Development of Iwasawa Theory - The Centennial of K. Iwasawa's Birth - Proceedings of the International Conference Iwasawa 2017

This volume is edited as the proceedings of the international conference 'Iwasawa 2017', which was held at the University of Tokyo from July 19th through July 28th, 2017, in order to commemorate the 100th anniversary of Kenkichi Iwasawa's birth. In total 236 participants attended the conference including 98 participants from 15 countries outside Japan, and enjoyed the talks and the discussions on several themes flourishing in Iwasawa theory. This volume consists of 3 survey papers and of 15 research papers submitted from the speakers and the organizers of the conference. We also included 4 essays on memories of Iwasawa to celebrate the Centennial of Iwasawa's birth. We recommend this volume to all researchers and graduate students who are interested in Iwasawa theory, number theory and related fields.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America Preface CONTENTS MEMORIES Memories of Kenkichi lwasawa -- John Coates My recollections of Kenkichi Iwasawa -- Ehud de Shalit My memories of lwasawa -- Ralph Greenberg Memories of Professor lwasawa -- Masato Kurihara RESEARCH ARTICLES Euler systems with local conditions -- David Loeffler and Sarah Livia Zerbes §1. Cohomology of Galois representations §2. Euler systems §3. The case d_(T) = 1 §4. Higher rank Euler systems §5. Euler systems with local conditions §6. A conjecture §7. Ordinarity conditions at p 7.1. Example A: Rankin-Selberg convolutions 7.2. Example B: The spin representation for GSp(4) §8. lwasawa theory and Greenberg Selmer groups 8.1. The rank 0 case 8.2. Higher ranks §9. Rank-lowering operators and reciprocity laws 9.1. The equal-rank case 9.2. Rank-lowering §10. Modular forms over an imaginary quadratic field §11. The non-ordinary case References Notes on noncommutative Fitting invariants -- Andreas Nickel with an appendix by Henri Johnston and Andreas Nickel §1. Introduction §2. The commutative case §3. Noncommutative Fitting invariants: Basic properties 3.1. Fitting domains and Fitting orders 3.2. Reduced norms and the integrality ring 3.3. Noncommutative Fitting invariants §4. Fitting invariants and annihilation 4.1. Generalised adjoint matrices 4.2. Denominator ideals 4.3. Fitting invariants and annihilation §5. p-adic group rings §6. Additivity of Fitting invariants §Appendix A. Fitting invariants and Morita equivalence by Henri Johnston and Andreas Nickel A.1. Preliminaries on Morita equivalence A.2. Rings that are Morita equivalent to commutative rings References Iwasawa theory for modular forms -- Xin Wan §1. Introduction §2. Modular Forms, Galois Representations and Assumptions §3. lwasawa Theory for Ordinary Modular Forms §4. Kato's Formulation for Main Conjecture §5. lwasawa Main Conjecture for Supersingular Elliptic Curves §6. Iwasawa Main Conjecture for Modular Forms §7. Ramified Cases References Construction of elliptic p-units -- Werner Bley and Martin Hofer §1. Introduction §2. Formulation of the Main Theorem 2.1. Elliptic units 2.2. Construction of elliptic p-units 2.3. Statement of the Main Theorem §3. Computing the constant term of a Coleman power series 3.1. Definition and basic properties of Seiriki's pairing 3.2. Relative Lubin-Tate groups and Coleman power series 3.3. Auxiliary results 3.4. Seiriki's theorem on the constant term of a Coleman power series §4. Proof of the Main Theorem References On Stark elements of arbitrary weightand their p-adic families -- David Burns, Masato Kurihara and Takamichi Sano §1. Introduction 1.1. Background and main results 1.2. Notation §2. Generalized Stark elements 2.1. The general set up 2.2. The period-regulator isomorphisms 2.3. The definition of generalized Stark elements §3. A Rubin-Stark Conjecture in arbitrary weight 3.1. Exterior power biduals and pairings 3.2. T-modified cohomology 3.3. Statement of the conjecture §4. Zeta elements and evidence for Conjecture 3.6 4.1. Perfect complexes 4.2. Zeta elements 4.3. The proof of Theorem 4.1 References Beilinson-Kato and Beilinson-Flach elements,Coleman-Rubin-Stark classes, Heegner pointsand a conjecture of Perrin-Riou -- Kâzim Büyükboduk §1. Summary of Contents and Background 1.1. Notation §2. Part I. Perrin-Riou's conjecture for elliptic curves over Q 2.1. Preliminaries 2.2. Main conjectures and Perrin-Riou's conjecture 2.3. Logarithms of Heegner points and Beilinson-Kato classes §3. Part II. Λ-adic Kolyvagin systems, Beilinson-Flach elements, Coleman-Rubin-Stark elements and Heegner points 3.1. The set up 3.2. Module of Λ-adic Kolyvagin systems 3.3. Example: Perrin-Riou's conjecture for Beilinson-Flach elements 3.4. CM Abelian Varieties and Perrin-Riou-Stark elements 3.5. Logarithms of Heegner points and Perrin-Rion-Stark elements References On primitive p-adic Rankin-Selberg L-functions -- Shih-Yu Chen and Ming-Lun Hsieh §1. Introduction 1.1. Galois representations attached to Hida families 1.2. Rankin-Selberg L-functions 1.3. The modified Euler factors at p and ∞ 1.4. Hida's canonical periods 1.5. Statement of the interpolation formula §2. Classical modular forms and automorphic forms 2.1. Notation 2.2. Hecke characters and Dirichlet characters 2.3. Classical modular forms 2.4. Automorphic forms on GL2(A) 2.5. 2.6. Preliminaries on irreducible representations of GL2(Q_v) 2.7. p-stabilized newforms §3. Λ-adic modular forms and Hida families 3.1. Ordinary Λ-adic modular forms 3.2. Galois representation attached to Hida families 3.3. Hecke algebras and congruence numbers §4. A three variable p-adic family of Eisenstein series 4.1. Eisenstein series 4.2. The Eisenstein series E^\pm_k(μ1, μ2, C) 4.3. Fourier coefficients of Eisenstein series 4.4. A three variable p-adic family of Eisenstein series §5. The construction of p-adic Rankin-Selberg L-functions 5.1. The construction of the p-adic L-function 5.2. The interpolation formula and Rankin-Selberg integral 5.3. Rankin-Selberg L-functions for GL2 × GL2 5.4. The interpolation formula and Rankin-Selberg L-values §6. The calculation of local zeta integrals 6.1. The p-adic place 6.2. The l-adic case with l I N §7. The interpolation formulae References On the rank one Gross-Stark conjecture for quadratic extensions and the Deligne-Ribet q-expansion principle -- Samit Dasgupta and Mahesh Kakde §1. Introduction §2. Group rings and characters §3. p-adic L-functions §4. Hilbert modular forms §5. Group-ring valued modular forms §6. Theta series §7. A congruence of modular forms §8. The Deligne-Ribet q-expansion principle References Iwasawa theory for Artin representations I -- Ralph Greenberg and Vinayak Vatsal §1. Introduction §2. Galois cohomology for Artin representations §3. Selmer groups for Artin representations 1 §4. Selmer groups for Artin representations 2 §5. The p-adic L-function of an Artin representation References Iwasawa-Cohen-Lenstra heuristics -- Cornelius Greither § Introduction and background §1. Calculating the sum S_Λ §2. Cyclic Λ-modules and a connection with Kähler's zeta function References The second syzygy of the trivial G-module, and an equivariant main conjecture -- Cornelius Greither, Masato Kurihara and Hibiki Tokio §1. Introduction 1.1. A second syzygy of Z 1.2. An equivariant main conjecture for X_{K_∞,S} §2. The matrix \tilde{M_s}s and the minors 2.1. A free resolution of Z 2.2. Admissible polynomials §3. Proof of Theorem 1.2 §4. Outline of the proof of Theorem 1.1 §5. Small τ-monomials §6. Introducing a power of v_j §7. Bigger τ-monomials §8. The final step for τ-monomials §9. Synthesis: General (τ, ν)-monomials References Cyclicity of adjoint Selmer groups and fundamental units -- Haruzo Hida §1. Big Hecke algebra §2. The R = T theorem and an involution of R §3. The Taylor-Wiles system §4. Taylor-Wiles primes §5. Galois action on unit groups §6. Proof of Theorem A §7. Proof of Corollary B References Fitting invariants in equivariant lwasawa theory -- Takenori Kataoka §1. Introduction §2. Fitting Invariants 2.1. Shift 2.2. Fitting Ideals over a Commutative Ring 2.3. Noncommutative Fitting Invariants §3. Quasi-Fitting Invariants 3.1. Duality 3.2. Shift 3.3. Modules over a Compact p-adic Lie Group §4. Properties 4.1. Functoriality 4.2. Dual and Shift 4.3. Computation §5. Applications to lwasawa Theory 5.1. Tate Sequences 5.2. Totally Real Fields 5.3. Result at Finite Level 5.4. Σ_p-Ramified lwasawa Modules 5.5. Imaginary Quadratic Fields References Height functions for motives, II -- Kazuya Kato §0. Introduction §1. Period domains and motives 1.1. The period domain D 1.2. The period domain X(C) 1.3. The set X(F) of motives 1.4. Examples §2. Curvature forms and Hodge theory 2.1. Reviews on curvature forms of line bundles 2.2. Review on the result of Griffiths on curvature forms 2.3. X(C) is like a hyperbolic space in the case G is reductive 2.4. Height pairings in Hodge theory and curvature forms §3. Height functions 3.1. The setting 3.2. Height functions h_Λ(H) and H_Λ(M) 3.3. Reviews on height functions in Nevanlinna theory 3.4. Some results on degeneration 3.5. Height functions T_{f,Λ} (r) 3.6. ♠-height functions, ♡-height functions, complements on height functions 3.7. Asymptotic behaviors 3.8. Toroidal partial compactifications and height functions §4. Speculations 4.1. Speculations on positivity 4.2. Speculations on Vojta conjectures 4.3. Speculations on the number of motives of bounded height References Anticyclotomic main conjecture for modular forms and integral Perrin-Riou twists -- Shinichi Kobayashi and Kazuto Ota §1. Introduction 1.1. Our setting of the main conjecture 1.2. Background 1.3. Difficulties in the non-ordinary higher weight case 1.4. The conjecture and our main result 1.5. Plan of our proof §2. The p-adic L-function and the Selmer group 2.1. Notation and settings 2.2. Hecke characters and p-adic L-functions 2.3. Selmer groups 2.4. The main theorem §3. Integral Perrin-Riou twists 3.1. Setup 3.2. The integral twist §4. Twists of generalized Heegner classes and Selmer groups 4.1. Generalized Heegner classes 4.2. Generalized Heegner classes and the p-adic L-function 4.3. Twists of generalized Heegner cycles 4.4. Local conditions of twisted Selmer groups 4.5. Bounding Selmer groups 4.6. Twisted Heegner classes and the p-adic L-function §5. Proof of the main result 5.1. Comparison of Selmer groups 5.2. Control theorem 5.3. Proof of the main theorem References Syntomic regulator of Asai-Flach classes -- David Loeffler, Christopher Skinner and Sarah Livia Zerbes §1. Introduction 1.1. Aims of the paper 1.2. Statement of results 1.3. Relations to other work §2. Preliminaries on elliptic modular forms 2.1. Nearly holomorphic modular forms 2.2. Geometric interpretation 2.3. Nearly overconvergent forms 2.4. Rigid cohomology 2.5. Overconvergent projection operators §3. A "compactification" of the GL2 Eisenstein class 3.1. The Eisenstein class of level N 3.2. Lifting to compact supports 3.3. The "Eisenstein period" 3.4. Small levels §4. Preliminaries on Hilbert modular forms 4.1. Nearly-holomorphic Hilbert modular forms 4.2. Overconvergent and nearly-overconvergent p-adic Hilbert modular forms 4.3. Theta operators and rigid cohomology 4.4. Rankin-Cohen brackets 4.5. P-depletion §5. Evaluation of the regulator 5.1. Cohomology classes from Hilbert eigenforms 5.2. Representatives over the ordinary locus 5.3. Choice of the g_i §6. An example for D = 13 6.1. The newform F 6.2. A basis for overconvergent modular forms 6.3. Numerical linear algebra 6.4. The result References lwasawa invariants and linking numbers of primes -- Yasushi Mizusawa and Gen Yamamoto §1. Introduction §2. Linking numbers and pro-p Galois groups §3. Proof of Theorem 1.1 via Theorem 2.2 §4. Linking matrices of number fields §5. A criterion of Greenberg's conjecture via capitulation §6. Examples References On spectral sequences for I wasawa adjoints à la Jannsen for families -- Oliver Thomas and Otmar Venjakob §1. Introduction §2. Conventions §3. A few facts on R-modules 3.1. Noncommutative rings 3.2. The Koszul Complex 3.3. Local Cohomology §4. (Avoiding) Matlis Duality §5. Tate Duality and Local Cohomology §6. lwasawa Adjoints §7. Local Duality for lwasawa Algebras §8. Torsion in Iwasawa Cohomology 8.1. Torsion in Local lwasawa Cohomology 8.2. Torsion in Global lwasawa Cohomology References Advanced Studies in Pure Mathematics Presents the proceedings of the international conference 'Iwasawa 2017', held to commemorate the 100th anniversary of Kenkichi Iwasawa's birth. This volume consists of three survey papers and 15 research papers, and is recommended to researchers and graduate students interested in Iwasawa theory number theory and related fields.