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El. knyga: Elementary Modular Iwasawa Theory

(Univ Of California, Los Angeles, Usa)
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Elementary Modular Iwasawa TheoryDidesnis paveikslėlis
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This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author's 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry.Starting with a description of Iwasawa's classical results on his proof of the main conjecture under the Kummer-Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation.The fundamentals in the first five chapters are as follows:Many open problems are presented to stimulate young researchers pursuing their field of study.
Preface v
Notes to the reader ix
1 Cyclotomic Iwasawa theory
1(48)
1.1 Cyclotomic fields
3(3)
1.1.1 Cyclotomic integers
3(1)
1.1.2 Cyclotomic character
4(1)
1.1.3 Decomposition group
5(1)
1.2 An outline of class field theory
6(2)
1.2.1 Ray class groups
6(1)
1.2.2 Main theorem of class field theory
7(1)
1.3 Class number formula and Stickelberger's theorem
8(2)
1.3.1 Class number formula
8(1)
1.3.2 Galois module structure of Cl-n
9(1)
1.4 Gauss sum
10(4)
1.4.1 Gauss sum over a finite field
10(2)
1.4.2 Jacobi sum
12(1)
1.4.3 Basic properties of Gauss sum
13(1)
1.5 Prime factorization of Gauss sum
14(6)
1.5.1 Integrality ideal
14(1)
1.5.2 Reformulation of the Stickelberger theorem
15(1)
1.5.3 Lemmas and key steps
15(4)
1.5.4 Proof of Stickelberger's theorem
19(1)
1.6 A consequence of the Kummer--Vandiver conjecture
20(2)
1.6.1 Conjecture
20(1)
1.6.2 Index calculation
21(1)
1.7 Kummer theory
22(2)
1.7.1 Kummer pairing
22(1)
1.7.2 Kummer theory for p-units
23(1)
1.8 Proof of cyclicity theorem
24(3)
1.8.1 Cyclicity for F0
24(2)
1.8.2 Proof in general
26(1)
1.9 Iwasawa theoretic interpretation
27(3)
1.9.1 Limit of Stickelberger's elements
28(1)
1.9.2 Cyclicity over Iwasawa algebra
28(1)
1.9.3 P-adic L-function
29(1)
1.10 Iwasawa's formula for |A-n|
30(5)
1.10.1 Iwasawa's formula
30(1)
1.10.2 Weierstrass preparation theorem
31(2)
1.10.3 Proof of Iwasawa's formula
33(1)
1.10.4 Iwasawa's heuristic
34(1)
1.11 Class number formula for F+n
35(6)
1.11.1 Cyclotomic units
35(1)
1.11.2 Class number as residue of Dedekind zeta function
36(2)
1.11.3 Specialization to F+n
38(1)
1.11.4 Class number as a unit index
39(1)
1.11.5 Key lemma
40(1)
1.12 Local Iwasawa theory
41(8)
1.12.1 A theorem of Iwasawa
41(5)
1.12.2 A version for a Z2p-extension
46(3)
2 Cuspidal Iwasawa theory
49(66)
2.1 Curves over a field
51(15)
2.1.1 Plane curves
51(3)
2.1.2 Tangent space and local rings
54(4)
2.1.3 Projective space
58(2)
2.1.4 Projective plane curve
60(2)
2.1.5 Divisors
62(2)
2.1.6 Theorem of Riemann--Roch
64(1)
2.1.7 Regular maps from a curve into projective space
65(1)
2.2 Elliptic curves
66(8)
2.2.1 Abel's theorem
66(1)
2.2.2 Weierstrass equations of elliptic curves
67(4)
2.2.3 Moduli of Weierstrass type
71(3)
2.3 Modular forms and functions
74(16)
2.3.1 Geometric modular forms
74(1)
2.3.2 Topological fundamental groups
75(2)
2.3.3 Fundamental group of an elliptic curve
77(1)
2.3.4 Classical Weierstrass Q-function
78(2)
2.3.5 Complex modular forms
80(1)
2.3.6 Weierstrass σ and σ functions
81(3)
2.3.7 Product q-expansion
84(2)
2.3.8 Klein forms
86(3)
2.3.9 Hubert's theorem
89(1)
2.4 Modular Stickelberger theory
90(25)
2.4.1 Siegel units
91(2)
2.4.2 Distribution on p-divisible groups
93(1)
2.4.3 Stickelberger distribution
94(1)
2.4.4 Rank of distribution
95(2)
2.4.5 Cusps of X(N)
97(1)
2.4.6 Finiteness of C1X(N)
98(1)
2.4.7 Siegel units generate A×pm
99(4)
2.4.8 Fricke--Wohlfahrt theorem
103(2)
2.4.9 Siegel units and Stickelberger's ideal
105(4)
2.4.10 Cuspidal class number formula
109(3)
2.4.11 Cuspidal class number formula for X1(N)
112(3)
3 Cohomological modular forms and p-adic L-functions
115(36)
3.1 The multiplicative group Gm and Dirichlet L-function
115(16)
3.1.1 Betti cohomology groups
116(3)
3.1.2 Cohomology of Gm(C) and Dirichlet L-values
119(3)
3.1.3 Relative cohomology
122(2)
3.1.4 GL(1) Hecke operators
124(2)
3.1.5 P-adic measure
126(3)
3.1.6 P-adic L-function of Kubota-Leopoldt
129(2)
3.2 Modular p-adic L-functions
131(20)
3.2.1 Elliptic modular forms
132(1)
3.2.2 Modular cohomology group
133(2)
3.2.3 GL(2) Hecke operators
135(3)
3.2.4 Duality
138(2)
3.2.5 Duality between Hecke algebra and cusp forms
140(3)
3.2.6 Modular Hecke L-functions
143(1)
3.2.7 Rationality of Hecke L-values
144(2)
3.2.8 P-old and p-new forms
146(2)
3.2.9 Elliptic modular p-adic measure
148(3)
4 P-adic families of modular forms
151(68)
4.1 P-adic family and slope
151(22)
4.1.1 P-adic L-functions as a power series
152(2)
4.1.2 Eisenstein series
154(3)
4.1.3 Eisenstein family
157(2)
4.1.4 Hecke operator
159(2)
4.1.5 Modular forms of level N
161(6)
4.1.6 Slope of modular forms
167(5)
4.1.7 Control of Hecke algebra
172(1)
4.2 Analytic families via cohomology groups
173(46)
4.2.1 Fundamental group, again
176(1)
4.2.2 Group cohomology
177(2)
4.2.3 Inflation and restriction
179(2)
4.2.4 Eichler--Shimura isomorphism
181(1)
4.2.5 Betti cohomology
182(2)
4.2.6 Duality of cohomology groups
184(2)
4.2.7 Hecke operator on cohomology groups
186(3)
4.2.8 P-Hecke operator and level lowering
189(3)
4.2.9 Weight independence of limit Hecke algebra
192(1)
4.2.10 Hecke operator on boundary
193(6)
4.2.11 Weight comparison
199(3)
4.2.12 Freeness and divisibility
202(2)
4.2.13 Proof of Theorem 4.2.17 for the ordinary part
204(3)
4.2.14 Proof of Theorem 4.2.17 in general
207(3)
4.2.15 Control theorem
210(1)
4.2.16 Control of cohomology
211(2)
4.2.17 Co-freeness over the group algebra
213(3)
4.2.18 Ordinary p-adic analytic families
216(3)
5 Abelian deformation
219(20)
5.1 Abelian deformation
220(2)
5.1.1 Deformation of a character
221(1)
5.1.2 Group algebra is universal
221(1)
5.1.3 Examples of group algebras
222(1)
5.2 A way of recovering the group and its application
222(11)
5.2.1 Ray class groups
223(2)
5.2.2 Differentials
225(3)
5.2.3 Algebra R[ M] = R M and derivation
228(1)
5.2.4 Congruence modules C0 and C1
229(2)
5.2.5 Congruence modules for group algebras
231(2)
5.3 Cohomology of induced representations
233(3)
5.3.1 Continuous group cohomology
233(1)
5.3.2 Induced representation
234(1)
5.3.3 Adjunction formula for Hom and ⊗
235(1)
5.3.4 Shapiro's lemma
235(1)
5.4 Class group as an arithmetic cohomology group
236(3)
5.4.1 Abelian Selmer groups
236(3)
6 Universal ring and compatible system
239(42)
6.1 Adjoint Selmer groups and differentials
241(13)
6.1.1 Ordinary deformation functor
241(2)
6.1.2 Tangent space of deformation functors
243(1)
6.1.3 Tangent space of local rings and generators
243(2)
6.1.4 Tangent space as adjoint cohomology group
245(2)
6.1.5 Mod p adjoint Selmer group
247(2)
6.1.6 P-finiteness condition
249(1)
6.1.7 Reinterpretation of the functor D
250(1)
6.1.8 General adjoint Selmer group
251(1)
6.1.9 Differentials and Selmer group
252(2)
6.1.10 Local condition at p
254(1)
6.2 Deformation rings of a compatible system
254(27)
6.2.1 Preliminary lemmas
258(1)
6.2.2 Consequence of vanishing of differentials
259(1)
6.2.3 Modular p-adic Galois representation
260(2)
6.2.4 Modular deformation
262(3)
6.2.5 "Big" ordinary Hecke algebra Tp
265(4)
6.2.6 Fitting ideals
269(2)
6.2.7 Algebraic p-adic L-function
271(4)
6.2.8 A detailed and stronger version of Tate's theorem
275(4)
6.2.9 Proof of Tate's Theorem 6.2.21
279(2)
7 Cyclicity of adjoint Selmer groups
281(60)
7.1 Basic set-up
283(5)
7.1.1 Greenberg's Selmer group
283(1)
7.1.2 Summary
284(4)
7.2 Upper bound of the number of Selmer generators
288(10)
7.2.1 Local theory
288(1)
7.2.2 Another example of local Tate duality
289(1)
7.2.3 An adjoint reflection theorem
290(2)
7.2.4 Proof of Theorem 7.2.3
292(2)
7.2.5 Details of H1(K, μp) = K× ⊗z Fp
294(1)
7.2.6 Restriction to the splitting field of Ad(p)
294(1)
7.2.7 Selmer group as a subgroup of F× ⊗z F
295(2)
7.2.8 Dirichlet's unit theorem
297(1)
7.3 Induced modular Galois representation
298(10)
7.3.1 Induced representation and self-twist
299(2)
7.3.2 Galois action on unit groups
301(3)
7.3.3 Ordinarity for induced representation
304(1)
7.3.4 Induction in three ways
305(3)
7.4 Cyclicity when p splits in K
308(5)
7.4.1 Identity of two deformation functors
308(2)
7.4.2 Decomposition of Sel(Ad(IndQK φ))
310(1)
7.4.3 An exotic identity of Galois groups
311(1)
7.4.4 Cyclicity of anticyclotomic Iwasawa modules
312(1)
7.5 Iwasawa theory over quadratic fields
313(5)
7.5.1 Galois action on global units
314(1)
7.5.2 Selmer group and ray class group
315(1)
7.5.3 Structure of Mp[ Ad] as a G-module in Case D
315(1)
7.5.4 Theorem for Sel(IndQK φ-)
316(2)
7.6 Selmer groups in the non-split case
318(11)
7.6.1 Set-up in Cases Ds and U_
318(1)
7.6.2 Local Galois action in non-split case
319(2)
7.6.3 Generality of Selmer cocycle in non-split case
321(3)
7.6.4 Adjoint Selmer groups in non-split case
324(3)
7.6.5 Comparison of Theorems 7.6.2 and 7.4.4
327(2)
7.7 Selmer group of exceptional Artin representation
329(12)
7.7.1 Classification of subgroups of PGL2(F)
329(2)
7.7.2 Minkowski unit
331(1)
7.7.3 Selmer group revisited
332(1)
7.7.4 Decomposition group as Galois module
333(1)
7.7.5 Structure of Selmer groups
334(1)
7.7.6 Proof of Theorem 7.7.3
335(3)
7.7.7 Proof of Corollary 7.7.4
338(1)
7.7.8 Concluding remarks and questions on cyclicity
339(2)
8 Local indecomposability of modular Galois representation
341(40)
8.1 One generator theorem of T over Λ
342(4)
8.1.1 Proof of one generator theorem
343(2)
8.1.2 Cyclicity of the adjoint Selmer group again
345(1)
8.1.3 Open questions
346(1)
8.2 Local structure at p of modular deformation
346(4)
8.2.1 Local indecomposability conjecture
347(1)
8.2.2 Inertia theorem
347(2)
8.2.3 Generic non-triviality of U
349(1)
8.3 Universal ring for IndQK φ for K real at split p
350(11)
8.3.1 Rord is a non-trivial extension of A if K is real
351(1)
8.3.2 Questions on structure of the universal ring
351(1)
8.3.3 Action of [ a] on Sel(Ad(p))
352(2)
8.3.4 Set-up for a structure theorem
354(1)
8.3.5 Structure theorems
355(3)
8.3.6 Wall-Sun-Sun primes
358(1)
8.3.7 Proof of Theorem 8.3.7
358(3)
8.4 Indecomposability of deformations of IndQK φ for K real
361(7)
8.4.1 Generalized matrix algebras
361(1)
8.4.2 Inertia embedding into T-
362(4)
8.4.3 Local indecomposability in the real case
366(2)
8.5 Local indecomposability for imaginary K
368(13)
8.5.1 CM and non-CM components
369(2)
8.5.2 Local indecomposability in the CM case
371(2)
8.5.3 Katz--Yager p-adic L-function
373(2)
8.5.4 Local indecomposability again
375(1)
8.5.5 Semi-simplicity and a structure theorem
376(4)
8.5.6 Open questions
380(1)
9 Analytic and topological methods
381(32)
9.1 Analyticity of adjoint L-functions
383(8)
9.1.1 L-function of modular Galois representations
383(1)
9.1.2 Explicit form of adjoint Euler factors
384(2)
9.1.3 Analytic continuation
386(5)
9.2 Integrality of adjoint L-values
391(9)
9.2.1 Eichler--Shimura isomorphism again
391(3)
9.2.2 Modified duality pairing
394(1)
9.2.3 Hecke Hermitian duality
395(2)
9.2.4 Integrality theorem
397(3)
9.3 Congruence and adjoint L-values
400(13)
9.3.1 T-freeness and the congruence number formula
400(2)
9.3.2 Taylor--Wiles system
402(2)
9.3.3 Level Q Hecke algebra
404(1)
9.3.4 Q-ramified deformation
405(3)
9.3.5 Taylor-Wiles primes
408(2)
9.3.6 Proof of T-freeness of cohomology groups
410(3)
Bibliography 413(8)
List of symbols and statements 421(4)
Index 425
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