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Algebraic functional equation for big Galois representations over multiple $\mathbb{Z}_p$-extensions

Zeping Hao, Meng Fai Lim

TL;DR

The paper develops a unifying axiomatic framework for proving algebraic functional equations for big Galois representations over multi-$ abla Z_p$-extensions, encompassing Selmer groups and Nekovář Selmer complexes. It introduces a datum $(T, igl"{T_v}igr brace_{v|p})_{R,F}$ with conditions (C1)-(C4) and (R1)-(R2) and proves that the Greenberg Selmer groups and Selmer complexes satisfy a two-sided functional-equation-type relation, including a pseudo-isomorphism of torsion parts and equality of characteristic ideals under finite generation. The results unify and generalize numerous arithmetic cases (Hida families, Rankin–Selberg, triple products, half-ordinary deformations) and provide tools to realize algebraic functional equations beyond cyclotomic extensions, resolving prior gaps and extending Greenberg’s questions. Applications cover abelian varieties, modular forms, and their $p$-adic $L$-functions, offering a broad, multi-variable Iwasawa-theoretic perspective on algebraic functional equations.

Abstract

We present a general approach to establish algebraic functional equations for big Galois representations over multiple $\mathbb{Z}_p$-extensions. Our result is formulated in both Selmer group and Selmer complex settings, and encompasses a broad range of Iwasawa-theoretic scenarios. In particular, our result applies to the triple product of Hida families in both balanced and unbalanced cases, as well as the half-ordinary Rankin-Selberg universal deformations recently studied by the first named author and Loeffler. Our result also significantly generalizes many previously known cases of algebraic functional equations and answers a question of Greenberg.

Algebraic functional equation for big Galois representations over multiple $\mathbb{Z}_p$-extensions

TL;DR

The paper develops a unifying axiomatic framework for proving algebraic functional equations for big Galois representations over multi--extensions, encompassing Selmer groups and Nekovář Selmer complexes. It introduces a datum with conditions (C1)-(C4) and (R1)-(R2) and proves that the Greenberg Selmer groups and Selmer complexes satisfy a two-sided functional-equation-type relation, including a pseudo-isomorphism of torsion parts and equality of characteristic ideals under finite generation. The results unify and generalize numerous arithmetic cases (Hida families, Rankin–Selberg, triple products, half-ordinary deformations) and provide tools to realize algebraic functional equations beyond cyclotomic extensions, resolving prior gaps and extending Greenberg’s questions. Applications cover abelian varieties, modular forms, and their -adic -functions, offering a broad, multi-variable Iwasawa-theoretic perspective on algebraic functional equations.

Abstract

We present a general approach to establish algebraic functional equations for big Galois representations over multiple -extensions. Our result is formulated in both Selmer group and Selmer complex settings, and encompasses a broad range of Iwasawa-theoretic scenarios. In particular, our result applies to the triple product of Hida families in both balanced and unbalanced cases, as well as the half-ordinary Rankin-Selberg universal deformations recently studied by the first named author and Loeffler. Our result also significantly generalizes many previously known cases of algebraic functional equations and answers a question of Greenberg.
Paper Structure (14 sections, 28 theorems, 124 equations)

This paper contains 14 sections, 28 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries
  3. Main results
  4. Comparing Selmer groups and Selmer complexes
  5. Proof of Theorems \ref{['main thm0']} and \ref{['main thm0 Selcomplex']}
  6. Proof of Theorems \ref{['main thm']} and \ref{['main thm Selcomplex']}
  7. Examples
  8. Abelian variety with good ordinary reduction at every $p$-adic prime
  9. $p$-ordinary modular forms
  10. Rankin-Selberg products
  11. Hida family
  12. Triple product of Hida families
  13. Half-ordinary Rankin-Selberg universal deformation
  14. Appendix

Key Result

Theorem 1.1

Suppose that $R=\mathcal{O}$ and that our datum $(T, \{T_v\}_{v|p})_{\mathcal{O},F}$ satisfies (C1)-(C4), (R1) and (R2). Write $\Gamma=\mathop{\mathrm{Gal}}\nolimits(F_\mathrm{cyc}/F)$. Then $X_{Gr}(A/F_\mathrm{cyc})$ and $X_{Gr}(A^*/F_\mathrm{cyc})$ have the same $\mathcal{O}\llbracket \Gamma \rrbr of $\mathcal{O}\llbracket \Gamma \rrbracket$-modules.

Theorems & Definitions (57)

  • Theorem 1.1: Theorem \ref{['main thm0']}
  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['main thm0 Selcomplex']}
  • Theorem 1.4: Theorem \ref{['main thm Selcomplex']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • ...and 47 more