i`;<mNZ mVo Ξ(M ) ^V_8`_; G;a^Vb& Z08q6O^N9  Hf0 (k, M ) := H 0 (k, M ) ≃ Q, ^VZ0{ ⊕v∈S∞ Hv (M )Gv ≃ (⊕v∈S∞ Z) ⊗Z Q. c0mN_^VZ 9ef=f_;iZfu A mVo Q[G] |0b<8l` A {f8lZ0mN`8q`608q;ij ^Vu‡84mVo A efZ0{f8l_`608 Q bi;iZ08 ^V_j ^Vx # # : Q[G] −→ Q[G] #  X X X ag g  := ag g −1 . ag g 7→  g∈G g∈G c0mN_^ A  jkmG{}efb<8 P |fb<8l` P = A ⊗ op # # Aop P g∈G 257608lZ |OmNZ08q6O^N9 Hf1 (k, M ∗ (1))∗ ≃ (US∞ ⊗Z Q)∗# . c0mN_n^VZ?> g0Zf;i`8K98l` T mVo*xf_;ijk8:9;iZ K |}b<8l` X {f8lZ0mN`84`608 ‡8l_Z08lb3mVo*`6084^Vefuvjk8lZ?`I^V`;<mNZ j ^Vx  T ⊕v∈T Z −→ Z X (av )v∈T 7→ av . 57608lZ |}oR_mNj `608K^V=m ‡8KmN=098l_ v^V`;<mNZ | 8qu‡8l`   v∈T  Ξ(M ) = [(US∞ ⊗Z Q)∗# ]  [(XS∞ ⊗Z Q)∗# ]−1 . 53mK_8lba^V`8`6089x8:wl;a^Vb v^Vbie08:9BmVo`608 LsoRefZ0wl`;<mNZk_8lba^V`; ‡8`m S | /8w:mNZ09;<{f8l_B`6087osmNbib<m ;iZfu_8lba^V`; ‡8 ‡8l_9;<mNZ mVo Ξ(M )       ΞS (M ) = [(US ⊗Z Q)∗# ]  [(XS ⊗Z Q)∗# ]−1 . nm K|`m w:mNZ09;<{f8l_ `608 ;iZT`8luv_I^Vbi;i` > xO^V_`mVo`608 8:rTef; v^V_;a^VZ?`]w:mNZ 8:wl`ef_8v| 8 7^VZ?` `m g0Z0{      .,E xf_m 8:wl`; ‡8 Z [G] 9`_e0wl`ef_8:9n`6O^V`uv; ‡84`608qZ08:w:8:99I^V_>]^Vbiu‡8l=f_I^V;<w48lb<8ljk8lZT`;iZ K (Z [G]; Q ) /57608 so mNbib<m ;iZfu 57608:mN_8lj mVo75d^V`8 ^Vbib<m 9qe09`m_8lxfba^Nw:8 U ^VZ0{ X =T> Z[G] jkmG{}efb<8:9KmVoBg0Zf;i`8 xf_m  8:wl`; ‡8K{};ijk8lZ09;<mNZ   0 l  l l         S S  374 -"   ( 3 #   ? ; 
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  +   '           ,  -   Z[G]     # / . † 0 → US → A → B → X S → 0  ! ) 3 )   
   ' Ext2G (XS , US ) " A B  > `6084^V= m ‡8K57608:mN_8lj]| 8qu‡8l`^VZ 8lb<8ljk8lZT`   [A, B; ψS,R ] ∈ K0 (Z[G]; R) 608l_8 ψ ;<97mN=f`I^V;iZ08:{)=T>z`6089Iw ^Vba^V_8}`8lZ09;<mNZmVo3`60880^Nwl`9I8:r?e08lZ0w:8  /  . † ^VZ0{)`608_8luvefba^V`mN_ j ^Vx  S,R RS : US ⊗Z R → XS ⊗Z R. 57608/;iZ?`8luv_I^Vbi;i`[>xO^V_`*mVo y mNZ 8:wl`ef_8 ~ oRmN_ `608BjkmN`; ‡8 M = h (Spec(K))(0) ;i`6zw:mG8  wl;<8lZ?`9 i; Z A = Q[G] w ^VZ Z0m = 8_8:9`I^V`8:{ ^N9  0   67-$89%,+ "  ? #$" b ∗ (0)# ) − [A, B; ψS,R ] = 0  K0 (Z[G]; R) 3 δ(L S > 38ljzj ^ =];iZ  „ | 8)6O^ ‡8 `6O^V``6088:rTef; N^V_;a^VZT`zw:mNZ 8:wl`ef_8 {fmG8:9Z0mN`{f8lx8lZ0{
   mNZ `6084w60mN;<w:8 mVo S 57608l_8EosmN_8;i`;<98lZ0mNefuv6 `m xf_m ‡8`6084w:mNZ 8:wl`ef_8oRmN_^zxO^V_`;<wlefba^V_q98l` S mVo fx _;ijk8:9         "  " 374 -"   2  + "    '#$ 4    T Ω(K/Q) = 0 3 "' 4 " (  22                  K/Q  < mf|`608 ;iZT`8l_8:9`ZO^V`ef_I^Vbibi>$;<9 `m ‡8l_;hoR> `608 w:mNZ 8:wl`ef_8ƒ;iZ Z0mNZO^V=8lbi;a^VZ w ^N9I8:9 Y[Z 0= |^VZ ^Vbiu‡mN_;i`6fj ;<9{f8 ‡8lb<mNx 8:{ƒ`m ‡8l_;hoR>]`608 w:mNZ 8:wl`ef_8 ;iZƒ`608 {};i608:{}_I^Vb#w ^N98# n9;iZfu`6f;<9q^Vbiu‡mN_;i`6fj]|   .„ ;i`z;<9z960m Z `6O^V`k`608 w:mNZ 8:wl`ef_8;<9k`_e08 oRmN_k^Vbib q^Vb<mN;<9k8}`8lZ09;<mNZ09 L/Q ;i`6 Gal(L/Q) `608 {};i608:{}_I^VbOuv_mNefx mVo mN_{f8l_ € * 57608^V_uvefjk8lZT`9/;iZ ‡mNb ‡8_8:{}e0wl;iZfu`608w:mNZ 8:wl`ef_8`m^VZ)^V= 8lbi;a^VZ w ^N98 0e 9;iZfuk`608j ^Vx     K0 (Z[Dn ], R)tors → K0 (Z[Cn ], R)tors , 6f;<w6 ;<9xf_m ‡8lZ `mz= 8;iZ 8:wl`; ‡8# < mf|f`608KZ08G `uv_mNefx]`mk= 8qw:mNZ09;<{f8l_8:{];<9`608 eO^V`8l_Zf;<mNZ uv_mNefx Q 2Y Z `6f;<9w ^N98qmNZ08q6O^N9    8    # (  + "    #$ 4           
  K0 (Z[Q8 ], R) → K0 (MQ8 , R) × K0 (Z[Q8 ]) × K0 (Z[Qab 8 ], R), "    M Q8             '  Z[Q8 ] )     3 576Te09B`608 N^VZf;<96f;iZfuxf_mN=fb<8lj ;iZ K (Z[Q ], R) _8:{}e0w:8:9B`m^ v^VZf;<96f;iZfu4xf_mN=fb<8lj&;iZk`608`6f_8:8 uv_mNfe x09mNZ `608n_;iuv6?`6O^VZ0{ 9;<{f8#257608 N^VZf;<96f;iZfuz;iZ K (M , R) ;<9 < `_mNZfu < `I^V_ |0^VZ0{;<9 GZ0m Z {}e08]`m 5d^V`8v|9;iZ0w:8 ^Vbib`608 w6O^V_I^Nwl`8l_9mVo Q ^V_8 _I^V`;<mNZO^Vb& 57608 v^VZf;<96f;iZfu ;iZ K (Z[Q ]) ;<9 y 6f;iZ?=fef_uGF 9 Ω w:mNZ 8:wl`ef_8# 576f;<9;<9 GZ0m Z osmN_^VZ ;iZ}g0Zf;i`84o ^Vjz;ibi>]mVo28} `8lZ09;<mNZ09nw:mNZ09`_e0wl`8:{ =?> y 6f;iZ?=fef_u ;iZ . ~   Y Z   | `608 v^VZf;<96f;iZfu;iZ K (Z[Q ], R) ;<9960m Z oRmN_`6f;<9;iZ}g0Zf;i`8o ^Vjz;ibi> mVo 8} `8lZ09;<mNZ09# 576Te09 |f`608qw:mNZ 8:wl`ef_8K;<9 GZ0m Z osmN_^VZ];iZ}g0Zf;i`8o ^Vjz;ibi>mVo Q  8G `8lZ09;<mNZ09  ! 8 ^V_8]`608l_8EosmN_8 ;iZT`8l_8:9`8:{$;iZ `608]Z08} `kZ0mNZO^V=8lbi;a^VZ w ^N98v|7`6O^V`);<9 |`608 w:mNZ 8:wl`ef_8 m ‡8l_  8G `8lZ09;<mNZ09  A 57608^V=m ‡8 8ljzj ^o ^V;ib<93`mn60mNb<{oRmN_ A  98:8 xfx 8lZ0{}; K osmN_*{f8l`I^V;ib<9 † * 57608l_8EoRmN_8v|v`6087rTe08:9`;<mNZ ^V_;<98:9 608l`608l_ /8Kw ^VZ {};i_8:wl`bi>w:mNjzxfef`8 T Ω(K/Q) =?>w:mNZ09`_e0wl`;iZfuk`60845d^V`8 < 8:rTe08lZ0w:8## 57608 ^VZ09 8l_;<9>‡8:9^VZ0{ `6f;<9;<9 6O^V` /8496O^Vbibd^Nw6f;<8 ‡84=?> `608K8lZ0{]mVo*`6f;<9 y 6O^Vxf`8l_ p8:9efbi`99;ijz;iba^V_`m 57608:mN_8lj ~ osmN_^V= 8lbi;a^VZ 8G `8lZ09;<mNZ09m ‡8l_;ij ^Vuv;iZO^V_>]rTeO^N{}_I^V`;<wKgO8lb<{f9n^V_8 GZ0m Z {}e08q`m  b<8l>]^VZ0{ ‡mN6fZ09mNZ  98:8 0/ ^VZ0{ ~ 1  †    0 8   0   Q8  8  0   8  0   ab 8  8  4 
4         

    II 
 I     !    "  !     8l` K/Q = 8q^g0Zf;i`8 q^Vb<mN;<98}`8lZ09;<mNZ ;i`6 q^Vb<mN;<97uv_mNefx G 2c0mN_^VZ?> xfba^Nw:8 v mVo K {f8lZ0mN`8q=T> `608 {f8:w:mNjzxmv9;i`;<mNZ uv_mNefx mVo v  < efxfx mv98`6O^V` S ;<9K^ g0Zf;i`8 98l`qmVo/xf_;ijk8:9;iZ K 9I^V`;<9 oR>G;iZfu G `608osmNbib<m ;iZfukxf_mNx8l_`;<8:9 
v  . .  Cl ;<9w:mNxf_;ijk8q`m #G   S ~  S ;<99`I^V=fb<8qefZ0{f8l_`608 G ^Nwl`;<mNZ /  S w:mNZT`I^V;iZ09n^Vbib3`608;iZ}g0Zf;i`8Kxf_;ijk8:9^VZ0{ ^Vbib3`608xf_;ijk8:9;iZ K 6f;<w6 _I^Vjz;hoQ>  1  57608l_848G ;<9`9 v ∈ S oRmN_ 6f;<wI6 G = G  8l` S = 8K^z9I8l`mVo _8lxf_8:98lZT`I^V`; ‡8:9osmN_`608 G mN_=f;i`9;iZ S 2 c0mN_ v ∈ S b<8l`   0 v0 
0 0 0→ v (−2) → v →  →Z→0 v =8B^VZ480^Nwl`398:rTe08lZ0w:8#mVo G  jkmG{}efb<8:93;iZ 6f;<wI64`608Bjz;<{f{}b<8#`8l_jk9d^V_82oQ_8:8B^VZ0{Kg0Zf;i`8lbi>u‡8lZ08l_I^V`8:{  Y[Z0{}e0wl;iZfuk`6084^V=m ‡8K98:rTe08lZ0w:8oR_mNj G `m G | /8qu‡8l`  v   v 0 → Yv (−2) → Av → Bv → Yv → 0, ^VZ 80^Nwl`98:rTe08lZ0w:84mVo G jkmG{}efb<8:975760848};<9`8lZ0w:84mVo v ;ijzxfbi;<8:9`6O^V` 84w ^VZ ;<{f8lZT`;hoQ> X ;i`6 2 576?e099efjzjz;iZfu)^Vbib `608q9I8:r?e08lZ0w:8:9n^V= m ‡8Km ‡8l_`608498l` S \ {v } /8qu‡8l` ⊕ Y   0  S   v∈S0 \{v0 } v 0 0 0 → X(−2) → ⊕v∈S0 \{v0 } Av → ⊕v∈S0 \{v0 } Bv → XS → 0, 608l_8 X(−2) = ⊕  Y (−2) c0mN_ v ∈ S b<8l` S(v) =8^)98l`mVoB_8lxf_8:98lZT`I^V`; ‡8:9oRmN_ G  mN_=f;i`9mVo S 9e0w6ƒ`6O^V` S ⊂ S(v)  8l` J {f8lZ0mN`8`608uv_mNefx mVo S  ;<{f8lb<8:9/^VZ0{ U {f8lZ0mN`8`608uv_mNefx mVo S  efZf;i`9#;iZ K *57608lZk=T> wlba^N9I9 Og 8lb<{ `608:mN_>  wEo  .  | 0/ ~  † /8K6O^ ‡8  v∈S0 \{v0 } v  0 v 0
S S   Ext2G (Yv , JS ) = H 2 (Gv , JS ) = ⊕w∈S(v) H 2 (Gv ∩ Gw , Kw∗ ) 608l_8 K ;<9 `608b<mTw ^Vbi; ^V`;<mNZkmVo K ^V` w  c0mN_#^9ef=fuv_mNefx H mVo G b<8l` inv(H, w) : H (H, K ) → =8`608q;iZ v^V_;a^VZ?`j ^Vx#57608q;ij ^Vu‡8KmVo*`6f;<9j ^Vx];<9u‡8lZ08l_I^V`8:{ =?>  Q/Z 57608q;iZ0wlbie09;<mNZ U → J ;iZ0{}e0w:8:9^VZ ;iZ 8:wl`;<mNZ  wEo  .  |}x € 1 †  2 w  1 #H S S Ext2G (Yv , US ) = H 2 (Gv , US ) → H 2 (Gv , JS ). ∗ w ~v 57608q;ij ^Vu‡8Kw:mNZ09;<9`9mVo#^Vbib β ∈ H (G , J ) = ⊕ 2 v X w∈S(v) S 9I^V`;<9 oR>G;iZfu 2 ∗ w∈S(v) H (Gv ∩ Gw , Kw ) inv(Gv ∩ Gw , w)(β) = 0. nm K|*osmN_z8 ^Nw6 v ∈ S \ {v } |Bw60mGmv98 ^]j ^Vx f : Y (−2) → U w:mN__8:9xmNZ0{};iZfu `m β ∈ 9e0w6 `6O^V` H (G , J ) = ⊕ H (G ∩ G , K )   0 2 v S 0 w∈S(v) 2 v v w v S ∗ w ;ho w = v | h; o w = v | inv(G ∩ G , w)(β) = −    Nm `608l_ ;<9I8#  0 y mN4j =f;iZf;iZfuz^Vbib9e0w 6 f F 9 /8 u‡8l`7^j ^Vx f : X(−2) → U * 576f<; 9#j ^Vx f _8lxf_8:9I8lZ?`97^VZ 8G`8lZ09;<mNZ wlba^N99 β ∈ Ext (X , U ) k57608lZ =T>  . ~  | \2_mNx /  ~  . | β w:mN__8:9xmNZ0{f9q`m `608 9I^Vjk8z8G `8lZ09;<mNZ wlba^N99^N9 `6O^V`mVo2^ 5 ^V` 8498:rTe08lZ0w:8 v  1     #(Gv ∩Gw ) 1 #(Gv ∩Gw ) w 0   v 2 G S S S 0 → US → A → B → XS → 0. nm K| b<8l` N = 8^zoR_8:8 Z[G] jkmG{}efb<8 9e0w6 `6O^V`n`608l_88};<9`9^zj ^Vx f˜ : X(−2) ⊕ N → U | 6f;<w 6 ;<99ef_ 8:wl`; ‡8##57608lZ 8q6O^ ‡8q`608qoRmNbib<m ;iZfu 8f^Nwl`98:r?e08lZ0w:8:9   S      0 → ker(f˜) → X(−2) ⊕ N → US → 0, 0 → X(−2) → AX → BX → XS → 0, 608l_8 A = ⊕ ^VZ0{ B = ⊕ A 0/%= |f576fj  =  . † /8qu‡8l`^ 5 ^V`8K98:rTe08lZ0w:8  X  v∈S0 \{v0 }  v X v∈S0 \{v0 } Bv )57608lZ osmNbib<m ;iZfu 5 ^V`8%F 9^V_uvefjk8lZT`  wEo  0 → US → (AX ⊕ N )/ker(f˜) → BX → XS → 0.      !
   ! H (V , K ) 2 ∗ v 4 Y[Z4`6f;<9*9ef=098:wl`;<mNZ /896O^Vbibf^N99efjk8/`6O^V` K/Q ;<9 ^ q^Vb<mN;<9*8G`8lZ09;<mNZ ;i`6 q^Vb<mN;<9 uv_mNefx G ≃ A  8l` V =8`608 D b<8l;iZ oRmNef_Buv_mNefx)9;i``;iZfuK;iZ09;<{f8 G  ! 896O^Vbibw:mNjzxfef`8`6088lb<8ljk8lZT`97mVo `608uv_mNefx ^N9jkmN_xf6f;<9jk9q^V_;<9;iZfu)oQ_mNj 9mNjk8 V  _8:9mNbief`;<mNZ mVo Z  / > b<mGw ^Vb*wlba^N99gO8lb<{ `608:mN_> H (V , K )  wEo 0/ ~  † /8 TZ0m `6O^V` H (V , K ) ;<9wl>Gwlbi;<w mVomN_{f8l_ #V = 4  ! 8 ;ibib/=8 ;iZT`8l_8:9`8:{ ;iZ   4
4 2 4 ∗ v  4   2 4 ∗ v  4 ~. GZ0m ;iZf)u ^z9e  wl;<l8 Z?`:w mNZ0}{ ;i`;<Nm Z mN]Z jkmN_ xf6f<; 9jk9 9m `6O^V``608l>w:mN__8:9x mNZ0{`mz`608K`_; T;a^Vb 8lb<8ljk8lZT` ^VZ0{ ^u‡8lZ08lI_ ^V`mN_nVm o H (V , K )  8l` V = {1, g , g , g }  y mNZ09;<{f8l_`608K80 ^Nwl`98:r?e08lZ0w:84mVo Z[V ] jkmT{}efb<8:9    2 ∗ v 4
4 1 2 3 4 δ δ δ 3 2 1 0→M → Z[V4 ] ⊕ Z[V4 ] → Z[V4 ] → Z → 0, {f8Eg0Z08:{ =T> • δ1 (1) = 1 • δ2 ((1, 0)) = 1 − g2 , δ2 ((0, 1)) = 1 − g3 ‡8l_ (δ ) ^VZ0{ δ ;<97`608q;iZ0wlbie09;<mNZ nm K|f`06 8 ‡l8 _ (δ ) ;<9u‡8lZ08l_I^V`8:{]=?> • M=    2 3  2 β1 = (−1 − g1 , 1 + g1 ), β2 = (1 + g2 , 0), β3 = (0, 1 + g3 ). 576?e09 M ≃ 608l_8 G ≃ V /{1, g }  ! 8q{f8lZ0mN`8=?> α `608nu‡8lZ08l_I^V`mN_  ;& 8v|G`608 4j efbi`;ixfbi;<w ^V`; ‡8;<{f8lZ?`;i`[> † mVo Z[G ] 257608lZ N ;<9u‡8lZ08l_I^V`8:{ =?> (1 + g )(α ) − (1 + g )(α + α )  8l` K /E =84^VZ 8G `8lZ09;<mNZ mVo2b<mTw ^Vb gO8lb<{f9 ;i`6]`608 q^Vb<mN;<9uv_mNefx V  8l` F , i = 1, 2, 3 =8K`608g } 8:{]gO8lb<{f9w:mN__8:9x mNZ0{};iZfu`m)`6089ef=fuv_mNefx09 {1, g }  ! 8 ^Vb<9m^N99efjk8`6O^V` F /E ;<9 I_ ^Vjz;hgO8:{ osmN_ i = 2, 3 ^VZ0{efZf_I^Vjz;hgO8:{]oRmN_ i = 1  8l` N {f8lZ0mN`8q`608KZ0mN_j j ^Vx N  nm K|Bb<8l` θ : M → K = 8 ^ jkmN_xf6f;<9j mVo V  jkmT{}efb<8:9  8l` x = θ(α )  57608lZ =?> `608 [ ^Nwl`;<mNZ mNZ M 8qu‡8l``608osmNbib<m ;iZfu)_8lba^V`;<mNZ09^VjkmNZfu x F 9 V | • x ∈F  Z[G1 ]⊕Z[G2 ]⊕Z[G3 ]  N i 4 i i i 1 3 3 
v 2 1 wi w
w 4 wi i
i   Fwi /Ew
∗ v 4  i i  4 i ∗ wi i • N3 (x3 ) = N2 (x2 )N1 (x1 )  nm K| θ {f8Eg0Z08:9^wlba^N997;iZ H (V , K ) #576f;<9wlba^N99mVo θ ;<97`_; T;a^Vb ;ho `608l_8q8G;<9`9 y , y ∈ K 9e0wI]6 `6O^V` x = N (y ) 608l_8 y = y y  y mNZ09;<{f8l_`608q8G `8lZ09;<mNZ K /F  ! 8 TZ0m `6O^V` F /N (K ) ;<9wl>Gwlbi;<wKmVo*mN_{f8l_ ~  y 60mGmv98  38l` β = # 576?e09 | N (β) = 1  β ∈ F \ N (K )   2  ∗ v 4  2 i Kv /Fwi i v ∗ w1 1
∗ v    # 2
 θ1  H 2 (V4 , Kv∗ )  −1 3 2   1 w1 β1 g3 (β1 ) θ1 : M → Kv∗  3 ∗ v     ∗ v ∗ w1 1  θ1 (α2 ) = θ1 (α3 ) = 1         " 43 0 θ1 (α1 ) = β 3    ~v~  3cd;i_9`#mVo^VbibfZ0mN`7 8 `6O^V` θ ;<9*;iZ0{f:8 8:{)^ jkNm _xf6f;<9j = 8l` /8:8lZ V  jkmG{}efb<8:9 b<9mf| (2θ )(α ) = ^VZ0{ N (β) = 1  < mf|f=?>w60mGmv9;iZfu y = 1 ^VZ0{ y = β | 8K98:8`6O^V` 2θ β =N (β) {f8Eg0Z08:9`608 `_; G;a^Vb3wlba^N99;iZ H (V , K ) # 576Te09 |f`m xf_m ‡8K`608b<8ljzj ^ ;i`;<98lZ0mNefuv6]`mz960m `6O^V` `608KwlbaN^ 99mVo θ ;<9Z0mN``_; G;aV^ b& < fe xfxmv982`6082w:mNZT`_I^V_>  57608lZ4`608l_8#8G ;<9`9 y , y ∈ K 9e0w6q`6O^V` N (y ) = N (y ) = ^1 VZ0{ N (y y ) = β k 576Te09 | 8kw ^VZƒg0Z0{ η , η ∈ K 9e0w6 `6O^V` y = , i = 1, 2  57608lZ |   1 4 1 1  2 Kv /Fw1 2 Kv /Fw3  2 ∗ v 4 3 1    1 2 2 2 Kv /Fw2  −1 3 2 Kv /Fw1 ∗ v 3 ∗ v 3 Kv /Fw3 ηi gi (ηi ) i β10 g2 (η2 )η3 g3 (η2 )g1 (η3 ) · = , η2 g3 (η3 ) g1 (η1 )g2 (η3 ) g3 (β10 ) β = (y2−1 y3 ) · g1 (y2−1 y3 ) = 3 608l_8 β = η g (η )g (η )g (η ) ∈ N (K ) 257608lZ β = δβ osmN_9mNjk8 δ ∈ E   ef` | F /E ;<9#^VZ efZf_I^Vjz;hgO8:{z8} `8lZ09;<mNZ ^VZ0{`608l_8EoRmN_8 δ ;<9#^9rTeO^V_8jkmT{}efb<mK^efZf;hosmN_jz; 8l_ mVo F   > `6f;<9 |/;i`zoRmNbib<m 9k`6O^V`k;i`z;<9)^ƒZ0mN_j mVoK^VZ$8lb<8ljk8lZ?` oQ_mNj K  576f;<9z;ijzxfbi;<8:9 `6O^V` ;<9^Z0mN_j mVo#^VZ]8lb<8ljk8lZ?`oQ_mNj K ^ w:mNZ?`_I^N{};<wl`;<mNZ 576f;<9w:mNjzxfb<8l`8:9`608xf_mGmVo*mVo `608 δβ = β 8ljzj ^  nm K|b<8l` π =8 ^$efZf;hosmN_jz; 8l_ mVo E | π = 8 efZf;hosmN_jz; 8l_9 mVo F _8:9x 8:wl`; ‡8lbi>  < ;iZ0w:8 ;<9)efZf_I^Vjz;hgO8:{ oRmN_ i = 2, 3 ;i`kosmNbib<m 9`6O^V` π ∈/ N (K ), i = 2, 3   cfef_`608l_ | K /F ;<9^ efZf;i`;iZ E  < ;iZ0w:8 F /E ;<9efZf_I^Vjz;hgO8:{ 8Kw ^VZ]g0Z0{ x ∈ F 9e0wI6 n = N (π )N (π ) `6O^V` N (x) = n   0 1 3 1 w1 3 2 2 3 2 ∗ v Kv /Fw1 0 1 1 w w  w1 v 0 1 1 v
   w i wi  v wi i −1 2 2 3 3 w w1 Kv /Fwi ∗ v  w w1 1   # ?        θ0  θ0 : M → Kv∗ H 2 (V4 , Kv∗ )    θ0 (α1 ) = x, θ0 (αi ) = πi  3    )   i = 2, 3 3       Y `n;<9wlb<8 ^V_`6O^V` θ ;<9q^ V  jkmG{}efb<8jkmN_xf6f;<9j  / > `608 w:mNZ09`_e0wl`;<mNZ |;i`;<9q^Vb<9m 8 T;<{f8lZT` `O6 ^V`` 608nwlba^N9I9/mVo θ ;<9/Z08l;i`608l_7`_; T;a^Vb Z0mN_78:r?eO^Vb`m`6O^V`mVo θ *576f;<9w:mNjzxfb<8l`8:97`608xf_mGmVodmVo `608 8ljzj ^    3  0 4  0 1
           " "    Cl(Z[A4 ]) 8l` N =8 ^  uvb<mN=O^Vb † gO8lb<{ | R =8 ^ t8:{f8 G;iZ0{ {fmNj ^V;iZ ;i`6 oQ_I^Nwl`;<mNZO^VbBgO8lb<{ N | =8 ^ g0Zf;i`8 {}i; jk8lZ09;<mNZO^Vb N [^Vbiu‡8l=f_I^k^VZ0{ Λ =8K^VZ R mN_{f8l_;iZ A 
 C, 57608     ;<9{f8Eg0Z08:{ =T> `608u‡8lZ08l_I^V`mN_9 Cl(Λ) 
 {[X]|X 
  ;<9^ o u *xf_m  Λ jkmG{}efb<8K9e0wI6]`6O^V` X ≃ Λ oRmN_^Vbib p ∈ < x 8:w (Λ)} p p ~/ ^VZ0{ _8lba^V`;<mNZ09  mN_8:rTef; N^Vb<8lZT`bi>‡| [X] = [Y ] ;iZ K (Λ)).  [X] = [Y ] ⇔ X ⊕ Λ ≃ Y ⊕ Λ 0 576084^N{f{};i`;<mNZ];<9{f8Eg0Z08:{]=?> [X] + [Y ] = [Z] ⇔ X ⊕ Y ≃ Z ⊕ Λ  Y o Λ ;<9/^nj ^%G ;ij ^Vb R mN_{f8l_#w:mNZT`I^V;iZf;iZfu Λ `608lZk`608l_88G ;<9`9B^nZO^V`ef_I^VbO9ef_ 8:wl`;<mNZkmVo `608wlba^N9I9 vu _mNefx09 0 Cl(Λ) → Cl(Λ0 ). 57608 ‡8l_Z08lb3mVo `6084^V= m ‡8qj ^Vx ;<9w ^Vbib<8:{ `608  C,
 D(Λ)    mVo Λ ^VZ0{ ;<9{f8lZ0mN`8:{ =?>   
   #$"* . cOmN_ A = N [G] ^VZ0{ Λ = R[G], G ^g0Zf;i`8quv_mNefx | Cl(Λ) ≃ ‡8l_ (K (Λ) → K (A))  0 0 ~ Y o N = Q, R = Z, A ;<9^VZ ^Vbiu‡8l=f_I^V;<w)8G`8lZ09;<mNZ mVo Q ^VZ0{ Λ = O |d`608lZ Cl(Λ) ;<9K`608 ;<{f8 ^Vbdwlba^N99uv_mNefx mVo O  /  57608 ‡8l_Z08lb uv_mNefx];<9;iZ0{f8lx8lZ0{f8lZ?`mVo `608qwI60mN;<w:84mVo Λ  1   Z08K6O^N9 Cl(M (Λ)) ≃ Cl(Λ) ^VZ0{ D(M (Λ)) ≃ D(Λ)  A A  0 n n 374 -"   ?      "  #$ #  2 (     " -,',- (         !  # Cl(Z[A4 ]) 0    D(Z[A4 ])      23      3 8l` M = 8^nj ^%};ij ^Vb Z mN_{f8l_2;iZ Q[A ] w:mNZT`I^V;iZf;iZfu Z[A ] d57608lZ =?>4`608^V=m ‡857608:mN_8lj  < ;iZ0w:8 Q[A ] ≃ Q×Q(ζ )×M (Q) mNZ08w ^VZz`I^ ‡8 M = Z×Z[ζ ]×M (Z) Cl(Z[A ]) ≃ Cl(M) ^VZ0{ `608l_8EosmN_8 Cl(M) ≃ Cl(Z) × Cl(Z[ζ ]) × Cl(M (Z)) = 0 =?> `608 ^V= m ‡8 _8lj ^V_ }9] 576f;<9 w:mNjzxfb<8l`8:9`608Kxf_mTmVo2mVo*\#_mNx mv9;i`;<mNZ .   3
  4 4  4 4 3 3  3 3  3 3 +$-
fO13-
- (7 O22
1  Y[Zk`f6 ;<9B9 8:wl`;<mNZ /896O^VbibOw:mNZ09`_e0wl`B`6088G`8lZ09;<mNZ K/Q osmN_ 6f;<wI6 /896O^Vbib ‡8l_;hoR>`608w:mNZ 8:wl`ef_8 ;iZƒ?r e08:9`;<mNZ | ^VZ0{ƒw:mNZ09`_e0wl`K^)5d^V`8 98:rTe08lZ0w:8e09;iZfu`6084j ^NwI6f;iZ08l_> {f8 ‡8lb<mNx8:{ ;iZ `608xf_8 G;<mNe09 98:wl`;<mNZ ! 8 ;ibib#=8ze09;iZfu]`6f;<945d^V`8 9I8:r?e08lZ0w:8 `m xf_m ‡8 `608 N^VZf;<96f;iZfu mVo T Ω(K/Q) osmN_K`6f;<9 xO^V_`;<wlefba^V_8} `8lZ09;<mNZ


                 ! 
  "   K/Q   ~1  8l` E =8q`608KefZf;<rTe08K{f8luv_8:85/ 9ef=8G`8lZ09;<mNZ mVo Q[ζ ] 576?e09 | E ;<9u‡8lZ08l_I^V`8:{ =?>]^ _mGmN`mVo*`608 xNm bi>TZ0mNjz;a^Vb X + X − 2X − 1 = 0  8l`  = ζ + ζ 57608lZ E = Q[ ] Y[`;<9 GZ0m Z `6O^V` ` 608zwlba^N99uv_mNefx mVo E ;<9q`_; G;a^Vb& m K| 9;iZ0w:8v| 43 ≡ 1 (mod 7) `608 xf_;ijk8 1/9xfbi;i`9;iZ E  8l`  57608lZ | θ u‡8lZ08l_I^V`8:94^ xf_;ijk8z;<{f8 ^VbB^V=m ‡8 43 ;iZ E  8l` θ ^VZ0{ θ =8`608 θ = 2 −  − 5 w:mNZ efu?^V`8:9mVo θ  < 8l` F = F = E[v ] | 608l_8 v = θ /θ > 0  8l` K = 8q`6084Z0mN_j ^Vbdwlb<mv9ef_8 mVo F 2 57608lZ G(K/Q) = A  c ;i_9`/b<8l`Be09Bg  9mNjk8Z0mN`I^V`;<mNZ09 38l` F ^VZ0{ F {f8lZ0mN`8`608gO8lb<{)8} `8lZ09;<mNZ09mVo E ;iZ K `6O^V` ^V_8;<9mNjkmN_xf6f;<w4`m F  Y Z mN`608l_ mN_{f9 |`608:984` /m)gO8lb<{f9^V_8mN=f`I^V;iZ08:{ =?> ^N{ mN;iZf;iZfu v ^VZ0{ v _8:9x8:wl`; ‡8lbi>‡| 608l_8 v = θ /θ ^VZ0{ v = θ /θ  8l` G = G(K/Q) = A  38l` g , g ^VZ0{ g = 8K`608qmN_{f8l_` m 8lb<8ljk8lZ?`9nmVo G 9e0w6 `6O^V``608 g f 8:{ gO8lb<{ w:mN__8:9x mNZ0{};iZfu)`m {1, g } ;<9 F  38l` h ∈ G = 84^VZ 8lb<8ljk8lZ?`mVo#mN_{f8l_`6f_8:849e0w6 `6O^V` j ^Vx09 F `m F  h 8l` S = {2, 7, 41, 43, ∞} ^VZ0{osmN_^VZ?> 8} `8lZ09;<mNZ N/Q b<8l` S = {xf_;ijk8:9nm ‡8l_ S ;iZ N }  8l` S = S  < 8l` p = 7, q = 43, l = 41, m = 2 2 57608K{f8:w:mNjzxmv9;i`;<mNZ mVo*`608:98xf_;ijk8:9;iZ K ^VZ0{ ;i`99ef=}gO8lb<{f9;<9^N97osmNbib<m 9
7 3
2 7   −1 7 7 

1 2  1  7  2 7 1  7 1 2 1 1 3
2 3 4
2 3   1   2 2 2 3 2 3 1 1 2 

3 4 1 2 3
i 1 i 3 
N Q Q
K  p31 p32 p33 p34 K F D  D D D  D D P13 P23 LL L Q l1 l2 · · · · · ·l11l12 m4 L1 · · · · · L 6 m2F q1 q2 q3 l1 l2 l3 m A A A EE  E E Q1 Q22 Q23 EE  E E EE  EE  EE  E E E E E E L p3 E q21 q22 q23 A


q = 43 p=7 A
A

m=2 l = 41 < mf|`608z98l`mVo/xf_;ijk8:9 6f;<wI6 _I^Vjz;hoQ> ;iZ K ;<9 {2, 7, 43} z57608 wlba^N99nuv_mNefx mVo K ;<9qmVomN_{f8l_ . € ^VZ0{ `608wl>Gwlbi;<w4o ^Nwl`mN_9n^V_8mVoBmN_{f8l_9 10| ~ ^VZ0{ ~ 57608 98l` S {f8Eg0Z08:{ƒ^V=m ‡89I^V`;<9 gO8:9q^Vbibd`608 w:mNZ0{};i`;<mNZ09#9`I^V`8:{z^V`2`608=8luv;iZfZf;iZfuKmVo`608xf_8 T;<mNe0929I8:wl`;<mNZ Y[Zos^Nwl` |N`6087xf_;ijk8 l = 41 ;<92w60mv98lZ 9 m4`6O^V`B`608 S  wlba^N99#uv_mNefx Cl ;<9#`_; G;a^Vb&257608efZf;<rTe08xf_;ijk8 m ;iZ K ^V= m ‡8 ~ 9I^V`;<9 gO8:9 G = G  ! 8)96O^Vbib#Z0m w:mNZ09`_e0wl``608kj ^Vx f : X(−2) → U e09;iZfu 6f;<w6 /8)96O^Vbib/8} xfbi;<wl;i`bi>ƒu‡8l` ` 6085d^V`898:r?e08lZ0w:8# p8:w ^Vbib3`6O^V` S = {2, 7, 41, 43, ∞} ^VZ0{ `6O^V` S = {xf_;ijk8:9m ‡8l_ S }  38l`    S m    S  Q
Q ~= =K8 ^z9l8 `mV*o _8lxf_8:98lZ?`I^V`; ‡8:9oRmN_ G mN_=f;i`9 S = {m, p = p , q = q , l = l , ∞} 9 K8 6O^ ‡898:8lZ ;iZ `60K8 xf_8 G;<mNe0998:wl`<; mNZ |O`608Kj ^Vx f ;<9`60849efj mVo f F 9osmN_ v = p, q, l, ∞  Z08Kw ^VZ];iZ0{}e0w:8K`06 8qj ^Vx09 f oR_mNj(o : (−2) → U 2 576f;<9j ^Vx ;<9`_; T;a^Vb oRmN_ v = l ^VZ0{ `6?e09 /48 mNZfbi>)Z08:8:{ `m :w mNZ09;<{f8l_`608qxf_;ijk8:9 p, q ^VZ0{ ∞   0 1  1 1   v  v v S v       "  v=q c ;i_9`/mVod^Vbib | /896O^Vbib0g )^498l`/mVo _8lxf_8:9I8lZ?`I^V`; ‡8:9/oRmN_ G  mN_=f;i`9  8l` S(v) = {m, p, q , q , q } ∪ `6f_8:8 xf_;ijk8:9^V=m ‡8 l} ∪ {`6f_8:8 ;iZ}g0Zf;i`8]xf_;ijk8:9 }  c0_mNj `608 xf_8 T;<mNe09)98:wl`;<mNZ | /8 6O^ ‡8 {  576Te09 | 8)Z08:8:{ `mƒw:mNZ09`_e0wl`k^ j ^Vx o : M → U 9e0wI6 `6O^V` `608w:mN__8:9xmNZ0{};iZfu = M 8} `8lZ09;<mNZ wlba^N99 β ∈ Ext (Y , U ) 9I^V`;<9 gO8:9  
v 1  2  3    v S v 2 G v S inv(Gv ∩ Gw , w)(β) = ! 8q6O^ ‡8           ;ho w = v | h; o w = m | h; o w ∈ S(v) \ {v, m}  0 1 #(Gv ∩Gw ) − #(Gv1∩Gw ) ;ho w = q , q , q , m | so mN_9mNjk8 i |};ho w|∞ | G ∩G = {1, g }    mN`608l_ ;<98#  {1} p8:w ^Vbibv`O6 ^V` P inv(G ∩G , w)(β) = 0 3 57608l_8EosmN_8v| ;ho inv(V , v)(β) = 1/4 ^VZ0{ inv(V , w)(β) = osmN_ w = q , q V^ Z0{kosmN_`608 ;iZ}g0Zf;i`8nxf_;ijk8:9 |G`608lZ;i`BosmNbib<m 9`6O^V` inv(V , m) = −1/4 2Y ZmN`608l_ 0 /Nm _{f9 /8Z0:8 8:{ `m w:mNZ09`_e0wl`o 9e0wI6 `6O^V` inv(V , v)(β) u‡8lZ08l_I^V`8:9 H (V , K ) | inv(V , w)(β) ;<9 `_; T;a^Vb ;iZ H (V , K ) osmN_ w = q , q ^VZ0{ 9e0w6 `6O^V` inv(C , w)(β) ;<9`_; G;a^Vb3;iZ H (C , C ) osmN_`608q;iZ}g0Zf;i`8qxf_;ijk8:9 w ;iZ S(v)  nmN`84`6O^V` M ;<9u‡8lZ08l_I^V`8:{ =T> α , α ^VZ0{ α ^N9;iZ0{};<w ^V`8:{ ;iZ `608Kxf_8 G;<mNe0998:wl`;<mNZ 576Te09 | `608kw:mNZ09`_e0wl`;<mNZ mVo7o _8:r?ef;i_8:9Kg0Z0{};iZfu `6f_8:8 S efZf;i`9 u , u ^VZ0{ u 9I^V`;<9 oR>T;iZfu]`608zoRmNbib<m ;iZfu xf_mNx8l_`;<8:9  608l_8 Q {f8lZ0mN`8:9`608qxf_;ijk8q;<{f8 ^Vb3;iZ F bi>G;iZfuk= 8lb<m q † .  u ∈ F osmN_ i = 1, 2, 3  ~  N (u )N (u ) = N (u )  /  |u | = 1 ^VZ0{ |u | = |u | = 1/q  1  u ;<9^ 9rTeO^V_8jkmT{}efb<m Q osmN_ i = 1, 2  v 2  w v w∈S(v)      V4 1 2 3 i  4 w 4  3 4   2 ∗ v 4  ∗ w 4 2 4 v 2 3 2  4 2 ∗ 2  1 2 3  1 v 2 3  ij i i F1 /E 1 Q11 i i 1 F2 /E 2 F3 /E 3 3 Q31 2 Q21 ii j ~v€ =  u , h(u ), h2 (u ) > 0  1 2 3 57608g0_9`^VZ0{ 98:w:mNZ0{ w:mNZ0{};i`;<mNZ0978lZ09ef_8`6O^V`B`608j ^Vx o ;<9/;iZ0{f8:8:{ ^ Z[G ] jkmG{}efb<8jkmN_xf6f;<9j  75 608 Z08G`)` /m w:mNZ0{};i`;<mNZ098lZ09ef_8 `6O^V`ko _8lxf_8:9I8lZ?`9 `608 `_; G;a^Vb8lb<8ljk8lZ?` ;iZ H (V , K ) osmN_ ^VZ0{ƒ^ u‡8lZ08l_I^V`mN_n;iZ H (V , K ) n 576f;<9oRmNbib<m 9oQ_mNj 8ljzj ^ /  57608ba^N9`w:mNZ0{};i`;<mNZ v = q ,q 8lZ09ef_8:9`6O^V`o _8lxf_8:9I8lZ?`9`608K`_; T;a^Vb38lb<8ljk8lZT`;iZ H (C , C ) # 576?e09 v v   2 v 2 2 3  ∗ q 4  2 v " -,',- 2  
 S v  &      # ! '  : M → US    #   (    
    ,!)       # ! inv(Gv ∩ Gw , w)(β) =      "  v=p  4 ∗ v
∗ 2      i = 1, 2, 3 " ui v (αi ) = ui !    !  
 '  β ∈ Ext2G (Yv , US )  1     #(Gv ∩Gw )   w=v − #(Gv1∩Gw )     0 fv ) w=m    ) w ∈ S(v) \ {v, m} 3 Y[Z `6f;<9w ^N98 G = {1, h, h }  8l` L =8q`608g f8:{gO8lb<{]mVo G  8l` S(v) = {m, p, p , q} ∪ {oRmNef_ fx _;ijk8:94^V=m ‡8 l} ∪ {` /m ;iZ}g0Zf;i`8kxf_;ijk8:9 } | ^98l`qmVo7_8lxf_8:98lZ?`I^V`; ‡8:94osmN_ G  mN_=f;i`9z57608 uv_mNefx ;<9`_; T;a^VbosmN_ w ∈ S(v) \ {p, m} | ^VZ0{ 8:r?eO^Vb<9 G osmN_ w = p, m  G ∩G y mNZ09;<{f8l_`608oRmNbib<m ;iZfu G  _8:9mNbief`;<mNZ mVo Z
2 v 
v  2  v  v w v  v δ δ δ 1 2 3 Z → 0, ZGv → ZGv → 0→Z→ {f8Eg0Z08:{ =T> δ (1) = 1, δ (1) = 1 − h, δ (1) = 1 + h + h  ef_ ^V;ij`608l_8EosmN_8 ;<9 `m {f8Eg0Z08 o : Z → U |0^VZ0{]w:mNjzxfef`8K;i`9;ij ^Vu‡8q;iZ `608Kw:mN60mNjkmNb<mNuv> e09;iZfuz`6084^V=m ‡8K_8:9mNbief`;<mNZ 57608Kw:mNZ09`_e0wl`;<mNZ mVo o _8:rTef;i_8:9^VZ S efZf;i` u 9I^V`;<9 oR>G;iZfu 1 2 2 3  S v p v .  u ∈ L p ~  u ;<9Z0mN`w:mNZfuv_e08lZT``m ±1 jkmG{}efb<mz`608qxf_;ijk8q=8lb<m p ;iZ L  5706 8g0_9`w:mNZ0{};i`;<mNZ ;<9`mzj ^ ‡8o ^VZ Z[G ] jkmT{}efb<8KjkmN_xf6f;<9j]|^VZ0{]9I8:w:mNZ0{ `mz8lZ09ef_8q`6O^V` u <; 9KZ0mN`;iZ N (K ) ^VZ0{ 608lZ0w:8zo _8lxf_8:9I8lZ?`9`608ku‡8lZ08l_I^V`mN_4;iZ H (G , K ) 576f;<9qosmNbib<m 9 Ro _mNj&`608no ^Nwl`7`6O^V`^VZ]8lb<8ljk8lZ?` α ∈ L ;<9`608Z0mN_j mVo ^VZ]8lb<8ljk8lZ?`7oR_mNj K ;ho*^VZ0{ mNZfbi> ;ho α ;<9 ^ wlef=8qjkmT{}efb<8 P | 6f;<w6 ;<98:rTef; v^Vb<8lZ?``m α Z0mN`=8l;iZfukw:mNZfuv_e08lZ?`n`m ±1 jkmG{}efb<m P # 576?e09 |  p  v v Kp /LP ∗ p  v  p 2 ∗ P ∗ v v ∗  ~:E " -,',- ?  
        
   !) $  v : Z → US      ,  -  ! (   v (1) = up ) "  β ∈ Ext2G (Yv , US )  1     #(Gv ∩Gw ) inv(Gv ∩ Gw , w)(β) = − #(Gv1∩Gw )     0   "  v=∞ up     w=v       # !   -
     0 &     # ! fv ) w=m ) w ∈ S(v) \ {v, m} 3 Y[Zq`6f;<9 w ^N98 G = {1, g }  8l` S(v) = {m, p, p , q, q , q } ∪ {9;  xf_;ijk8:9m ‡8l_ l} ∪ {`6f_8:8K;iZ}g0Zf;i`8qxf_;ijk8:9 } | ^98l`mVod_8lxf_8:9I8lZ?`I^V`; ‡8:9osmN_ G  mN_=f;i`9 257608uv_mNefx G ∩ G ;<97Z0mNZ?`_; G;a^Vb3mNZfbi> oRmN_ w = xf_;ijk8:9 m ‡8l_ q ^VZ0{ w = m, v  y mNZ09;<{f8l_`608oRmNbib<m ;iZfu G  _8:9mNbief`;<mNZ mVo Z 
v 1 2 2 3   v v w   v δ δ δ 3 2 1 0→Z→ ZGv → ZGv → Z → 0, {f8Eg0Z08:{]=T> δ (1) = 1, δ (1) = 1 − g , δ (1) = 1 + g  ! 8qZ08:8:{]`mkw:mNZ09`_e0wl`o : Z → U 9e0wI6 ` 6O^V`;i`9;ij ^Vu‡8K;iZ]`608qw:mN60mNjkmNb<mNuv> uv_mNefx09^V_8K^N9{f8:9;i_8:{  < mf|f`6084w:mNZ09`_e0wl`;<mNZ o _8:r?ef;i_8:9n^ S  efZf;i` u 9I^V`;<9 oR>T;iZfu 1 2 1 3 S v ∞ v .  u ∈F 1 ∞ ~  |u | = 1 osmN_ i = 1, 2, 3  /  u ;<9^ 9rTeO^V_8qjkmG{}efb<m Q  1  u < 0 ^VZ0{ σ(u ) > 0 | 608l_8 σ ;<9`608Z0mNZT`_; T;a^Vb 8lb<8ljk8lZ?`nmVo Gal(F/E)  5706 8 g0_9`)w:mNZ0{};i`;<mNZ 8lZ09ef_8:9)`6O^V`ko ;<9^ Z[G ] jkmG{}efb<8 jkmN_xf6f;<9j 57608 9I8:w:mNZ0{ ^VZ0{$`6f;i_{ :w mNZ0{};i`;<mNZ09 8lZ09ef_8#`6O^V`d`608B8lb<8ljk8lZ?` _8lxf_8:98lZT`8:{4=T>o ;iZ H (C , K ) ;<93`_; T;a^Vb?oRmN_ w = q, q , q  75 608 g0ZO^Vbw:mNZ0{};i`;<mNZ 8lZ09ef_8:9z`6O^V` o _8lxf_8:98lZ?`9 `608Z0mNZ?`_; G;a^Vb8lb<8ljk8lZ?`k;iZ H (C , K ) |/^VZ0{ ` _; G;a^Vb38lb<8ljk8lZ?`;iZ H (C , K ) oRmN_;iZ}g0Zf;i`84xf_;ijk8:9 w 6= v # 576Te09 | ∞ Qi ∞ 1  ∞  ∞ v v v 2 2 ∗ w  2 " -,',-    
2 2 2 v   2 3 ∗ v ∗ w v : Z → US #      v (1) = u∞ ) "   u∞    # !   -
     ~„  - !)     ! β ∈ Ext2G (Yv , US ) inv(Gv ∩ Gw , w)(β) =  
  !   -
 '   1     #(Gv ∩Gw )  w=v  − #(Gv1∩Gw )     0  ) w=m  ) w ∈ S(v) \ {v, m} !"   !          "          # ! fv 3  \ pY  0/   † 7^N9 e098:{ `m 8}xfbi;<wl;i`bi> w:mNjzxfef`8 `608 S  efZf;i`9uv; G;iZfu _;<98 `m `608 5d^V`898:rTe08lZ0w:8# n ; ‡8lZƒ`6O^V`q`608 S  efZf;i`9q960mNefb<{ƒ9I^V`;<9 oR> w:8l_`I^V;iZ w:mNZ0{};i`;<mNZ09K^N9;iZƒ`608xf_8 T;<mNe09q9ef=09I8:wl`;<mNZ |`608 98 ^V_wI6 oRmN_9e0w6]efZf;i`9 ^N9{fmNZ08;iZ `608oRmNbib<m ;iZfuzo ^N96f;<mNZ 57608 efZf;i`9 u ^VZ0{ u 6O^ ‡8 N^VbieO^V`;<mNZ 1/q ;i`6 _8:9x8:wl`q`m`608 xf_;ijk8z;<{f8 ^Vb<9 Q ^VZ0{ Q _ 8:9x8:wl`; ‡8lbi> $ 576f;<9)8:998lZ?`;a^Vbibi> 6f;iZT`9k`6O^V` 8]w:mNefb<{$wI60mTmv98]`608lj `m =8u‡8lZ08l_I^V`mN_9 mVoq`608:98 fx _;ijk8;<{f8 ^Vb<9  6f;<w6 ^V_8xf_;iZf;<wl;ixO^Vb †  m 8 ‡8l_ |T;ho u ^VZ0{ u ^V_8w60mv98lZ)`mq=8`608u‡8lZ08l_I^V`mN_97^N9 ^V=m ‡8v|`608lZ `608w:mNZ0{};i`;<mNZ N (u )N (u ) = N (u ) ;<9Z0mN`9I^V`;<9gO8:{ osmN_n^VZ?> S  efZf;i`  57608l_8EoRmN_8v| 8j4efbi`;ixfbi> u =?>^u‡8lZ08l_I^V`mN_2mVo ^xf_;ijk8;<{f8 ^VbG;iZ F bi>G;iZfuK^V= m ‡8 l  57608lZ | u ∈F mNZ08w ^VZ]g0Z0{ u ∈ F 9I^V`;<9 oR>T;iZfu)^Vbib `608w:mNZ0{};i`;<mNZ09_8:rTef;i_8:{  y b<8 ^V_bi>‡| u ;ibib2^Vb<9mku‡8lZ08l_I^V`8z^ xf_;ijk8q;<{f8 ^Vbd^V=m ‡8 l ;iZ F  c0mN_ u |mNZ084w ^VZ wI60mTmv9I8K;i``m = 8 ~  m 8 ‡8l_ |O`m j ^ ‡8K`6084;ij ^Vu‡8mVo2`608Kj ^Vx f =f;iuvu‡8l_ 8 w60mGmv98n^qu‡8lZ08l_I^V`mN_ α mVod^qxf_;ijk8;<{f8 ^Vb;iZ L bi>G;iZfu^V=m ‡8 l |T^VZ0{ /898l` u = 2α * 576f;<9/9 ^V`;<9 gO8:9 `608q_8:rTef;i_8:{]w:mNZ0{};i`;<mNZ09 c ;iZO^Vbibi>‡|GZ0mNZ08mVo `608u‡8lZ08l_I^V`mN_9mVo `608xf_;iZ0wl;ixO^Vbxf_;ijk8;<{f8 ^Vb<9B;iZ F bi>T;iZfu= 8lb<m $`608xf_;ijk8:9 ;iZ S 9 ^V`;<9 oQ> `608 w:mNZ0{};i`;<mNZ09nosmN_ u  ! 8`608l_8EoRmN_8 b<mGm ^V``608xf_mG{}e0wl`9mVoBxf_;ijk8 ;<{f8 ^Vb<9n`6O^V` ^V_8qZ0mN`xf_;iZ0wl;ixO^Vb& 576084wlba^N99ZTefj4=8l_mVo F ;<9 ~  =?>]\ pY † ^VZ0{]`608l_8EosmN_8K`608qxf_mG{}e0wl`mVo#^VZ?> ` m ;<{f8 ^Vb<9q`6O^V`4^V_8 Z0mN`Kxf_;iZ0wl;ixO^VbB;<9xf_;iZ0wl;ixO^Vb&  Z08zw ^VZƒg0Z0{ ^)xf_;ijk8z;<{f8 ^Vb l bi>T;iZfu ^V=m ‡8 l ;iZ F 9e0w6]`6O^V`^u‡8lZ08l_I^V`mN_mVo*`608;<{f8 ^Vb l m 9 ^V`;<9 gO8:9`608_8:rTef;i_8:{]w:mNZ0{};i`;<mNZ09 | 608l_8 m ;<9 `608qxf_;ijk8q;<{f8 ^Vb3;iZ F bi>G;iZfu)^V=m ‡8 ~  ! 8K9I8l` u `mz=8`6f;<9u‡8lZ08l_I^V`mN_ ! ;i`6 `608n608lbix]mVo \ pY#`608:98 S  efZf;i`9 8l_8noRmNefZ0{]^VZ0{ ;iZ `608oRmNbib<m ;iZfu /8q96O^Vbib8} xfbi;<wl;i`bi> _;i`8K{fm Z mNZ08q9e0wI6]x mv9I9;i=fb<8q98l`mVo S  efZf;i`9 8l`  = ζ + ζ  < mf| E = Q[ ]  < 8l`        2   3 21  31      2 3  1 F1 /E 2 F2 /E F3 /E 3  1  1 3 3  1 1 1  1      p   p l l  1  ∞ 1   F  1 F F F  1   ∞ 
7 p −1 p 7 θ1 = 227 − 7 − 5,   ~  θ2 = −327 − 27 + 3, θ3 = 27 + 37 − 2. 57608lZ θ u‡8lZ08l_I^V`8:9 ^ xf_;ijk8 ;<{f8 ^Vb^V=m ‡8 q oRmN_ 8 ^Nw6 i = 1, 2, 3  8l` v = pθ /θ  < ;iZ0w:8 |`608 8} `8lZ09;<mNZ F = E[v ] ;<9n_8 ^Vb& 8l` v = pθ /θ ^VZ0{ v = pθ /θ  ! 8 96O^Vbib θ /θ > 0 8} xf_8:99K^Vbib2`608 S  efZf;i`9q;iZ `8l_jk9qmVo  ^VZ0{ v F 9 57608oRmNbib<m ;iZfu];<9K^ 98l`KmVo S efZf;i`9q9I^V`;<9oQ>G;iZfu `608q_8:rTef;i_8:{]w:mNZ0{};i`;<mNZ09 | • u = (3 + 7 + 2) + (2 + 7 + 5)v | • u = (1472 − 1182 − 816) + (−1697 + 1359 + 943)v | • u = (782 + 435 − 1756) + (901 + 503 − 2021)v | • u = 8 + (6 + 4 − 6)v + (−2 − 6 + 4)v + (4 − 2 − 10)v 
i 1 2 3
2 3 1 2 3 1 3 1 2  7 2 7 1 2 7 7 2 7 2 7 1 2 7 7 2 7 3 i 2 7 7 2 7 p 7 7 2 7 1 2 7 7  • u∞ = (527 + 77 − 2) + (−227 − 77 − 5)v1 3 2 7 2 7 3 576f;<93uv; ‡8:9d^j ^Vx f : X (−2) → U 6f;<w6_8lxf_8:98lZ?`9d`608#w ^VZ0mNZf;<w ^VbTwlba^N9I9 ;iZ Ext (X , U )  nm K| ;i`6`608^V=m ‡898l``;iZfu0| 8nw ^VZ g0Z0{^ S  efZf;i` u |f^Vu?^V;iZ e09;iZfu\ pY |G9e0wI6 `6O^V`7`608j ^Vx | 608l_8 N ≃ Z[G] |8} `8lZ0{};iZfu f =?> f˜((0, 1)) = u ;<99ef_ 8:wl`; ‡8#57608 f˜ : X (−2) ⊕ N → U ;ij ^Vu‡8KmVo f 6O^N9 Z _I^VZ 21 6f;ib<8K`608 Z _I^VZ mVo U ;<9 ~ = 2 c0ef_`608l_ |0`608 v^VbieO^V`;<mNZ ;i`6]_8:9x8:wl` `m p mVod^VbibO`608 S  efZf;i`9/`6O^V`7^V_8nw60mv98lZ9mKos^V_/;<9 . 2 576?e09 | 8wI60mTmv98 u 9e0w6 `6O^V`;i`9 v^VbieO^V`;<mNZ ;i`6 _8:9x 8:wl``m p ;<9 1/p  576f;<9 ;ibib j ^ ‡8K`608K_I^VZ ]8:rTeO^Vb ~ = 7 cfef_`608l_n`608qosmNbib<m ;iZfuw60mN;<w:8mVo 9e0wI6 ^efZf;i`^Vb<9m j ^ ‡8:9`608qj ^Vx f˜ 9ef_ 8:wl`; ‡8   S     2 G S S S  0  S  S  0     S   0        • u0 = (4983/227 + 55977 + 3991/2) + (−5735/227 − 64447 − 2300)v1 + (232227 + 10435/27 + 1862)v2 + (−5087/227 − 11435/27 − 2043)v3 . nmN`8 `6O^V`K=?>ƒ`608zw:mNZ09`_e0wl`;<mNZ mVo f˜| ;i`q;iZ0{}e0w:8:9^VZ ;<9mNjkmN_xf6f;<9j mNZ `608 w:mN60mNjkmNb<mNuv> uv_mNefx09 :w mN__8:9xmNZ0{};iZfu]`m X (−2) ^VZ0{ U 576?e09 |3;ho P {f8lZ0mN`8:94`608 ‡8l_Z08lbBmVo f˜| `608lZ P ;<94w:mN60mNjkm  <b mNuv;<w ^Vbibi>]`_; G;a^Vb*^VZ0{ `mN_9;<mNZ oQ_8:8# 576f;<9960m 9n`6O^V` P ;<9xf_m 8:wl`; ‡8#  ef` |O=T>]`608\2_mNx mv9;i`;<mNZ . 8 GZ0m `6O^V`8 ‡8l_> xf_m 8:wl`; ‡8 Z[A ] jkmT{}efb<8;<9oQ_8:8# 8lZ0w:8 P ;<9oQ_8:8# 576Te09 | /86O^ ‡8mNef_ 5d^V`8498:r?e08lZ0w:8   S S          4 0 → US → (A ⊕ N )/P → B → XS → 0.   /    
 O   fe _#^V;i,j mVo`6f;<9298:wl`;<mNZ ;<9 `mnxf_m ‡87`608 y mNZ 8:wl`ef_8 ~ ;iZ `608xO^V_`;<wlefba^V_#w ^N987mVo K/Q `6O^V` 87^V_8 w:Nm Z09;<{f8l_;iZfu  576f;<9 /896O^Vbib ^NwI6f;<8 ‡8=?>]w:mNjzxfef`;iZfu `608Kb<8 ^N{};iZfuw:mT8  wl;<8lZT`9nmVo2`608 LsoQefZ0wl`;<mNZ09 ^VZ0{]8:rTeO^V`;iZfuk`6O^V` ;i`6 ^VZ]8lb<8ljk8lZT`w:mNjz;iZfu oQ_mNj(`608K5 ^V`8K98:rTe08lZ0w:8#

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   8l` k = Q ^VZ0{ K/Q =87`608 q^Vb<mN;<9#8}`8lZ09;<mNZ {f8Eg0Z08:{k;iZz`6088 ^V_bi;<8l_B98:wl`;<mNZ | ;i`6 q^Vb<mN;<92uv_mNefx  57608l_8z^V_84oRmNef_;i__8:{}e0wl;i=fb<8zw:mNjzxfb<8 w6O^V_I^Nwl`8l_9qmVo A n 57608wI6O^V_I^Nwl`8l_ G = Gal(K/Q) ≃ A `I^V=fb<8q;<9=8lb<m  
4 4  1 g1 h h2 χ0 1 1 1 1 χ1 1 1 ζ3 ζ32 χ01 1 ζ32 ζ3 .   57608]wI6OV^ _I^Nwl`8l_ χ ;<9k`06 8 `_; T;a^VbwO6 ^V_IN^ wl`8l_ $57608l_8 ^V_8` /m mN`608l_)^V=8lbi;a^VZ$w6O^V_I^Nwl`8l_9 χ V^ Z0{ χ   57608 ab ^N9` wI6OV^ _I^Nwl`8l_ χ :w mN__8:9xmNZ0{f9)`m `608 %/  {};ijk8lZ09;<mNZO^Vbq;i__8:{}e0wl;i=fb<8ƒ_8lxf_8:98lZT`I%^  `;<mNZ  cOmN_ χ | 8 6O^ ‡8 L(χ , 0) = ζ (s) ^VZ0{ `608l_8EoRmN_8 L (χ , 0) = −1/2 ^VZ0{ L (χ , 0) =  − (log m)(log p)(log q)(log l) 57608KwI6O^V_I^Nwl`8l_ χ ;<9^V= 8lbi;a^VZ ^VZ0{]608lZ0w:8os^Nwl`mN_9`6f_mNefuv6 G ≃ C  ! 8q6O^ ‡8  wEo  0/ E  † χ2 / 1   0 1 0 1 2   0 0 ∗ Q ∗ S 0 1 2 0  1 3 6 1X L (χ1 , 0) = − log |1 − ζ7a |χ1 (a), 2 ∗ a=1 608l_8 ζ = e ;<9 ^ xf_;ijz;i`; ‡8 9I8 ‡8lZ?`6 _mGmN`kmVonefZf;i`[> ^VZ0{ χ ;<9k{f8Eg0Z08:{ mNZ (Z/7Z) =?> i; Z ^V`;iZfu T;a^ (Z/7Z) → C → C  8l` α = (1 − ζ )(1 − ζ ) = 2 −  ∈ E  57608lZ α u‡8lZ08l_I^V`8:9`608KefZf;<r?e08qxf_;ijk8K;<{f8 ^Vb ;iZ E ^V=m ‡8 p 2Y Z `8l_jk9mVo α /8q6O^ ‡8   2πi/7 7  × 3 χ1  × 1 −1 7 
7 7  × 7  7 1 L∗ (χ1 , 0) = − (log |α7 | + ζ3 log |h(α7 )| + ζ32 log |h2 (α7 )|) 2  h2 (α7 ) α7 1 | + ζ32 log | | . log | = − 2 h(α7 ) h(α7 ) 7 /.
8l` w = ( + 1) ^VZ0{ w =  q57608lZ | {w , w } u‡8lZ08l_I^V`8 `608efZf;i`uv_mNefxƒ;iZ E qcfef_`608l_ | ^VZ0{ 2576084^V= m ‡8K8:rTeO^V`;<mNZ]_8:{}e0w:8:9`m =w =w 1 −1 7 α7 h(α7 ) 2 h(α7 ) h2 (α7 ) 2 1 7 1 2  2 2 L∗ (χ1 , 0) = −(log |w1 | − ζ32 log |w2 |). cfef_`608l_ | /8q6O^ ‡8 L (χ , 0) = (1 − ζ )(log l)(log q)L (χ , 0)  57608KwI6O^V_I^Nwl`8l_ χ {f8:w:mNjzx mv98:9^N9 χ = Ind 1 − Ind 1 | 608l_8 C = Gal(K/F ) ^VZ0{ K 576Te09 | L(χ , s) =  576Te09 | ^Vu?^V;iZƒ=?> wlba^N9I9ZTefj4=8l_oRmN_jefba^}| V = Gal(K/E) = /8nu‡8l` L (χ , 0) = | 608l_8 h ^VZ0{ R {f8lZ0mN`8`608wlba^N997ZTefj4=8l_^VZ0{ `608_8luvefba^V`mN_mVo L _8:9x8:wl`; ‡8lbi>  ! 8 GZ0m `6O^V` h = 1 # cfef_`608l_ | /8q6O^ ‡8 h = 2 |0=?> \ pY nmN`8z`6O^V` E ;<94^`mN`I^Vbibi> _8 ^Vb K ^Vb<mN;<948} `8lZ09;<mNZ mVo Q ^VZ0{ `6Te09 | `608l_8 ^V_8z`6f_8:8k_8 ^Vb/8lj  =8:{f{};iZfu‡94;iZ E 9I^ > σ = id, σ ^VZ0{ σ  8lZ0w:8k`608 efZf;i`4uv_mNefx ;<94mVo Z _I^VZ ~  8l` w ^VZ0{ = 8`608 Z u‡8lZ08l_I^V`mN_9qmVoB`6f;<9nuv_mNefx ^N9{f8Eg0Z08:{ ^V=m ‡8#q Y[`;<98 ^N9> `m ‡8l_;hoR>]`6O^V` F 6O^N9n` /m w _8 ^Vb8lj4=8:{f{};iZfu‡9^VZ0{zosmNef_w:mNjzxfb<8) 8lj= 8:{f{};iZfu‡9 # 576?e09 |G`608efZf;i`/uv_mNefx mVo F ;ibib =8mVo Z _I^VZ /   Z08w ^VZ wI60mTmv98 ^zefZf;i` w ;iZ F 9e0wI6 `6O^V` {w , w , w } ;<9^z=O^N9;<9nmVo2`6084efZf;i`nuv_mNefx ;iZ c  576f;<97osmNbib<m 9oQ_mNj `608no ^Nwl``6O^V` F 6= E[p(u)] oRmN_^VZT>efZf;i` u ∈ E  8l` id, σ =8`608q` mz_8 ^Vb 8lj4=8:{f{};iZfu‡9nmVo F ^VZ0{ b<8l` σ , σ = 8` /m)w:mNjzxfb<8 8lj= 8:{f{};iZfu‡9 B 57608lZ |   ∗ S 2 3 1 2 G C2 C2 L(1C2 ,s) L(1V4 ,s) ζF (s) ζE (s) 2 4 2  ∗ 2   hF R F hE R E   ∗ L 1  G V4 V4 2 L   E F    1 2
3 1    2  3 1 2  3  
r  c0 c  RE = det  log |w1 | log |σ2 (w1 )|  log |w2 | log |σ2 (w2 )|   , log |w1 | log |σr (w1 )| 2 log |σc (w1 )|      RF = det  log |w2 | log |σr (w2 )| 2 log |σc (w2 )|  .   log |w3 | log |σr (w3 )| 2 log |σc (w3 )| > ^N99efjzxf`;<mNZ | F ;<9 ^ _8 ^Vb/gO8lb<{ |#^VZ0{ `608l_8EosmN_8 σ ;<9 ^VZ 8lb<8ljk8lZ?`zmVo Gal(F/E)  576Te09 | osmN_ i = 1, 2  b<9mf| /8 w ^VZ wI60mTmv9I8 σ 9e0w6 `6O^V` σ (w ) = σ (w ) oRmN_ i = 1, 2  σ (w ) = w 57?6 e09 | R = 2| log | ||R |d^VZ0{ `608l_8EosmN_8 8k6O^ ‡8 L (χ , 0) = 4| log | || )cfef_`608l_ | r  r i i F c w3 σr (w3 ) E L∗S (χ2 , 0) = 8(log p)(log l)3 | log | σrw(w3 3 ) ||   c  ∗ 2 i 2 i w3 σr (w3 ) ! /~     !  " 
cf_mNj `608 w:mNZ09`_e0wl`;<mNZ ;iZ `608 xf_8 T;<mNe09 98:wl`;<mNZ /8]6O^ ‡8 ^ w:mNjzj4ef`I^V`; ‡8ƒ{};a^Vuv_I^Vj mVo480^Nwl` 9 8:rTe08lZ0w:8:9  0  0 0  P  / λ(P )  / XS (−2) ⊕ N λ  /A⊕N  θ  µ /B / XS /0 / XS /0 h  608l_8 A := (  S A⊕N λ(P ) )  / AS / US 0   0 0 / BS ^VZ0{ B := B 257608lZ /8q6O^ ‡8   S Det(AS,R ) ⊗ Det−1 (BS,R ) ≃ Det−1 (λ(P )R ) ⊗ Det(AR ⊕ NR ) ⊗ Det−1 (BR ) ≃ Det−1 (λ(P )R ) ⊗ Det(XS (−2)R ⊕ NR ) ⊗ Det−1 (XS,R ) ≃ Det−1 (λ(P )R ) ⊗ Det(PR ) ⊗ Det(US,R ) ⊗ Det−1 (XS,R ) ≃ Det(0) ≃ ζ(R[G]). 57608g0_9`;<9mNjkmN_xf6f;<9j osmNbib<m 9qoQ_mNj `608zw:mNZ09`_e0wl`;<mNZ mVo A | `6089I8:w:mNZ0{ oQ_mNj `608z80^Nwl`Z08:99 mVo/`608 jz;<{f{}b<8k_m K| `608 `6f;i_{ G;a^)`608 j ^Vx h ^VZ0{ƒ`608 ba^N9`;<9mNjkmN_xf6f;<9j T;a^)`608 j ^Vx09 λ ^VZ0{ 2 57608Kw:mNjzxmv9;i`;<mNZ mVo#^Vbib `608:98;<9mNjkmN_xf6f;<9jk9;<9`608qj ^Vx ;iZ0{}e0w:8:{ =?> ψ : A ≃ B  R nmN`8 `6O^V``608);ij ^Vu‡8 mVo (A , B ; ψ ) ;iZ K (Z[G]) ;<9  8l_m  576?e09 |#mNZ08 w ^VZ b<mTm ƒosmN_4`608 ;iZ ‡8l_9I8q;ij ^Vu‡8qmVo `6f;<98lb<8ljk8lZT`;iZ K (R[G]) G;a^ δ  oR`8l_w60mGmv9;iZfu Z[G] =O^N98:9osmN_ A, N, B ^VZ0{ | /84w ^VZ g0Z0{ ^u‡8lZ08l_I^V`mN_osmN_ P  S    S S  S,R S,R  S S  S 0  1  Det−1 (λ(P )R ) ⊗ Det(AR ⊕ NR ) ⊗ Det−1 (BR ) ^V_;<9;iZfu oR_mNj `608:98=O^N98:9K57608;ij ^Vu‡8zmVo/`6f;<9;iZ ζ(R[G]) efZ0{f8l_q`608z^V= m ‡8 ;<9mNjkmN_xf6f;<9j ;<9;iZ  /%/ `608;iZ ‡8l_98;ij ^Vu‡8 δ ((A , B ; ψ )) ef_7^V;ij Z0m ;<9B`m4w:mNjzxfef`8`6f;<9/8lb<8ljk8lZT`/;iZ)`608;iZ ‡8l_9I8 ;ij ^Vu‡8# c ;i_9` 8K96O^Vbib3wI60mTmv98K=O^N98:97osmN_ A, B ^VZ0{ N 2 5760848f^Nwl`98:rTe08lZ0w:8   −1 S S  S  λ µ θ 0 → XS (−2) ⊕ N → A ⊕ N → B → XS → 0 9xfbi;i`9;iZT`mk{}; $ 8l_8lZ?`xf_;ijk84w:mNjzx mNZ08lZT`9n^N9 λ µv θ v 0 → Yv (−2) →v Av → Bv → Yv → 0 ^VZ0{mVodw:mNef_98 λ| : N → N ;<9`608n;<{f8lZ?`;i` > j ^Vx ! 896O^Vbib g  `608nu‡8lZ08l_I^V`mN_9oRmN_ A, B ^VZ0{ N N^ 97osmNbib<m 9 c0mN_ v = ∞ |}b<8l` A = Z[G]a , B = Z[G]b ;i`6 λ (1) = (1 + g )a , θ (a ) • N    = (1 − g )b , µ (b ) = 1 v 1 • 1 v v 1 v 1 1 v 1 1 c0mN_ v = p |}b<8l` A = Z[G]a , B = Z[G]b ;i`6 λ (1) = (1 + h + h )a , θ (a )   = (1 − h)b , µ (b ) = 1 v 2 • 1 v 2 v 2 2 v 2 v 2 2 c0mN_ v = q |}b<8l` A = Z[G]a ⊕ Z[G]a , B = Z[G]b ;i`6 λ (α ) = (−1 − g )a  v 3 4 v 3 v 1 1 3 + (1 + g1 )a4 , λv (α2 ) = (1 + g2 )a3 , λv (α3 ) = (1 + g3 )a4 , θv (a3 ) = (1 − g2 )b3 , θv (a4 ) = (1 − g3 )b3 , µv (b3 ) = 1  c0mN_ v = l | A = 0 ^VZ0{ B = Z[G]b ;i`6 µ (b ) = 1  c ;iZO^Vbibi>‡|fb<8l` N = Z[G]a ;i`6 f˜(a ) = u  • ! 84`608lZ w60mGmv98^zu‡8lZ08l_I^V`mN_ c oRmN_ P ≃ Z[G] | ^VZ0{ `608K;ij ^Vu‡84mVo*`6f;<9u‡8lZ08l_I^V`mN_;iZ X (−2) ^VZ0{ N ;<9uv; ‡8lZ]=?>  • v v 4 v 4  5 5 0 v  G G c = (c∞ , cp , cq , cN ) ∈ IndG C2 Z ⊕ IndC3 Z ⊕ IndV4 M ⊕ N, /1 608l_8  c∞ = (−3 − g2 + 5h + 5hg2 − 3h2 − 4h2 g2 ), cp = 0, cq = (−5 − 5g1 + 3h + 4hg1 + 3h2 + h2 g1 )(α3 − α2 ) + (h2 − h2 g1 )α2 , = (g1 + g2 + g3 )(1 − h2 ). cN c ;iZ0{};iZfu 9e0w6 ^ c ;<9^ xf_mN=fb<8lj ;iZ bi;iZ08 ^V_q^Vbiu‡8l=f_I^}|^VZ0{ \ pY ^N9ne098:{ `m w ^V__> mNef`n`608w:mNj  fx ef`I^V`;<mNZ09 ! 8k9I8l` z = (∧ a ) ∈ Det(A ⊕ N ) | z = [∧ b ] ∈ Det (B) ^VZ0{ z = [λ(c)] ∈  ! 8496O^Vbib w:mNjzxfef`8K`608q;ij ^Vu‡84mVo z ⊗ z ⊗ z ;iZ ζ(R[G])  Det (λ(P )) ! 8Z0m b<mTm ^V`7`608;<{f8ljzx mN`8lZT`9mVo R[G]  8l` e = P g, e = (2e − h − h )(e + ^VZ0{ e = (3e−g −g −g )   Z0{f8l_d`608BZO^V`ef_I^VbG;<9ImNjkmN_xf6f;<9j ρ : R[G] → R×C× g +g +g ), |N`608:9I8B;<{f8ljzxmN`8lZ?`9*j ^Vx`m (1, 0, 0), (0, 1, 0) ^VZ0{ (0, 0, id) _8:9x 8:wl`; ‡8lbi>  M (R) =: R ×R ×R 576?e09 | /846O^ ‡8 e + e + e = 1  cfef_`608l_ |OoRmN_n^ Z[G] jkmT{}efb<8 M | /849I8l` M := e M | 6f;<wI6 /8n96O^Vbib _8lu?^V_{ ^N97^4jkmT{}efb<8nm ‡8l_ R * c0mN_^4jkmN_xf6f;<9j f ∈ Hom (M, N ) |Gb<8l` f {f8lZ0mN`8`608 w:mN__8:9xmNZ0{};iZfu)8lb<8ljk8lZT`;iZ Hom (M , N )   5 i=1 i 0 −1 00   2
2 1 4 1 2 0 1 3 3 0   1 2 00 −1 0 0 1 −1 4 −1 i=1 i 1 12 1 g∈G 1 12 2 3    2  i i R  i Ri ! 
      e  i R[G] i i 0 c ;i_9` Z0mN`8 `6O^V``608j ^Vx θ : A ⊕ N → B ;<9`608  8l_mƒj ^Vx 57608l_8EosmN_8v| λ ^VZ0{ µ ^V_8 <; 9mNjkmN_xf6f;<9jk9 257608jkmT{}efb<8 A ⊕ N ;<9#u‡8lZ08l_I^V`8:{ =?> {e a } , B ;<92u‡8lZ08l_I^V`8:{ =?> {e b }  57608;ij ^Vu‡8nmVo e (z ⊗ z ) = (∧ e a ) ⊗ [∧ e b ] ;iZ Det(X (−2) ⊕ N ) ⊗ Det (X ) ;<9 0 0 0 0 0 z 0 ⊗ z 0 0 := 0 0 5 0 i i=1 0 5 i=1 0 i 0 4 −1 j=1 0 j S 0 4 0 j j=1 0 0 0 −1 S,0 1 5 (∧ e0 di ) ⊗ [e0 (∞ − m) ∧ e0 (p − m) ∧ e0 (q − m) ∧ e0 (l − m)]−1 , 24 i=1 608l_8;iZ X (−2) ⊕ N ≃ (Ind R) ⊕ (Ind R) ⊕ (Ind M ) ⊕ N , `6088lb<8ljk8lZT`9 d F 9^V_8 {f8Eg0Z08:{ =T>  S R • d1 = (1, 0, 0, 0) • d2 = (0, 1, 0, 0), • d3 = (0, 0, α2 , 0), • d4 = (0, 0, α3 , 0), R G C2 G C3 G V4 R R i • d5 = (0, 0, 0, a5 ) /%=  nm K|f`608qj ^Vx (R ◦ f˜ ) : X (−2) ⊕ N → X ;<9uv; ‡8lZ]=T>    S,0 0 S 0 0 e0 di 7→ −e0 < mf|f`608;ij ^Vu‡8:9mVo e d ^V_8 X v∈S S,0 log |f˜0 (di )|v (v − m). 0 i | • e0 d1 7→ − log |N (u∞ )|e0 (∞ − m) + 2(log l)e0 (l − m) • e0 d2 7→ − log |N (up )|e0 (∞ − m) + 3(log l)e0 (l − m) | • e0 d3 7→ − log |N (u2 )|e0 (∞ − m) + 2(log q)e0 (q − m) | • e0 d4 7→ − log |N (u3 )|e0 (∞ − m) + 2(log l)e0 (l − m) + 2(log q)e0 (q − m), • e0 d5 7→ − log |N (u0 )|e0 (∞ − m) + (log p)e0 (p − m) | 608l_8 N = N ;<9`608Z0mN_j j ^VxoQ_mNj K `m Q  57608jkmT{}efb<8 P ;<9u‡8lZ08l_I^V`8:{ =?> c 60mv984;ij ^Vu‡84;iZ X (−2) ⊕ N ;<9 e (−d − d + d ) | _ 8:9x8:wl`; ‡8lbi> # 576?e09 |  K/Q  0 0 S 0 0 0 1 3 4  z0 1 5 (∧ e0 di ) 24 i=1 1 = − (im(c0 ) ∧ e0 d1 ∧ e0 d2 ∧ e0 d3 ∧ e0 d5 ) 24 1 7 → − (e0 c) ⊗ (∧i=1,2,3,5 (RS,0 ◦ f˜0 )(e0 di )) 24 1 = (e0 c) ⊗ 2(log α)(log l)(log q)(log p)(z 00 )−1 , 24 = 608l_8 α = |N ( )| ef`7`6f;<978:r?eO^Vb<9 m ^VZ0{ `6?e09 |G`608;ij ^Vu‡8nmVo e (z ⊗ z ⊗ z ) ;iZ ζ(R[G]) ;<9  u2p u3∞ ! 

6 0 00 0 1 (log m)(log l)(log q)(log p)e0 = −e0 L∗S (χ0 , 0) = −e0 L∗S (χ0 , 0)# . 2       e 1 c0mN_8 ^NwI6 v ∈ S w:mNZ09;<{f8l_`608K80^Nwl`98:rTe08lZ0w:8 λv,1 θv,1 µv,1 0 → Yv (−2)1 → Av,1 → Bv,1 → Yv,1 → 0. /€ 576087j ^Vx09 θ , θ ^VZ0{ θ ^V_8  8l_mj ^Vx09 6f;ib<8 θ ;<9#^VZz;<9mNjkmN_xf6f;<9j uv; ‡8lZz=?> θ (e a ) =  57608)jkmT{}efb<8 A ⊕ N ;<9u‡8lZ08l_I^V`8:{ =T> {e a } ^VZ0{ B =?> {e b }  57608 (1 − ζ )e b ;ij ^Vu‡8KmVo e (z ⊗ z ) = (∧ e a ) ⊗ [∧ e b ] ;iZ Det(X (−2) ⊕ N ) ⊗ Det (X ) ;<9  ∞,1 3 q,1 1 2 1 0 1  p,1 l,1 p,1 5 1 i i=1 1 5 i=1 1 i 4 −1 i=1 1 j S 4 1 j j=1 1 1 1 2 −1 1 S,1 z 1 ⊗ z 0 1 := 1 − ζ3 4 (∧i=1 e1 di ) ⊗ [e1 (∞ − m) ∧ e1 (q − m) ∧ e1 (l − m)]−1 , 8 S R 608l_8n;iZ X (−2) ⊕ N ≃ Ind R ⊕ Ind R ⊕ Ind M ⊕ N |?`6088lb<8ljk8lZ?`9 d F 9^V_8n{f8Eg0Z08:{ ?= > | • d = (1, 0, 0, 0) | • d = (0, 0, α , 0) | • d = (0, 0, α , 0)  R G C2 G C3 G V4 R i R 1 2 2 3 3 • d4 = (0, 0, 0, a5 )  nm K|f`608qj ^Vx (R ◦ f˜ ) : X (−2) ⊕ N → X ;<9uv; ‡8lZ]=T>    S,1 1 < mf|f`608;ij ^Vu‡8:9mVo e d ^V_8 S 1 e1 di 7→ −e1 1 X v∈S S,1 log |f˜1 (di )|v (v − m). 1 i • e1 d1 7→ −e1 ( • e1 d2 7→ −e1 ( • e1 d3 7→ −e1 ( • e1 d4 7→ −e1 ( P −1 g log |u∞ |g)(∞ − m) + 2(log l)e1 (l − m) | P log |u2g |g)(∞ − m) + 2(log q)e1 (q − m) P log |u0g |g)(∞ − m) P −1 | −1 log |u3g |g)(∞ − m) + 2(log l)e1 (l − m) + 2(log q)e1 (q − m) −1 |  c0mN_ u ∈ K | /826O^ ‡8 e (P log |u |g) = (log |N (u)|+ζ log |h (N (u))|+ζ log |h(N 608lZ w:mNZ09;<{f8l_8:{ ^N9^VZ 8lb<8ljk8lZ?`nmVo R |fefZ0{f8l_`608Kxf_m 8:wl`;<mNZ R[G] → R  8l`   1 g −1 K/E 3 2 2 3 K/E 
1 1 α(u) := (log |NK/E (u)| + ζ3 log |h2 (NK/E (u))| + ζ32 log |h(NK/E (u))|). K/E (u))|)e1 | /E 57608jkmT{}efb<8 P ;<9u‡8lZ08l_I^V`8:{ =?> e c 60mv984;ij ^Vu‡84;iZ X (−2) ⊕ N ;<9 e ((3 + 17ζ )d + 2 57608l_8EosmN_8v| (17 + 3ζ )d + (−17 − 3ζ )d + (6 + 3ζ )d )  1 3 1 2 3 2 z1 3 S 3 1 1 1 3 1 4 (1 − ζ3 ) 4 (∧i=1 e1 di ) 8 (1 − ζ3 ) (im(e1 c) ∧ e1 d2 ∧ e1 d3 ∧ e1 d4 ) 8(3 + 17ζ3 ) (1 − ζ3 ) (e1 c) ⊗ (∧i=2,3,4 (RS,1 ◦ f˜1 )(e1 di )) 8(3 + 17ζ3 ) (1 − ζ3 ) (e1 c) ⊗ −4α(u0 )(log l)(log q)(z10 )−1 . 8(3 + 17ζ3 ) = = 7→ = 53m w ^Vb<wlefba^V`8 α(u ) 8zw:mNZ09;<{f8l_K`608 8lb<8ljk8lZ?` β = N (u ) ∈ E  ! 8 6O^ ‡8z`6O^V` ^VZ0{ ^V_8qefZf;i`9;iZ E 2576?e09 /8Kw ^VZ _;i`8q`608:98q;iZ `8l_jk9mVo `608=O^N9;<9 {w , w } ^N97osmNbib<m 9   0 p  h(βp ) h2 (βp ) βp h(βp )  0  1 2 βp h(βp ) = w1−28 w26 , 2 = w1−6 w2−34 . h(βp ) h (βp ) 8lZ0w:8v|  K/E α(u0 ) = (log |βp | + ζ3 log |h2 (βp )| + ζ32 log |h(βp )|)   h(βp ) βp | − ζ3 log | 2 | = log | h(βp ) h (βp ) = −28 log |w1 | + 6 log |w2 | + 6ζ3 log |w1 | + 34ζ3 log |w2 | = −ζ32 (6 + 34ζ3 )(log |w1 | − ζ3 log |w2 |) = ζ32 (6 + 34ζ3 )L∗ (χ1 , 0)# . < ;iZ0w:8 L∗ (χ , 0)# = (1 − ζ )(log l)(log q)L∗ (χ , 0)# ;i`*oRmNbib<m 9#`6O^V`#`6087;ij ^Vu‡8mVo e (z00 ⊗ z ⊗ z0 ) ;iZ R ;<9 −ζ e L (χ , 0)  1 S × 2 ! 

3 2 3 1  ∗ S  e  1 # 1    1 2 8l` e = (e + g − g − g ) *57608lZ e ;<9^VZ);iZ0{f8:w:mNjzx mv9I^V=fb<8;<{f8ljzxmN`8lZ?`7mVo R[G] 6f;<w6 bi;<8:9/;iZ  Y o V ;<9n^kg0Zf;i`8lbi>]u‡8lZ08l_I^V`8:{ xf_m 8:wl`; ‡8jkmG{}efb<8m ‡8l_ R |O`608lZ /8w ^VZƒw:mNZ09;<{f8l_ R = M (R) `608 e V ^N9^zjkmG{}efb<8Km ‡8l_ ζ(R ) ≃ R / 57608lZ | /8K6O^ ‡8K`6O^V``6084{f8l`8l_jz;iZO^VZ?`mVo V m ‡8l_ R ;<9 `608K9I^Vjk8K^N9`608K{f8l`8l_jz;iZO^VZ?`mVo e V m ‡8l_ ζ(R ) 
1 4 3 2  1 2 3 2   3  3  2  2   2  3 2 /v„ nm K|}osmN_8 ^Nw6 v ∈ S |Ow:mNZ09;<{f8l_   0 λv,3 θv,3 µv,3 0 → Yv (−2)3 → Av,3 → Bv,3 → Yv,3 → 0, 6f;<w6;<9^VZ8f^Nwl`798:rTe08lZ0w:8nmVo R jkmG{}efb<8:9257608:98n98:rTe08lZ0w:8:9^V_8n{f8Eg0Z08:{ ;iZ `8l_jk97mVo3`608=O^N9I8:9 mVo A , B ^N97osmNbib<m 9   3 3 v=∞ ;i`6]=O^N9;<9 {e h a } ;i`6]=O^N9;<9 {e h b } ≃ R u‡8lZ08l_I^V`8:{]=T> e ≃ Y (−2) ≃ R  Av,3 ≃ R3 Bv,3 Yv,3 3 3  3 v i i 1 i=0,1,2 1 i=0,1,2 3 3 λv,3 (e3 ) = 2e3 a1 θv,3 (αe3 a1 + βe3 ha1 + γe3 h2 a1 ) = 2(βe3 hb1 + γe3 h2 b1 ) µv,3 (αe3 b1 + βe3 hb1 + γe3 h2 b1 ) = αe3 v=p ;i`6]=O^N9;<9 {e h a } ;i`6]=O^N9;<9 {e h b } ≃ R u‡8lZ08l_I^V`8:{]=T> e ≃ Y (−2) ≃ R Av,3 ≃ R3 Bv,3 Yv,3  3 3  v 3 i i 2 i=0,1,2 2 i=0,1,2 3 λv,3 (e3 ) = (e3 a2 + e3 ha2 + e3 h2 a2 ) θv,3 (αe3 a1 + βe3 ha1 + γe3 h2 a1 ) = (α − γ)e3 b2 + (β − α)e3 hb2 +(γ − β)e3 h2 b2 µv,3 (αe3 b2 + βe3 hb2 + γe3 h2 b2 ) = (α + β + γ)e3 3 /  v=q ;i`6]=O^N9;<9 {e h a } ;i`6]=O^N9;<9 {e h b } B ≃ R ;i`6]=O^N9;<9 {e h α } Y (−2) ≃ R  Av,3 ≃ R6 3  v,3 v 3 3 3  3 3 j i i=3,4,j=0,1,2 i 3 i=0,1,2 i i+1 i=0,1,2 Yv,3 = 0 λv,3 (αe3 α1 + βe3 hα2 + γe3 h2 α3 ) = 2αe3 (−a3 + a4 ) + 2βe3 ha3 +2γe3 h2 a4 θv,3 (αe3 a3 + βe3 ha3 + γe3 h2 a3 ) = 2(αe3 b3 + γe3 h2 b3 ) θv,3 (α0 e3 a4 + β 0 e3 ha4 + γ 0 e3 h2 a4 ) = 2(α0 e3 b3 + β 0 e3 hb3 ) µv,3 = 0 v=l Av = 0 ;i`6]=O^N9;<9 {e h b } ;i`6]=O^N9;<9 {e h } ≃ R  Bv,3 ≃ R3 3 3  Yv,3 3 i i 4 i=0,1,2 i=0,1,2 µv,3 (αe3 b4 + βe3 hb4 + γe3 h2 b4 ) = αe3 + βe3 h + γe3 h2 nm K|?b<8l` x ^VZ0{ y = 87u‡8lZ08l_I^V`mN_9BmVo Det(A ) ^VZ0{ Det(B ) _8:9x8:wl`; ‡8lbi>‡|T^V_;<9;iZfuoR_mNj `60K8 =O^N9I8:9jk8lZ?`;<mNZ08:{ ^V=m ‡8# b<9mf|fb<8l` x =8q`608Ku‡8lZ08l_I^V`mN_mVo Det(N ) ;i`6 _8:9x8:wl``mk`608 O= ^N9;<9jk8lZT`;<mNZ08:{ 8 ^V_bi;<8l_ 8l`    v,3 v,3 v,3 v,3   N,3 3
(z3 ⊗ z30 ) := (x∞,3 ∧ xp,3 ∧ xq,3 ∧ x1,3 ∧ xN,3 ) ⊗ [y∞,3 ∧ yp,3 ∧ yq,3 ∧ yl,3 ]−1 . 576f;<9;<9^zu‡8lZ08l_I^V`mN_mVo Det(A ⊕ N ) ⊗ Det (B )  8l` z ⊗ z =8`608q;ij ^Vu‡84mVo z ⊗ z ;iZ Det(X (−2) ⊕ N ) ⊗ Det (X ) B57608lZ | z = 608l_8 d F 9^V_88lb<8ljk8lZ?`9mVo X (−2) ⊕ N ≃ Ind R ⊕ Ind R ⊕ Ind M ⊕ N 2(∧ e d ) {f8Eg0Z08:{ =T> | • d = (1, 0, 0, 0) 3
3 0 8 i=1 3 i 1 3 −1 3 3 3 0 3 S 3  i S R R 3 −1 G C2 G C3 S,3 3 G V4 R R • d2 = (0, 1, 0, 0) 1 | • d3 = (0, 0, α1 , 0) | • d4 = (0, 0, h2 α2 , 0) • d5 = (0, 0, hα3 , 0) • d6 = (0, 0, 0, a5 ) | • d7 = (0, 0, 0, ha5 ) | | | • d8 = (0, 0, 0, h2 a5 )  cfef_`608l_ | z = [e (∞ − m) ∧ e (p − m) ∧ ∧e (l − m) ∧ e h(l − m) ∧ e h (l − m)]  57608qj ^Vx (R ◦ f˜ ) : X (−2) ⊕ N → X ;<9uv; ‡8lZ]=T> 0 3 3 3 3 3 3 2 −1  S,3 3 < mf|f`608;ij ^Vu‡8:9mVo e d ^V_8 S 3 3 e3 di 7→ −e3 X v∈S S,3 log |f˜3 (di )|v (v − m). 3 i • e3 d1 7→ β(u∞ )e3 (∞ − m) + 2(log l)e3 (l − m), • e3 d2 7→ β(up )e3 (∞ − m) + (log l)e3 (1 + h + h2 )(l − m), • e3 d3 7→ β(u1 )e3 (∞ − m) + 2(log l)e3 (l − m), • e3 d4 7→ β(h2 (u2 ))e3 (∞ − m), • e3 d5 7→ β(h(u3 ))e3 (∞ − m) + 2(log l)e3 h(l − m), • e3 d6 7→ β(u0 )e3 (∞ − m) + (log p)e3 (p − m), • e3 d7 7→ β(h(u0 ))e3 (∞ − m) + (log p)e3 h(p − m), • e3 d8 7→ β(h2 (u0 ))e3 (∞ − m) + (log p)e3 h2 (p − m), 608l_8 β(u) = − log  57608qjkmG{}efb<8 P ;<9u‡8lZ08l_I^V`8:{]=T> e h c, j = 0, 1, 2 /57608;ij ^Vu‡8:9mVo*`608:9I8;iZ X (−2) ⊕ N <; 9uv; ‡8lZ]=T>  ug2 ·ug3 u·ug1 3 3 j  • e3 c 7→ e3 (−2d1 + d4 − 2d5 − d6 + d8 ), S 3 3 1. • e3 hc 7→ e3 (d1 − d5 + d6 − d7 ), • e3 h2 c 7→ e3 (d7 − d8 ). 8l` z {f8lZ0mN`8 `608u‡8lZ08l_I^V`mN_KmVo Det(P ) ;i`6ƒ_8:9x8:wl`n`m `608z^V=m ‡8 =O^N9;<9457608lZ | ;i`6 `608:98 vV^ bie08:9mNZ08q6O^N9
 00 3   3  z3 = −2(∧j ;ij (e h c) ∧ e d ∧ e d ∧ e d ∧ e d ∧ e d ) 3 j 3 2 3 3 3 4 3 5 3 7 7→ 8([z300 ]−1 ) ⊗ β(h2 (u2 ))(log l)3 (log p)[z 0 3 ]−1 . 57608 efZf;i` u ;<9 `608 u‡8lZ08l_I^V`mN_]mVo^VZ ;<{f8 ^Vb;iZ F bi>T;iZfu ^V=m ‡8 q ,57608 w60mN;<w:8 mVo`6f;<9uv; ‡8:9 B 576?e09 |f`608;ij ^Vu‡8qmVo e (z ⊗ z ⊗ z ) ;iZ β(u ) = −| log |w || = −| log | || = − L (χ , 0) ;<9 2 576f;<9960m 97`6O^V``608n;ij ^Vu‡8mVo e (z ⊗ z ⊗ z ) ;iZ ζ(R[G]) −e L (χ , 0) = −e L (χ , 0) ;<9  ζ(R[G]) −e L (χ , 0)  2 w3 σc (w3 ) 2 3 2 ! 

 374 -"    2 ∗ 3 S 2 ∗ 2 S 2 3 ∗ S 1 4 ∗ 2 00 3 3 0 3 2 00 2 2 0 2  # 2 3 # !                K/Q   
 K/Q   /'    2   ,       3             "  /'            '    2  2   1       3 K/Q   3  + /> `608w:mNjzxfef`I^V`;<mNZ09;iZ `6084xf_8 G;<mNe0998:wl`;<mNZ09;i`osmNbib<m 9n`6O^V``608;ij ^Vu‡8;iZ ζ(R[G])   × mVo7`608zu‡8lZ08l_I^V`mN_mVo Det (P ) ⊗ Det(A ⊕ N ) ⊗ Det (B ) |3mN=f`I^V;iZ08:{ =?>ƒ`608 Z[G] =O^N98:9 so mN_ A, B, P ^VZ0{ N ;<9 −1 R R R −1 R −e0 L∗S (χ0 , 0)# − ζ32 e1 L∗S (χ1 , 0)# − e2 L∗S (χ2 , 0)# = (−1, −ζ32 , −1)L∗S (0)# . ef``6f;<9;<9;iZ δ ((A , B ; ψ )) #57608l_8EosmN_8v| −1 S S S T Ω(K/Q) = (AS , BS ; ψS ) − δ(L∗S (0)# ) = δ((−1, −ζ32 , −1)L∗S (0)# /L∗S (0)# ) = δ((−1, −ζ32 , −1)) = 0, 608l_8`608ba^N9`8:rTeO^Vbi;i` > oRmNbib<m 97oR_mNj&`608no ^Nwl`7`6O^V``608q8lb<8ljk8lZ?` (−1, −ζ , −1) ;<97`608;ij ^Vu‡8qmVo # 576f;<9w:mNjzxfb<8l`8:9`608qxf_mGmVo*mVo*`608K57608:mN_8lj  −h ∈ K (Z[G])   1 2 3 #$" 1~ Z08 w:mNefb<{ x mv99;i=fbi> 8G`8lZ0{ `6089I^Vjk8);<{f8 ^]osmN_ ‡8l_;hoQ>G;iZfuƒ`608)w:mNZ 8:wl`ef_8 oRmN_z^VZ  ;iZ}g0Zf;i`8os^Vjz;ibi> mVo A  8}`8lZ09;<mNZ0957608j ^V;iZƒ;<{f8 ^k608l_8 mNefb<{ =89;ijz;iba^V_`m `6O^V`qmVo y 6f;iZT=fef_u   . ~ | . / † |}`6O^V`7;<9 |}`mb<mTm oRmN_/o ^Vjz;ibi>)mVo 8}`8lZ09;<mNZ09 ;i`6
[9ef=w:mNZfuv_e08lZ?`qefZf;i`7uv_mNefx09^VZ0{  }` wlba^N998:9   4   1/
   



   O  8l` K/Q = 87^g0Zf;i`8 q^Vb<mN;<928G`8lZ09;<mNZ ;i`6 q^Vb<mN;<9*uv_mNefx G  8l` E =8^VZk8lbibi;ixf`;<wwlef_ ‡8{f8Eg0Z08:{ m ‡8l_ Q 2 c0mN_^VZ?>)gO8lb<{]8} `8lZ09;<mNZ L mVo Q | /8Kb<8l` E := E × Spec(L)  ef_;iZT`8l_8:9`;<9;iZ ` 608jkmN`; ‡8 M = h (E )(1)  mN`8n`6O^V``608 K ^Vb<mN;<9/uv_mNefx G ^Nwl`97mNZ M ^VZ0{ 608lZ0w:8 A := Q[G] ^Nwl`9mNZ]`608K_8 ^Vbi; ^V`;<mNZ09mVo M  9 Z0mN`8:{ ;iZ y 6O^Vxf`8l_ ~ |#`608 jkmN`; G;<w Ls oQefZ0wl`;<mNZ$^N9I9mTwl;a^V`8:{ `m M ;<9k^ `efxfb<8 mVo` ;<9`8:{ ^N998&! 8l;ib Ls oQefZ0wl`;<mNZ092 5dmK=8xf_8:wl;<98v|?osmN_^qtn;i_;<w6fb<8l`wI6O^V_I^Nwl`8l_ χ b<8l` L(E ⊗ χ, s) {f8lZ0mN`8`608 ` ;<9`8:{ ^N9I98&! 8l;ib Ls oRefZ0wl`;<mNZ# 57608lZ 8q6O^ ‡8
  

   L  1 Spec(Q)  K        L(M, s) = (L(E ⊗ χ, s + 1)) χ∈Gb . ! 8K96O^Vbib {f8lZ0mN`8K=?> L (1) `608b<8 ^N{};iZfu)w:mG8  wl;<8lZ?` ∗ L∗ (M, 0) = (L∗ (E ⊗ χ, 1))χ∈Gb . ef_K^V;ij ;<9n`m)_8lba^V`8`6f;<9 L v^Vbie08`m `608z^V_;i`6fjk8l`;<w v^Vbie08:9K^V_;<9;iZfu oR_mNj ^ w:mNjzxfb<8 ^N9n;iZ   `608w:mNZ 8:wl`ef_8#B53mzw:mNjzxfef`8`608q^V_;i`6fjk8l`;<wK9;<{f8v| /8j ^ ‡8w:8l_`I^V;iZ ^N99efjzxf`;<mNZ09`6O^V`9;ijzxfbi;hoQ> ` 608Kw ^Vb<wlefba^V`;<mNZ09 
,+ ,' ( K/Q
,+ ,' 2 X(K)
,+ ,' ? E(K)    #      0      #       3     -23  
  43 1%1 nmN`87`6O^V` K/Q ;<9 `I^Vjk8lbi>_I^Vjz;hgO8:{z;ho^VZ0{zmNZfbi>;ho`6087_;iZfu O ;<92^nb<mGw ^Vbibi>KoR_8:8 Z[G] jkmT{}efb<8 mVo0_IV^ Z .  wEo  .%.  † dY[Z4`608/8f^Vjzxfb<8 /8w:mNZ09;<{f8l_ |vmNZ08#6O^N9 Cl(Z[G]) = 1 ^VZ0{4`608l_8EoRmN_8/;i` osmNbib<m 9 ` 6O^V` O ;<9;<9ImNjkmN_xf6f;<w `m Z[G] ^N9q^ Z[G] jkmT{}efb<8#4 Y Z `608oRmNbib<m ;iZfu0| /8z^N99efjk8;iZ ^N{f{};i`;<mNZ ` m `608g0_9`n^N99efjzxf`;<mNZ]`6O^V``608l_848G ;<9`9 α ∈ O 9e0w6]`6O^V` O ≃ Z[G]α   K      K 0

 (    1       " 
  ! " K K 0   57608nx 8l_;<mG{)j ^Vx α ;<97^VZ;<9mNjkmN_xf6f;<9j(= 8l` /8:8lZ `608 78l``; ^VZ0{{f8:p6O^Vj&_8 ^Vbi; ^V`;<mNZ09 #57608l_8 so mN_8 ` /m mVo/`608 `8l_jk9q;iZ `608z{f8Eg0Zf;i`;<mNZ mVo Ξ(M ) ^V_8z`_; G;a^Vb&cfef_`608l_ |dmNZ08 6O^N9 H (Q, M ) ≃ |}^VZ0{ H (Q, M (1)) ≃ E(K) ⊗ Q | 6f;<wI6 ^V_8=mN`6)`_; G;a^Vb=T>k`608n98:w:mNZ0{ ^N9I9efjzx  E(K) ⊗ Q `;<mNZ ^V=m ‡8## 57608l_8EosmN_8 8K6O^ ‡8  M   1 f  ∗  ∗ 1  ∗   1 . † Ξ(M ) = v∈S∞ [Hv (M )Gv ]  [HdR /F 0 ]. 8l` E =8 ^ l_mNZ jkmG{f8lb#osmN_ E m ‡8l_ Z  8l` ω = 8 ^ u‡8lZ08l_I^V`mN_mVo H (E, Ω ) )57608lZ `608 ;ij ^Vu‡K8 mVo*`608qj ^Vx 

0 1 E H1 (E(C), Z) → C Z ω γ 7→ γ ;<9B^ Z ba^V``;<w:8;iZ C  ! 87b<8l` Ω `m= 87`608b<8 ^N9`2x mv9;i`; ‡8_8 ^Vb0Z?efj= 8l_#;iZ `6f;<92;ij ^Vu‡8^VZ0{ 8{f8Eg0Z08 608l_8 r ;<9n`608Z?efj= 8l_KmVoBw:mNZfZ08:wl`8:{ w:mNjzxmNZ08lZ?`9q;iZ E(R) q576f;<9n;<9n`6084_8 ^Vb*x8l_;<mT{ Ω = rΩ ^N99mGwl;a^V`8:{`m E  wEo 0/1 †   0  0  " -,',-B;   ϑ∞ !   -
   ')     Ω−1   X   Ξ(M )  g(α0 )g −1  .  R[G] &
    -  / # p8:w ^Vbib0`6O^V`#`608;<9mNjkmN_xf6f;<9j ϑ : R[G] ≃ Ξ(M ) ⊗ R ;<9#w:mNZ09`_e0wl`8:{ke09;iZfuK`608x 8l_;<mG{ j V^ x α n 57608l_8EoRmN_8v| /896O^Vbibdg0_9` _;i`8{fm Z `608 7 8l``;*^VZ0{ƒ{f8:p6O^Vj _8 ^Vbi; ^V`;<mNZ09 | ^VZ0{ `608 w:Nm __8:9xmNZ0{};iZfu x8l_;<mT{j ^Vx = 8l` /8:8lZ `608lj  8l` H = ⊕  ! 8 96O^Vbib;<{f8lZ?`;hoR>$8 ^Nw6 9efjzj ^VZ0{$mNZ `608 H (σE (C), 2πiQ)  3    M 
B σ∈Hom(K,C) 1 K ∞ g∈G Q  1= _;iuv6?`6O^VZ0{]9;<{f8 ;i`6]`608q{}eO^Vb360mNjkmNb<mNuv> T;a^`608;<9mNjkmN_xf6f;<9j   57608l_8EosmN_8v| /846O^ ‡8  H 1 (σEK (C), 2πiZ) ≃ Hom(H1 (σEK (C), Z), 2πiZ).  HB ≃ Hom (⊕σ H1 (σEK (C), Z), (2πi)Z) . ! 8796O^Vbib}{f8lZ0mN`8/=T> H | `608/g f8:{9ef=fjkmT{}efb<87mVo H efZ0{f8l_*`6087^Nwl`;<mNZ mVo w:mNjzxfb<8w:mNZ efu?^V`;<mNZ nmN`8`6O^V` H ;<9`608qg0_9``8l_j ;iZ  1  . †  8l` γ ^VZ0{ γ =8 Z u‡8lZ08l_I^V`mN_9*mVo H (E (C), Z) 9e0wI64`6O^V` γ ;<9 _8 ^Vb |V`6O^V`3;<9 g }8:{qefZ0{f8l_d`608 ^Nwl`;<mNZ mVo/`608w:mNjzxfb<8 ƒw:mNZ efu?^V`;<mNZ K57608lZ | σγ ^VZ0{ σγ ^V_8u‡8lZ08l_I^V`mN_9osmN_ H (σE (C), Z)  < ;iZ0w:8 K ;<9`mN`I^Vbibi>_8 ^Vb |f;i`7osmNbib<m 9`6O^V` σγ ;<97_8 ^V&b  nm {f8Eg0Z08 + B  B + B
1 2 1 K 1 1 2   1 K  1 γ˜1 : ⊕σ H1 (σEK (C), Z) → (2πi)Z =?> 98l``;iZfu ;ho σ ;<{ | j = 1 γ˜ (σγ ) = mN`608l_ ;<98#  0 nmN` 8 `6O^V`48 ‡8l_> 8lj= 8:{f{};iZfu mVo K ;iZ C w:mN__ 8:9xNm Z0{f9q`m ^ _8 ^Vb2;iZ}g0Zf;i`8 xfba^Nw:8#z57608l_8EoRmN_8v|3`608 ^Nwl`;<mNZ mVoB`60z8 w:mNjzxfb<8ƒw:mNZ efu?^V`;<mNZ mNZ `608:98 8lj= 8:{f{};iZfu‡9q;<9n`_; G;a^Vb&4 576Te09 | ;i`noRmNbib<m 9`6O^V` γ˜ u‡8lZ08l_I^V`8:9 H ^N9^ Q[G] jkmG{}efb<84^ oR` 8l_;<{f8lZ?`;hoR>T;iZfu T;a^ `608K^V=m ‡8K;<9mNjkmN_xf6f;<9j  nm K| 8Kw:mNZ09;<{fl8 _`6084{f8:p6O^Vj _8 ^Vbi; ^V`;<mNZ mN`8`6O^V`=T> < 8l__8K{}eO^Vbi;i` > mNZ08q6O^N9 1  j   2πi      1    + B    HdR /F 0 = H 1 (EK , OEK ) ≃ H 0 (EK , Ω1EK )∗ 608l_8 Ω ;<97`60849608 ^ o*mVo2{}; $ 8l_8lZ?`;a^Vb<9 | ^VZ0{ ∗ {f8lZ0mN`8:9`608K{}eO^Vb&  1 EK    #   E -
 E      3 /              3 OK 
     
-      E -
      E   Z 
       -    K  EK   OK         )  ! 
   EOK ≃ E ×Spec(Z) Spec(OK ).  mN`8`6O^V``608xf_;ijk8:9mVoB=O^N{ _8:{}e0wl`;<mNZƒmVo E ^VZ0{ `6084xf_;ijk8:9`6O^V`n_I^Vjz;hoR> ;iZ K/Q {fm Z0mN` ;iZ?`l8 _98:wl`ƒ57608l_8EosmN_8v| E × Spec(O ) ;<9 ^VZ ^V=8lbi;a^VZ 9Iw608ljk8m ‡8l_ Spec(O )  57608 l8 jzj ^kZ0m osmNbib<m 9oQ_mNj y mN_mNbiba^V_> .  k1 ;iZ  ~     3  
Spec(Z)   K K 1€ cf_mNj `6084^V= m ‡8 8ljzj ^ /8q6O^ ‡8  

H 0 (EK , Ω1EK ) ≃ H 0 (EOK , Ω1EO ) ⊗OK K ≃ H 0 (E, Ω1E ) ⊗Z OK ⊗OK K. K 8l` ω =8/^u‡8lZ08l_I^V`mN_doRmN_ H (E, Ω )  57608lZ | ((ω ⊗α )⊗1) ;<9 ^ Q[G] u‡8lZ08l_I^V`mN_ oRmN_ H / Fil  ! 8K96O^Vbib wI60mTmv9I8 ω 9e0wI6]`6O^V``608qx8l_;<mT{ R ω ;<9xf_8:wl;<9I8lbi> Ω  oQ`8l_;<{f8lZT`;hoQ>G;iZfu `608 7 8l``;d^VZ0{ {f8:p6O^Vj _8 ^Vbi; ^V`;<mNZ09n^N9^V= m ‡8v|0`608x8l_;<mT{]j ^Vx
0 0 1 E 0 0 ∗ 0 0 dR 0 γ1  + αM : HB,R → HdR,R / Fil0 ;<9uv; ‡8lZ]=T>  αM (γ ⊗ α) : ω 7→ Z ω γ −1   X  g(α0 )g −1  . < ;iZ0w:8 Q[G] ;<998ljz;<9;ijzxfb<8 `608 {f8l`8l_jz;iZO^VZ?`oRefZ0wl`mN_;<9uv; ‡8lZ =?> `608)j ^Vx Det {f8Eg0Z08:{ ;iZ y 6O^Vxf`8l_ . 257608 Q[G] u‡8lZ08l_I^V`mN_9 γ˜ ^VZ0{ ((ω ⊗ α ) ⊗ 1)∗ osmN_ H + ^VZ0{ H / Fil0 _8:9x8:wl`; ‡8lbi>‡| g∈G ^Vbib<m 9ke09`m ;<{f8lZT`;hoQ> Ξ(M ) ;iZ09;<{f8 Ξ(M ) ≃ R[G]  53mƒ=8 xf_8:wl;<98v|B`608 ;ij ^Vu‡8 mVo (γ˜ ) ⊗ ;iZ R[G] ≃ Ξ(M ) ;<9 Ω P g(α )g  · Q[G]  576f;<9 ((ω ⊗ α ) ⊗ 1) · Q[G] = Ξ(M ) w:mNjzxfb<8l`8:9`608Kxf_mTmVo2mVo `6084\2_mNx mv9;i`;<mNZ 1 0 0 dR B  1 R 0 0    −1 R   "    " 0 g∈G −1 −1 ϑl 8l` T (E) := lim E(Q̄)/l ^VZ0{qb<8l` V (E) := T (E)⊗ Q  57608lZ V (E) ;<93`608 l[^N{};<w#_8 ^Vbi; ^V`;<mNZ mVo M = h (E ←−)(1) / cfef_`608l_ | T (E) ;<9^ K^Vb<mN;<99`I^V=fb<8 Z [G] ba^V``;<w:849;i``;iZfuz;iZ09;<{f8 V (E) #57608 ^Vbiu‡8l=f_I^V;<w 9;<{f8ƒmVo y mNZ 8:wl`ef_8 ~  1 † ;iZ ‡mNb ‡8:9 RΓ (O [ ], T (E)) |q^ x8l_ oR8:wl` Z [G] w:mNjzxfb<8| 6f;<w6 efx mNZ `8lZ09ImN_;iZfu ;i`6 Q ;<9r?eO^N9;  ;<9ImNjkmN_xf6f;<wK`m RΓ (O [ ], V (E))  5dmkw:mNjzxfef`8 [RΓ (O [ ], T (E))] 8Kw:mNZ09;<{f8l_`608K{};<9`;iZfuvef;<9608:{ `_;a^VZfuvb<8
n l l n 1 K l Zl l  c  1 K S l 1 K S l c l c l l   l  l 1 K S l l 1 RΓc (OK [ ], Tl (E)) → RΓf (K, Tl (E)) → ⊕v∈S RΓf (Kv , Tl (E)), S 608l_8 RΓ ;<97`608   w:mNjzxmNZ08lZ?`  wEo  „ † mVod`608ne09eO^Vb w:mNjzxfb<8 RΓ mVo w:mGw6O^V;iZ09 #c0mNbib<m ;iZfu   f  1E `608K{f8Eg0Zf;i`;<mNZ09oQ_mNj  E d^VZ0{  „ | /8q6O^ ‡8q`6O^V`osmN_ v - l∞, RΓ (K , T ) ;<9r?eO^N9;  ;<9ImNjkmN_xf6f;<wq`m   v f l 1−Fr Tl (E)Iv −→v Tl (E)Iv 608l_8`608jkmT{}efb<8:97^V_8xfba^Nw:8:{);iZ){f8luv_8:8  ^VZ0{ .  c0ef_`608l_ |?osmN_ v|∞ |‡`608w:mNjzxfb<8 RΓ (K , T ) <; 9{f8Eg0Z08:{ `mk= 8`608Kw:mNjzxfb<8 RΓ(K , T ) #c ;iZO^Vbibi>‡|}osmN_ v|l | /8qZ0mN`8K`6O^V`  v f l  v l Hf1 (Kv , Vl (E)) ≃ (lim E(Kv )/ln ) ⊗Zl Ql ≃ tE ⊗Q Ql ←− n 608l_8 t ;<9B`608n`I^VZfu‡8lZT`79xO^Nw:8nmVo E ^VZ0{ `608ba^V``8l_7;<9mNjkmN_xf6f;<9j&;<9/uv; ‡8lZ=?>k`608n8}x mNZ08lZT`;a^Vb j ^Vx 57608 Z [G] 9ef=fjkmG{}efb<8 lim E(K )/l w ^VZ 6O^ ‡8 `mN_9;<mNZ xmN;iZ?`9 |3^VZ0{ƒ`608l_8EosmN_8 ;<9Z0mN`4^ u‡mGmT{ wI60mN;<w:8kosmN_ H (K , T (E))←− 38l` E =8 ^ l_mNZ jkmG{f8lb2osmN_ E 57608lZ `608l_8 ;<9^oR_8:8 .  {};ijk8lZ09;<mNZO^Vb Z[G] ba^V``;<w:8 H (E , Ω ) 9;i``;iZfuk;iZ09;<{f8 t ≃ H (E , Ω )  ! 8K9I8l`   E K l  n v n 
1 f v OK l 0 OK K 1 EO K 0 E 1 EK K Hf1 (Kv , Tl (E)) = H 0 (EOK , Ω1EO ) ⊗Z Zl . K 576f;<9uv; ‡8:9n^xf_m 8:wl`; ‡8 Z [G] ba^V``;<w:89;i``;iZfuz;iZ09;<{f8l_ H (K , V (E))    1 f l   # ;   3  !  3   3   
 3 Kv ≃ C  . 
"      σ #  RΓf (Kv , Tl (E)) 
  +    #     ' v - l∞ #Hf1 (Kv , Tl (E))  v|l l    '    )  v|∞ Kv ≃ R   
v  $      /    !     H1 (σE(C), Z) ⊗ Zl   H1 (σE(C), Z)+ ⊗ Zl 
 '       
    /                  v Hf1 (Kv , Tl (E)) ≃ H 0 (EOK , Ω1EO ) ⊗Z Zl K ⊂ 0 H (EK , Ω1EK ) ⊗Q Ql ≃ Hf1 (Kv , Vl )     !)   v ∈ S       '   RΓf (Kv , Tl (E)) → RΓf (Kv , Vl (E)), "     -         / !       & "   Q 3 l  . 
"   3  " $/' - 1     ! 3 1?„ 57608g0Zf;i`84w:mN60mNjkmNb<mNuv> ;iZ `608quvb<mN=O^Vb3w ^N98;<9uv; ‡8lZ =?>`608osmNbib<m ;iZfu \2_mNxmv9;i`;<mNZ     #
!!&     X(EK )  Hfi (K, Tl (E)) ≃ 
      
       0   l 6= 2, 3 3  ,)  E(K) ⊗Z Zl     Hom (E(K) ∞ , Q /Z ) l Z l l i=0  i=1  i=3 0 → X0l∞ → Hf2 (K, Tl (E)) → HomZ (E(K), Zl ) → 0            (   3        limn E(Kv )/ln ←− 3 X0 # % 
-  < 8:8  E 3mN_  ~v~  oRmN_xf_mGmVos9mVo `608K^V=m ‡8   #$"* . 576084r?eO^N9;  ;<9ImNjkmN_xf6f;<9j
                  4       8ljzj ^N9 ⊕v∈S∞ RΓf (Kv , Tl (E)) ⊗ Ql → ⊕v∈S∞ RΓf (Kv , Vl (E)) ;iZ0{}e0w:8:9n^N9;<9mNjkmN_xf6f;<9j ⊕σ:K→CH1 (σE(C), Z)+ ⊗ Ql ≃ DetQl [G] ⊕v∈S∞ RΓf (Kv , Vl (E)), 6f;<wI6 ;<9xf_8:wl;<98lbi> `608zw:mNjzxO^V_;<9mNZ ;<9mNjkmN_xf6f;<9j =8l` 8:8lZ `608 l`I^Vb<8k^VZ0{ƒ9;iZfuvefba^V_Kw:mN60m  kj mNb<mNuv;<8:9  ~  Y o E(K) ^VZ0{ X(E/K) ^V_8 `_; T;a^Vb*`608lZ RΓ (K, T ) ;<9r?eO^N9;  ;<9mNjkmN_xf6f;<wz`m `608  8l_m w:mNjzxfb<8 ^VZ0{ 608lZ0w:8    l∞ f l 1 [RΓc (OK [ ], Tl (E))] = [X0l∞ ]  [⊕v∈S RΓf (Kv , Tl (E))]. S nmN`8q`6O^V``608qw:mNZT`_;i=fef`;<mNZ mVo [X ] ;<9 e09``608q`mN_9;<mNZ mVo lim E(K )/l  ←− ! ;i`6]`608K^V=m ‡8 8ljzj ^N9nmNZ08qrTef;<w Gbi>{f8:{}e0w:8:9`608osmNbib<m ;iZfu   0 l∞  " -,',-    
! 
[RΓc (OK [ S1 ], Tl (E))]     n     &$       [RΓc (OK [ S1 ], Vl (E))] v n # X(K)  -
 Zl [G]   E(K)        -23       (ω0 ⊗ α0 ) 3  1   21dI /  - ') I
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  8l` H = {z ∈ C|Im(z) > 0} {f8lZ0mN`8`608efxfx 8l_6O^Vbho xfba^VZ08K^VZ0{b<8l` H = H ∪ Q ∪ {∞} {f8lZ0mN`8 `06 88G`8lZ0{f8:{efxfx8l_ 6O^Vbhoxfba^VZ08# 57608/uv_mNefx SL (Z) ^Nwl`9 {};<9Iwl_8l`8lbi>^VZ0{{};<9w:mNZ?`;iZTe0mNe09bi>zmNZ H  c0mN_4^VZT> w:mNZfuv_e08lZ0w:8k9ef=fuv_mNefx Γ mVo SL (Z) b<8l` X = 8`608jkmG{}efba^V_K9ef_ os^Nw:8 Γ\H Kc0ef_`608l_ | b<8l` S (Γ) =8`608K9xO^Nw:8KmVo#^Vbib wle09x osmN_jk9mVo /8l;iuv6T` ~ osmN_ Γ 2 57608l_8q;<9^=f;ibi;iZ08 ^V_xO^V;i_;iZfu
∗ ∗ 2 2 ∗ Γ  2 S2 (Γ) × H1 (XΓ , C) → C (f, γ) → 2πi Z f (z)dz γ `6O^V`;iZ0{}e0w:8:9n^VZ ;<9mNjkmN_xf6f;<9j S (Γ) ≃ H (X , C) mVo2w:mNjzxfb<8 ‡8:wl`mN_9xO^Nw:8:9 c0mN_wle09x09 α, β ∈ Q ∪ {∞} w:mNZ09;<{f8l_^ 9jkmGmN`6xO^V`6];iZ H oR_mNj α `m β 2c0mN_n^w:mNZfuv_e08lZ0w:8 9ef=fuv_mNefx Γ mVo SL (Z) b<8l` {α, β} {f8lZ0mN`8`608K;ij ^Vu‡8mVo#`608KxO^V`6 ;iZ X  nmN`84`6O^V`;ho α ^VZ0{ ^V_8 Γ 8:r?ef; v^Vb<8lZ?` |B`608lZ {α, β} ;<9k^ wlb<mv98:{ xO^V`6 ;iZ X ^VZ0{ 608lZ0w:8{f8Eg0Z08:9k^VZ 8lb<8ljk8lZ?`kmVo β  576084‚ƒ^VZf;iZ  tn_;iZ}oR8lb<{ƒ57608:mN_8lj 9I^:>}9`6O^V`osmN_n^VZT> wle09x09 α, β ∈ Q ∪ {∞} mNZ0846O^N9 H (X , Z)  mN`8K`6O^V``6f;<98lb<8ljk8lZ?`;<9;iZ0{f8lx8lZ0{f8lZ?`mVo*`608KwI60mN;<w:84mVo*`608qxO^V`6 oR_mNj {α, β} ∈ H (X , Q) ^VZ0{ β d 57608l_8EosmN_8v| 8w ^VZ  ^VZ0{ ;ibib † ;<{f8lZ?`;hoR> {α, β} ^N9B^VZz8lb<8ljk8lZ?`/mVo H (X , Q)  ! 896O^Vbib α ^Vb<9mK{}_mNx `6089ef=09wl_;ixf` Γ 608lZ08 ‡8l_B`608Z0mN`I^V`;<mNZ);<92efZO^Vj=f;iuve0mNe09# c0mN_7^qwle09xzosmN_j f ∈ S (Γ) b<8l` 2 1  +∧ Γ ∗  2 Γ Γ  Γ 1 Γ Γ  Γ 1 Γ   Γ  1 Γ  2 h{α, β}Γ , f i := 2πi Z β f (z)dz u?^V;iZ |fZ0mN`8q`6O^V``6f;<9{f8Eg0Zf;i`;<mNZ ;<9;iZ0{f8lx8lZ0{f8lZ?`mVo*`608qw60mN;<w:8mVo `608KxO^V`6 oQ_mNj α `m β  c0mN_ γ ∈ SL (Z) b<8l` (γ) = {γ(0), γ(∞)}  38l` C(Γ) = 8^ Q ‡8:wl`mN_9xO^Nw:8K`6089>Tj= mNb<9 (γ) ^Nq9 =O^N9;<9 | 608l_8 γ _efZ09Km ‡8l_^98l`KmVo_8lxf_8:98lZ?`I^V`; ‡8:9osmN_q`608zw:mv98l`9KmVo SL (Z)/Γ  8l` R = 8 ` 608qb<8EoR`;<{f8 ^Vb3mVo Z[SL (Z)] u‡8lZ08l_I^V`8:{ =?> I + S ^VZ0{ I + (T S) + (T S) | 608l_8 α  
2   
2 2  2
 S= 0 −1 1 0   , T =  1 1 1 0   8l` B(Γ) = C(Γ)R  8l` H (Γ) {f8lZ0mN`8`6084oQ_8:8 ^V= 8lbi;a^VZ uv_mNefx mNZ `608wle09x09mVo Γ  8l` Z(Γ)
0 
= {f8lZ0mN`8q`608 ‡8l_Z08lbdmVo `608qj ^Vx C(Γ) → H (Γ) {f8Eg0Z08:{ =?>  0 (γ) → [γ(∞)] − [γ(0)] 608l_8 [α] {f8lZ0mN`8:9`608 Γ mN_=f;i`mVo `6084wle09x α #57608lZ | B(Γ) ⊂ Z(Γ)  38l` H(Γ) = Z(Γ)/B(Γ)  " -,',-

 Q      ! H(Γ) 
 H (X , Q)  - / ) -
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  / ! #  3 <       1          Γ (γ) 7→ {γ(0), γ(∞)}Γ . 8:8  ~v€             "  c0mN_^zxf_;ijk8 p b<8l` T {f8lZ0mN`8`608 8:w ‡8mNx8l_I^V`mN_576?e09 |OoRmN_n^kwle09x oRmN_ f ∈ S (Γ) |`608 8:w ‡8 mNx8l_I^V`mN_^Nwl`9^N9    p  2  f |Tp = f  p 0 0 1  + p−1 X  f  1 k 0 p  . nm K|0{f8Eg0Z08q`608^Nwl`;<mNZ mVo `608:9I8 8:w ‡8KmNx8l_I^V`mN_9mNZ H (X , Q) =?>     k=0 1 57608lZ mNZ08q6O^N9   L Γ  p−1  X α+k β+k Tp {α, β} = {pα, pβ} + . , p p k=0 h{α, β}, f |Tp i = hTp {α, β}, f i .     "  8lZ0w:8EoRmN_`6 | 8b<8l` Γ = Γ (N ) ^VZ0{ f ∈ S (Γ)  ! 8g0_9`nZ0mN`8`6O^V`n`608‚ 8lbibi;iZ `_I^VZ09oRmN_j osmN_ `06 8 LsoRefZ0wl`;<mNZ ^V``I^NwI608:{]`mk^ Z08 osmN_j f ;<9uv; ‡8lZ]=T>   0 2   s L(f, s) = (2π) Γ(s) Z i∞ 0 < ef=09`;i`ef`;iZfu s = 1 ;iZ]`608K^V=m ‡8 /8qu‡8l` L(f, 1) = −2πi −1 Z i∞ 0 (−iz)s f (z) dz . z f (z)dz = − h{0, ∞}, f i . =. nmN`8`6O^V`^z9;ijz;iba^V_;<{f8lZT`;i` > 60mNb<{f9osmN_` ;<9`8:{ osmN_jk9 f ⊗ χ    " -,',-    0   l      f    "      -
 N 3   "    χ     4  (l, N ) = 1 l L(f ⊗ χ, 1) = "   g(χ) = n  n=1 χ(n)ζl Pl  g(χ) X χ̄(a) h{0, a/l}, f i , l a=1  4! !&$3 mN`8`6O^V` |f=T>`608K{f8Eg0Zf;i`;<mNZ mVo*`608 q^Ve0999efj mNZ08q6O^N9  3   χ̄(−a)g(χ) = l X χ(−a −1 n)ζln = χ(n)ζl−an . n=1 n=1 57608l_8EosmN_8 /8qu‡8l` l X  l χ(n) = nm K|f;ho f (z) = P   n≥1 an e 2πinz g(χ) X χ̄(−a)ζlan . l a=1 ;<9`608 q 8}xO^VZ09;<mNZ]mVo f `608lZ (f ⊗ χ)(z) = = X χ(n)an e2πinz n≥1 l X g(χ) X n≥1 l a=1 l = g(χ) X χ̄(−a)f |γa l a=1   608l_8 γ =  l a   a 0 l χ̄(−a)an e2πi( ! lz+a l (z), )    
    =~ nm 9;iZ0w:8 L(f ⊗ χ, 1) = −2πi R (f ⊗ χ)(z)dz = − h{0, ∞}, f ⊗ χi | 8Ku‡8l`    ∞ 0 L(f ⊗ χ, 1) = − h{0, ∞}, f ⊗ χi l = − g(χ) X χ̄(a) h{0, ∞}, f |γa i l a=1 = − l g(χ) X = − g(χ) χ̄(a) h{a/l, ∞}, f i l a=1 l a=1 l X χ̄(a) h{γa (0), γa (∞)}, f i l g(χ) X χ̄(a) h{0, a/l}, f i . l a=1 = 576f;<9w:mNjzxfb<8l`8:9`608xf_mTmVo2mVo*`608K\2_mNxmv9;i`;<mNZ p8:w ^Vbib3`6O^V` /86O^ ‡8 ^VZ ;<9mNjkmN_xf6f;<9j S (Γ) ≃ H (X , C) 576Te09 | ^z_I^V`;<mNZO^Vb Z08 osmN_j w:mN__8:9xmNZ0{f9/`m ^VZ8lb<8ljk8lZ?` {α, β} ∈ H (X , Q) G;a^K`6f;<9;<9mNjkmN_xf6f;<9j  mN`8`6O^V` f ∈ S (Γ) `6f;<9q8lb<8ljk8lZ?`4;<9efZf;<r?e08efx `m9;iuvZƒ;ho 8oQef_`608l__8:9`_;<wl`q;i``m=8 ^VZ 8lb<8ljk8lZT`4mVo H (X , Z)  ! 8z96O^VbibB{f8lZ0mN`8k`6f;<9 .  wl>Gwlb<8 =T> γ  < ;iZ0w:8z`608zxO^V;i_;iZfu h, i ;<9qw:mNjzxO^V`;i=fb<8 ;i`6 `608k^Nwl`;<mNZ mVo `608 8:w ‡8mNx 8l_I^V`mN_9 |};i`/osmNbib<m 9`6O^V` γ ;<9^w:mNjzjkmNZ]8l;iu‡8lZ ‡8:wl`mN_7osmN_^Vbib `608 8:w ‡8mNx8l_I^V`mN_9 ;i`6 8l;iu‡8lZ N^Vbie08:9 9I^Vjk8)^N9K`6O^V`mVo`608 Z08 osmN_j ) 576Te09 | =?> b<mGm T;iZfu ^V`4`608 8:w ‡8)^Nwl`;<mNZ mNZ 8w ^VZƒw:mNjzxfef`8 γ  38l` V =8K`608 Q ‡8:wl`mN_9xO^Nw:84u‡8lZ08l_I^V`8:{ =T> γ  mN`84`6O^V` H (X , Z) /8w ^VZkw:mNZ09`_e0wl`2`608w:mNjzxfb<8ljk8lZT`I^V_>k9ef=09xO^Nw:8 V mNZ 6f;<w6 `608/xO^V;i_;iZfu h·, f i ;<9*`_; T;a^Vb&* 57608lZ | osmN_^VZ?> γ ∈ H (X , Q) 8q6O^ ‡8   2 1 2 1  +∧ Γ +  Γ   1 + Γ  f       f   1  +  Γ    
 f f   0 1 +  Γ   Z f= Z f 608l_8 γ| ;<9`608Kxf_m 8:wl`;<mNZ mVo γ mNZT`mk`6089ef=09xO^Nw:8 V 7Y Zo ^Nwl` | γ| ;<9^z_I^V`;<mNZO^Vb3jefbi`;ixfb<8mVo  < mf|?;ho 8b<8l` Ω := R f `608lZ 898:8`6O^V` R f ;<9B^_I^V`;<mNZO^Vbj4efbi`;ixfb<8mVo Ω d57608oRmNbib<m ;iZfu γ /8lbib  GZ0m Z \#_mNx mv9;i`;<mNZƒuv; ‡8:9n`608 8:r?eO^Vbi;i`[>]=8l` /8:8lZ `608x8l_;<mT{f9mVo/^VZƒ8lbibi;ixf`;<wwlef_ ‡8z^VZ0{ `608 w:mN__8:9xmNZ0{};iZfu jkmG{}efba^V_osmN_j  γ  γ|V V V  f f        -         γf  " -,',- "   d  E         '   f γ   !      "  !    !  -
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 '      =%/ nmN`8K`6O^V` | {0, a/l} + {0, (l − a)/l} ∈ H (X , Z) ^VZ0{ {0, ∞} ∈ H (X , Z)  8lZ0w:8 8 w ^VzZ w:mNjzxfef`8`608 L v^Vbie08:92mVo ^_I^V`;<mNZO^VbfZ08 oRmN_j ^VZ0{;i`9*` ;<9`9*e09;iZfuq`6087^V=m ‡8w:mNjzxfef`I^V`;<mNZ Y[jzxfb<8ljk8lZ?`I^V`;<mNZƒmVo*9e0wI6 ^ w:mNjzxfef`I^V`;<mNZ ;<99`e0{};<8:{ ;iZ {f8l`I^V;ib =T> y _8ljkmNZO^  98:8  . €  †  Y o τ ;<94^ Z0mNZO^V=8lbi;a^VZ ;i__8:{}e0wl;i=fb<8 _8lxf_8:98lZ?`I^V`;<mNZ mVo G | `608lZ ^)osmN_j4efba^ 9e0w6 ^N9  1  . † {fmG8:9 0Z mN`z8G ;<9` m 8 ‡8l_ |B=T> ^ 57608:mN_8ljmVo  _I^Ve08l_kmNZ08)w ^VZ _;i`8 τ ^N9z^ bi;iZ08 ^V_zw:mNj4=f;iZO^V`;<mNZ$mVo _ 8lxf_8:98lZT`I^V`;<mNZ09z;iZ0{}e0w:8:{ =?> ^V=8lbi;a^VZ _8lxf_8:98lZ?`I^V`;<mNZ09kmVo`608)9ef=fuv_mNefx09zmVo G ƒ 57608l_8EosmN_8mNZ08 O6 ^N9  1    + Γ 1    + Γ     Y L(f ⊗ τ, s) = i  1  ~‡† L(fbi ⊗ τi , s) 608l_8 fb ;<9B`608=O^N98w6O^VZfu‡8nmVo f `mK`608g f8:{zgO8lb<{ mVo3^49ef=fuv_mNefx G mVo G |}^VZ0{ τ ;<9^VZ ^V=8lbi;a^VZ _ 8lxf_8:98lZT`I^V`;<mNZ mVo G k576?e094;ho7mNZ08 w:mNefb<{ u‡8l`^ osmN_j4efba^ oRmN_ L(fb ⊗ τ , s) ^VZO^Vb<mNu‡mNe09K`m  1  . † | ` 608lZ 8qw ^VZ w:mNjzxfef`84^Vbib `608q9x8:wl;a^Vb N^Vbie08:9mVo `608` ;<9`8:{ LsoRefZ0wl`;<mNZ09 m 8 ‡8l_ |0`608`608:mN_> mVo jkmT{}efba^V_9>Gj4=mNb<9#;<92b<mN`/jkmN_8w:mNjzxfbi;<w ^V`8:{ 8 ‡8lZzoRmN_^qr?eO^N{}_I^V`;<wgO8lb<{ * c0mN_/80 ^Vjzxfb<8v|GmNZ08w ^VZ _;i`8n{fm Z]^VZ);iZ}g0Zf;i`898l`mVo u‡8lZ08l_I^V`mN_9oRmN_ H (X, Q)  608l_8 X ;<9`608jkmT{}efba^V_79ef_ o ^Nw:8 † |?=fef` ` 608q_8lba^V`;<mNZ09n^VjkmNZfu‡9``608:98u‡8lZ08l_I^V`mN_9^V_8Kw:mNjzxfb<8l`8lbi>efZ TZ0m Z  wEo  ~  †  57608)9x 8:wl;a^Vb N^Vbie08:9mVo7` ;<9`8:{ Ls oQefZ0wl`;<mNZ09 ^N99mGwl;a^V`8:{ `m ^ =O^N98kw6O^VZfu‡8 osmN_j fb mVo f ^V_8 _ 8lba^V`8:{ `mz`608Kx 8l_;<mG{f9mVo fb2 57608K8:rTeO^V`;<mNZ  1  ~‡† `608l_8EosmN_8q_8lba^V`8:9`608K9x8:wl;a^Vb N^Vbie08KmVo#^` ;<9` mVo f ^VZ0{`608x8l_;<mT{f9mVo =O^N98qw6O^VZfu‡8qoRmN_jk9 nmN`8`6O^V``608K^V_;i`6fjk8l`;<wK9;<{f8qmVo `608K8:rTef; N^V_;a^VZT` w:mNZ 8:wl`ef_8B;iZ ‡mNb ‡8:9 mNZfbi>`608/x 8l_;<mG{f9d^N9I9mTwl;a^V`8:{4`m f 3 576Te09 |V`608B8:rTef; N^V_;a^VZT` w:mNZ 8:wl`ef_8B;ijzxfbi;<8:9 x8l_;<mT{ _8lba^V`;<mNZ09)=8l` /8:8lZ `608]x8l_;<mG{f9 mVo f ^VZ0{ x8l_;<mT{f9)mVo`608]=O^N98 wI6O^VZfu‡8 osmN_jk9 57608:9I8 _8lba^V`;<mNZ09^V_8Kxf_8:wl;<98lbi>)`6084mNZ08:9w:mNZ 8:wl`ef_8:{ =?> tmN; | ;<{0^ ^VZ0{]Y[96f;i; ;iZ ~v    i i i i i   i         2              

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  8l` τ = 8K^VZ  ;i__8:{}e0wl;i=fb<8 † 98lbho  {}eO^Vb _8lxf_8:98lZT`I^V`;<mNZ mVo G ^VZ0{]b<8l` N (E, τ ) =8q`6084w:mNZ0{}e0wl`mN_nmVo  ! 8q9efxfxmv98`6O^V``608 Ls oRefZ0wl`;<mNZ L(E ⊗ τ, s) 6O^N9^jk8l_mNjkmN_xf6f;<w4w:mNZT`;iZ?eO^V`;<mNZ `m `608 E ⊗τ 60mNb<8 s xfba^VZ084^VZ0{ `6O^V`;i`9I^V`;<9gO8:9^oQefZ0wl`;<mNZO^Vb 8:r?eO^V`;<mNZ
 608l_8  b ⊗ τ, s) = ±L(E b ⊗ τ, 2 − s) L(E b ⊗ τ, s) = As γ(s)L(E ⊗ τ, s) L(E  1 0/ † =1 osmN_9mNjk8Kw:mNZ09`I^VZT` A = A(E, τ ) ^VZ0{ ^ Γsos^Nwl`mN_ γ(s) = Γ  s + λ1 2      s + λ2 s + λr Γ · · · Gamma . 2 2 57608:98^N99efjzxf`;<mNZ09^V_8 GZ0m Z `m 60mNb<{];iZ j ^VZ?>]w ^N98:97c0mN_;iZ09`I^VZ0w:8v|;ho τ ;<9^ztn;i_;<w6fb<8l`qwI6O^V_ ^Nwl`8l_ |f`608lZ]`608^N99efjzxf`;<mNZ09^V_8 TZ0m `m = 8K`_e08K{}e08`m < 6f;ij4ef_I^  7 mNefu?^VZf;<9n^VZ0{ tm ‡w6f;i`9I8l_ O6 ^ ‡8960m Z ;iZ  1 d `6O^V``608^V=m ‡8 ^N99efjzxf`;<mNZ ;<9`_e084oRmN_`608KZ0mNZO^V=8lbi;a^VZƒ_8lxf_8:98lZ?`I^V`;<mNZ mVo/^ o ^Vb<985 ^V`8z8G `8lZ09;<mNZq Y Z `608:98 w ^N98:9 |mNZ086O^N9 A = √ | 608l_8 d ;<9n`608{};ijk8lZ09;<mNZ mVo τ  b<9mf|fmNZ08q6O^N9          N (E,τ ) πd γ(s) = Γ  s d 2 Γ  s+1 2 d . nm K|fb<8l` φ(s) = 8`608q;iZ ‡8l_98K‚ 8lbibi;iZ]`_I^VZ09 osmN_j mVo γ(s) |f`6O^V`;<9 |    γ(s) =
Z ∞ φ(t)ts 0 8l` Gs (t) = t−s Z ∞ dt . t φ(x)xs dx x =8`608q;iZ0w:mNjzxfb<8l`84‚ 8lbibi;iZ `_I^VZ09 osmN_j mVo φ(t) B57608lZ mNZ08q6O^N9 t " -,',- (   b ⊗ τ, s) = L(E ∞ X an Gs n=1 n A ± ∞ X an G2−s n=1 n A  1 1 † . 8:8  . 3mN_ 0/ €   c0Nm _7g f 8:{ s |G`60898l_;<8:9  1  1 † w:mNZ ‡8l_u‡8:98GxmNZ08lZ?`;a^Vbibi> ;i`6 t ^VZ0{ `608l_8EoRmN_8 8qw ^VZe098n`6f;<9 9 8l_;<8:49 `m]u‡8l`4ZTefjk8l_;<w ^Vb^Vxfxf_m } ;ij ^V`;<mNZ09mVo`608 N^Vbie08 L(E ⊗ τ, 1)  57608k_I^V`8 mVow:mNZ ‡8l_u‡8lZ0w:8 Vm o`608 98l_;<8:9 {f8lx 8lZ0{f9mNZ `608 w:mNZ0{}e0wl`mN_ N (E ⊗ τ )  ! 8)_mNefuv6fbi> Z08:8:{ `m 9efj pN (E ⊗ τ ) ` 8l_jk9;iZ `608498l_;<8:9`m)mN=f`I^V;iZ ^VZ ^Vxfxf_mG ;ij ^V`;<mNZ nmN`8q`6O^V`;ho*`608K=O^N{]xf_;ijk8:9mVo E ^VZ0{ τ {fm 0Z mN`;iZ?`8l_9I8:wl` |f`608lZ `608qw:mNZ0{}e0wl`mN_mVo E ⊗ τ ;<9 N (E, τ ) = N (E) N (τ ) 2 57608l_8EosmN_8v|fmN=f`I^V;iZf;iZfu Z?efjk8l_;<w ^Vb^Vxfxf_mG ;ij ^V`;<mNZ09k`m `608 N^Vbie08 L(E ⊗ τ, 1) ;<9w:mNjzxfef`I^V`;<mNZO^Vbibi> ;iZ}os8 ^N9;i=fb<8 oRmN_ba^V_u‡8 Og 8lb<{]8} `8lZ09;<mNZ09 Y[Z  .  tnm ‡wI6f;i`98l_ 6O^N9)8} xfba^V;iZ08:{$60m `m w:mNjzxfef`8 G (t) 8  wl;<8lZ?`bi> ^VZ0{$`6f;<9 6O^N9)=8:8lZ ;ijzxfb<8ljk8lZ?`8:{ ;iZ  . =   ef_ZTefjk8l_;<w ^Vb ^Vxfxf_mG ;ij ^V`;<mNZ09ne098`6f;<9xO^V_`;<wlefba^V_;ijzxfb<8ljk8lZ?`I^V`;<mNZ   3 <        d     s 2 =%= 57608z\2_mNxmv9;i`;<mNZ .  ;<9;iZ o ^Nwl`4^)9x 8:wl;a^Vb2w ^N98zmVo^ jkmN_8 u‡8lZ08l_I^VbB_8:9efbi`q`6O^V`q60mNb<{f9 osmN_V^ Z?> L 9l8 _;<8:96O^ T;iZfu ^ jk8l_mNjkmN_xf6f;<w4w:mNZT`;iZ?eO^V`;<mNZ ^VZ0{ 9I^V`;<9 oR>T;iZfu ^ oQefZ0wl`;<mNZO^Vb38:rTeO^V`;<mNZ mVo ` >Gx8  1 0/ †  #$" 

K  8l` p(x) = x − 4x + 1 |0^VZ0{b<8l` K =8n`608n9xfbi;i``;iZfu gO8lb<{ mVo p(x) m ‡8l_ Q  57608q{};<9Iwl_;ijz;iZO^VZ?`mVo ;<9 ~v~ }|0^VZ0{ 608lZ0w:8 K ;<9^`mN`I^Vbibi> _8 ^Vb S  8G`8lZ09;<mNZ257608efZf;<rTe08qrTeO^N{}_I^V`;<wq9ef=}gO8lb<{ mVo K p(x) ;<9 Q[√229]  8l` r ^VZ0{ s = 8 8lb<8ljk8lZ?`9mVo G := S mVoKmN_{f8l_ ~ ^VZ0{ ƒ/ _8:9x8:wl`; ‡8lbi>  57608l_8 ^V_8 `6f_8:8 ;i__8:{}e0wl;i=fb<8 wI6O^V_I^Nwl`8l_9mVo G | ` /m mVo 6f;<w6 ^V_8 ^V= 8lbi;a^VZ ^VZ0{ƒ`608z`6f;i_{ 6O^N9K{};ijk8lZ09;<mNZ ~ z 57608 w6O^V_I^Nwl`8l_`I^V=fb<8q;<9960m Z]=8lb<m 


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3     . . . r . s . . . ~  −1 8l_8 χ ;<9 `608ƒ0Z mNZT`_; T;aV^ b ^V= 8lbia; ^VZ wI6O^VI_ ^Nwl`8l_ ^VZ0{ ψ ;<9`608 w6O^V_I^Nwl`8l_ mVo`608ƒ;i__8:{}e0wl=f;ib<8 Z0Nm ZO^V= 8lbi;a^VZ _8lxf_ 8:98lZT`I^V`;<Nm Z ρ mVo G uv; ‡8l]Z =T> χ −1 ψ     r 7→  0 −1 −1 0    , s 7→   −1 1 −1 0  Y[Z o ^Nwl` | ψ ;<9/;iZ0{}e0w:8:{ =T> `608n`_; T;a^Vb wI6O^V_I^Nwl`8l_mNZ C = {1, s, s } ⊂ S |T`6O^V`7;<9 | ψ = Ind 1  < m osmN_^VZ?> Z08 osmN_j f ∈ S (Γ (N )) mNZ08q6O^N9  2 3 S3 C3 C3 3  2 0 L(f ⊗ ψ, s) = L(fb, s) 608l_8 fb ;<9`608=O^N984w6O^VZfu‡8KmVo f `m Q[√229]  8l` E =8`608K8lbibi;ixf`;<wKwlef_ ‡8K{f8Eg0Z08:{ =?> y + y = x − 8x − 9 #576f;<9;<9`608Kwlef_ ‡8 307A1 ;iZ ` 60898lZ09I8mVo  . €   nmN`8`6O^V`B`6f;<9#;<9/^q9`_mNZfu ! 8l;ibOwlef_ ‡8^VZ0{z608lZ0w:8^Vbib0`608^V=m ‡8w:mNjzxfef`I^V`;<mNZ09 ^V_8)^Vxfxfbi;<w ^V=fb<8 `m `6f;<9wlef_ ‡8# 57608)w:mNZ0{}e0wl`mN_ mVo`608)wlef_ ‡8 ;<9 / :E |2^VZ0{ ^N94Z0mN`8:{ ;iZ  . €  `608 vu _mNefx E(Q) ;<92`_; T;a^Vb& ! 87Z0mN`8`6O^V`B`608xf_;ijk8:9 / E ^VZ0{ =%q/ 9xfbi;i`#w:mNjzxfb<8l`8lbi>z;iZ K  Y `2;<9B8 ^N9>`m w608:w `6O^V` E(F ) = 35 ^VZ0{ E(F ) = 64 |T^VZ0{ `608l_8EosmN_87;i` oRmNbib<m 92`6O^V` E(K) ;<9*`_; T;a^Vb  wEo  
2   3        37 53  tors =€ 0/1  †  9 /8 ;ibib 98:8;iZ `608Z08}`98:wl`;<mNZ |f`608 LsoRefZ0wl`;<mNZ09 L(E ⊗ τ, s) {fm Z0mN` v^VZf;<96 ^V` s = 1    osmN_n^VZ?> τ ∈ Gb ^VZ0{ `608l_8EoRmN_8mNZ084w ^VZƒw:mNZ0wlbie0{f84`6O^V` E(K) ;<9`_; T;a^Vb&576Te09 | M = h (E )(1) 9I^V`;<9 gO8:9`608g0_9`` m)^N99efjzxf`;<mNZ09j ^N{f8q;iZ < 8:wl`;<mNZ 1  .   1 K   !       "
 57608 9x 8:wl;a^Vb N^Vbie08)mVo`608 L oQefZ0wl`;<mNZ ^V``I^Nw608:{ `m M ;<9 L (M, 0) := (L (E ⊗ φ, 1))  Y[Z o ^Nwl` | /84w ^VZ G;<8 `6f;<9^N9^VZ 8lb<8ljk8lZ?`mVo ζ(R[G])   ∗   ∗ b φ∈G × L∗ (M, 0) = X b φ∈G L∗ (E ⊗ φ, 1)eφ 608l_8 e = P φ(g)g ;<9`608K;<{f8ljzx mN`8lZT`w:mN__8:9xmNZ0{};iZfuk`mz`608qwI6O^V_I^Nwl`8l_ φ ∈ Gb  Y[Z mNef_w ^N98v| /846O^ ‡8 G = S |O^VZ0{ Gb = {1, χ, ψ} 57608 L N^Vbie08:9 L(E ⊗ φ, 1) ^V_84Z0mNZ v^VZ  <; 96f;iZfuqosmN_B^Vbibf`608wI6O^V_I^Nwl`8l_9 φ ∈ Gb ^VZ0{k`6?e09 | L (E ⊗ φ, 1) = L(E ⊗ φ, 1)  < ;iZ0w:8`608 E(K) 6O^N9 0Z m`mN_9;<mNZ ^VZ0{z`608 Ls oQefZ0wl`;<mNZ {fmG8:9*Z0mN` v^VZf;<96 ^V` s = 1 |‡;i`*oRmNbib<m 9#`6O^V` E(K) ;<92`_; G;a^Vb& 576Te09 | ^VZ0{ K 9 ^V`;<9 oQ> `608Z08:w:8:99I^V_> 6T>TxmN`608:98:9q9`I^V`8:{ƒ;iZ < 8:wl`;<mNZ 1  .  c0mN_ φ = 1, χ `608 L N^Vbie08 E w ^VZƒ= 8 w:mNjzxfef`8:{ e09;iZfu jkmG{}efba^V_K9>Gj4=mNb<9q^N9q{f8:9wl_;i= 8:{ ;iZ < 8:wl`;<mNZ1 0/  .  57608w:mNjzxfef`I^V`;<mNZ09 >G;<8lb<{  1 |G| φ −1 g∈G     3 ∗     √ L(E, 1) = ΩE , L(E ⊗ χ, 1) = −ΩE / 229 608l_8 Ω ;<97`608K_8 ^Vb x 8l_;<mG{]^V``I^Nw608:{ `m E  ! 8K^Vxfxfbi>)jk8l`60mT{f9{f8:9Iwl_;i= 8:{ ;iZ `608 < 8:wl`;<mNZ 1  1z`m w:mNjzxfef`8 L(E ⊗ ψ, 1) B57608w:mNZ0{}e0wl`mN_ mVo E ⊗ ψ ;<9 N (E) · N (ψ) | 608l_8 N (E) = 307 ^VZ0{ N (ψ) = 229 ^V_8n`608w:mNZ0{}ef`mN_9mVo E ^VZ0{ _8:9x8:wl`; ‡8lbi> 2 57608w:mT8  wl;<8lZT`9mVo3`60898l_8l;<9 L(E × ψ, s) w ^VZ)=8w:mNjzxfef`8:{=?>kb<mGm T;iZfuk^V``608 ψ b<mGw ^Vbos^Nwl`mN_9  ;ho p = 307  1+T +T     ;ho p = 229    1 − a T + pT   ;ho p 6= 307, 229 ^VZ0{ f (p) = 1 L (E ⊗ ψ, T ) = (1 − a + pT )    ;ho p 6= 307, 229 ^VZ0{ f (p) = 2   1 + (2p − a )T + p T     ;ho p 6= 307, 229 ^VZ0{ f (p) = 3  1 + a T + (a − p)T + a pT + p T 57608Kw:mNjzxfef`I^V`;<mNZ09>G;<8lb<{ L(E ⊗ ψ, 1) ∼ 0.1330438452   E 2 2    2 2 p p 2 2 T 2 p p 2 2 p 2 2 4 p 3 2 4 =E             "
 ! 84g0_9`w:mNZ09;<{f8l_`608x 8l_;<mG{];<9ImNjkmN_xf6f;<9j p8:w ^Vbib3oR_mNj < 8:wl`;<mNZ 1  ~ `6O^V``608x8l_;<mT{ ;<9mNjkmN_ xf6f;<9j + αM : HB,R → HdR,R /F 0 ;<9uv; ‡8lZ]=T>  αM (γ ⊗ α) : ω 7→    #
,      )  θ '     / ! Det(Φ) ≃ ζ(R[G]) "   ρη    
 3 Z ω γ −1   θ Φ : R[G] → R[G]        X g∈G    1 0= † ' !        .  -
 g(α)g −1  . R[G]      
 
   &
      / !    -    X eη (detρη (θ(1))) b η∈G   !     - "         3 η 8l` ρ = 8 ^VZ$;i__8:{}e0wl;i=fb<8 w:mNjzxfb<8$_8lxf_8:98lZT`I^V`;<mNZ mVo G ^VZ0{ η =8 `608 w:mN__8:9x mNZ0{};iZfu w 6O^V_IN^ wl`8l_ 8l` e = P η(g)g |}`608K;<{f8ljzx mN`8lZT`n^N99mGwl;a^V`8:{`m η 257608lZ e C[G] ≃ M (C) osmN_9 mNjk8 n ≥ 1  8l`  3

η −1 g∈G η n
 1 0 ... 0   0   e = ..   ..  0    1 . . . 0   .. . . . ..   .. . . . ..  0 ... 1 57608lZ eM (C) ;<9*^VZ n {};ijk8lZ09;<mNZO^Vb ‡8:wl`mN_29xO^Nw:8m ‡8l_ C  nmN`8B`6O^V` e θ(1) = θ(e ) ∈ e C[G] ≃ |9m b<8l` ρ(θ(1)) = e θ(1) = (α )  57608lZ =?> `608ƒw:mNZ09`_e0wl`;<mNZ mVoK`608 {f8l`8l_jz;iZO^VZ?` M (C) oRefZ0wl`mN_osmN_98ljz;<9;ijzxfb<84;iZ < 8:wl`;<mNZ ~  .  ~ | /846O^ ‡8 e Det Φ ;<9`608 C bi;iZ08K=fefZ0{}b<84u‡8lZ08l_I^V`8:{ =T> {f8l`8l_jz;iZO^VZ?`mVo*`608j ^Vx    n η n η ij  η η  η θ̄ : eMn (C) −→ eMn (C). 57608 {f8l`8l_jz;iZO^VZT` mVo θ̄ ;<9 uv; ‡8lZ =T> det(α ) = det(ρ(θ(1))) ^VZ0{ `608l_8EosmN_8 Det(Φ) 8:rTeO^Vb<9 P   e det(ρ(θ(1))) ij b η η∈G nmN`8 `6O^V` (ω ⊗ α ) ^VZ0{ (γ ⊗ 1) ^V_8 `608 Q[G] u‡8lZ08l_I^V`mN_9 6f;<w6 {f8Eg0Z08 Ξ(M ) ;iZ09;<{f8   0 0 0 =v„ dc0_mNj,`608^V= m ‡8 8ljzj ^K^VZ0{  1 0= † |‡;i` osmNbib<m 92`6O^V`2`608;ij ^Vu‡8mVo Ξ(M ) ;iZ Ξ(M )⊗R ;<9vu ; ‡8l]Z =T>`608 Q[G] u‡8lZ08l_I^V`mN_  Ξ(M )⊗R 
 RΩ := X det(ρη (Ω−1 E X g(α0 )g −1 ))eη . g∈G b η∈G oQ`8l_zw60mGmv9;iZfu ^ xO^V_`;<wlefba^V_ Z[G] u‡8lZ08l_I^V`mN_ α mVo O  e09;iZfuƒ\ pY † | /8 u‡8l`z`608)oRmNbib<m ;iZfu w:Nm jzxmNZ08lZ?`9  0  K −1 −1 e1 RΩ = Ω−1 E · trK/Q (α0 ) = ΩE = L(E, 1) √ −1 √ eχ RΩ = Ω−1 · tr (α − g (α )) = Ω · 229 = −L(E ⊗ χ, 1)−1 0 1 0 E E K/Q[ 229] √ 229) ∼ (0.7047640380)2 (15.131274594) = 7.516319136 eψ RΩ ∼ Ω−2 · ( E ∼ L(E ⊗ ψ, 1)−1 (1.00000000)
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   ! 8K96O^Vbib xf_m ‡8K608l_8q`608nosmNbib<m ;iZfu)b<8ljzj ^    #       K0 (Z[A4 ], R) → K0 (MA4 , R) × K0 (Z[A4 ]) × K0 (Z[Aab 4 ], R)      !) "    M A4          Q[A4 ]      ' Z[A4 ] 3 c0mNbib<m ;iZfu)`608Z0mN`I^V`;<mNZ09;iZ `608xf_mGmVo mVo 8ljzj ^1kmVo   |}`608K^V=m ‡8Kj ^Vx ;<9;iZ 8:wl`; ‡8;ho V^ Z0{]Nm Zfbi> ;ho `608Kj ^Vx D(A ) → D(C ) ;<9;iZ 8:wl`; ‡8 608l_8oRmN_n^g0Zf;i`8quv_mNefx Γ mNZ084{f8Eg0Z08:9  3  
 4     3 D(Γ) := nr(Uf (MΓ )) ∩ im(nrQ[Γ] ) , nr(Uf (Z[Γ])) ∩ im(nrQ[Γ] ) ;i`6 M =8l;iZfu ^ j ^%G;ij ^VbdmN_{f8l_;iZ Q[Γ] w:mNZ?`I^V;iZf;iZfu Z[Γ]  57608uv_mNefx D(C ) ;<9;<9ImNjkmN_xf6f;<w`m M /Z[C ] 6f;<w6ƒ;<9mVoBmN_{f8l_ ~ n57608Z?efjk8l_I^V`mN_KmVo ;<9 9;iZ0w:8 Q[A ] ≃ Q × Q[ζ ] × M (Q) *576?e09 |?`608Z?efjk8l_I^V`mN_mVo D(A ) D(A ) Z × Z[ζ ] × Z 6O^N9mN_{f8l_ ~ 1 q 57608{f8lZ0mNjz;iZO^V`mN_q;<9 {nr(u) : u ∈ Z[A ] } q Y o /8 960m `6O^V``6f;<9n6O^N9qmN_{f8l_ € | `608lZ ;i`4oRmNbib<m 9`6O^V` D(A ) 6O^N9mN_{f8l_ 01 |d9m `608kj ^Vx D(A ) → D(C ) ;<9KZ0mN`;iZ 8:wl`; ‡8 ^VZ0{ 608lZ0w:8q`608qb<8ljzj ^ oRmNbib<m 9 nm ,w:mNZ09;<{f8l_4^)efZf;i` u ∈ Z[A ] K 57608;ij ^Vu‡8 mVo/`6f;<9nefZ0{f8l_q`608Z0mN_j _8:{}e0wl`;<mNZ j ^Vx nr bi;<8:9;iZ09;<{f8 Z × Z[ζ ] × Z  8l` nr(u) = (a, b, c) efZ0{f8l_n`6f;<9n;<{f8lZ?`;hgOw ^V`;<mNZ ! 896O^Vbib 960m `6O^V` a = c  nmN`8k`6O^V` nr(u) = (ρ (u), ρ (u), det(ρ (u))) | 608l_8 ρ ;<94`608)`_; T;a^Vb/_8lxf_8:98lZ?`I^V`;<mNZ | ρ ;<94^Z0mNZT`_; T;a^Vb^V=8lbi;a^VZ _8lxf_8:98lZ?`I^V`;<mNZ ^VZ0{ ρ ;<9K`608k{f8luv_8:8 `6f_8:8z_8lxf_8:98lZ?`I^V`;<mNZ mVo A  / > _8lxfba^Nwl;iZfu u =?> −u ;ho3Z08:w:8:99 ^V_>‡| /8qw ^VZ]^N99efjk8n`6O^V` a = 1 2 cfef_`608l_ | /8qw ^VZ]^N99efjk8`6O^V`7`608  Γ × C3 3 × 4 3 × × ×  3 4 3 3 4 4  ×    4 4 3    4 × 3 × × × 
  0 1 2  0 1  2  4  €. ;ij ^Vu‡87mVo u efZ0{f8l_ `608/xf_m 8:wl`;<mNZ Z[A ] → Z[C ] ;<9 `_; T;a^Vb |N=T>Kj4efbi`;ixfbi>G;iZfuq=?>^9ef;i`I^V=fb<8/efZf;i`  ±h † ;ho Z08:w:8:99I^V_> 2 576?e09 |T;ho u = P a · g 608l_8 a ∈ Z |?`608lZ P a = ρ (u) = a = 1  cfef_`608l_ |`608 ;ij ^Vu‡8ƒmVo u ;iZ Z[C ] ;<9 `_; G;a^Vb;ijzxfbi;<8:9)`6O^V` a + a + a + a = 1 ^VZ0{ P oRmN_ i = 1, 2  a + a =0 57608q_8lxf_8:9I8lZ?`I^V`;<mNZ ρ ;<9uv; ‡8lZ]=T> `608nosmNbib<m ;iZfu 4 3  ×  i g g∈A4 3 3 j=1 hi × g  × e g∈A4 g g1 g2  0 1 0 0 g3 hi g j   2  1 0 0   −1 0       g1 7→  0 −1 0  , g2 7→  0    0 0 0 −1 576?e09 | c = ρ (u) 8:rTeO^Vb<9`608K{f8l`8l_jz;iZO^VZT`nmVo (r ) 2 0         0  , h 7→  0 0 1  .    1 0 0 0 −1 1 608l_8  ij 1≤i,j≤3 rij = 2(ah−i+j + ah−i+j gj ) − (ah−i+j + ah−i+j g1 + ah−i+j g2 + ah−i+j g3 ). 57608l_8EosmN_8v|f=?>`6084^V=m ‡8q;<{f8lZT`;i`;<8:9 | c 8:rTeO^Vb<9`608K{f8l`8l_jz;iZO^VZ?`mVo   2(ae + ag1 ) − 1    2(ah2 + ah2 g1 )  2(ah + ahg1 ) 2(ah + ahg2 ) 2(ah2 + ah2 g3 ) 2(ae + ag2 ) − 1 2(ah + ahg3 ) 2(ah2 + ah2 g2 ) 2(ae + ag3 ) − 1    .  mGm T;iZfu ^V``608z{f8l`8l_jz;iZO^VZT`4mVo/`608k^V=m ‡8 jkmG{}efb<m 10| /8z98:8`6O^V` c ≡ 1 (mod 4) ^VZ0{ƒ608lZ0w:8 2 57608l_8EosmN_8v| 8q6O^ ‡8 c=a 
    #{nr(u) : u ∈ Z[A4 ]× } = #{(ρ0 (u), ρ1 (u)) : u ∈ Z[A4 ]× } = #Z[C3 ]× = 6. 576f;<9w:mNjzxfb<8l`8:9`608xf_mTmVo €v~   


      
 
 

     

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