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Horizontal $p$-adic $L$-functions

Daniel Kriz, Asbjørn Christian Nordentoft

TL;DR

The paper constructs horizontal $p$-adic $L$-functions attached to twists of elliptic curves by $p$-power order characters and develops a horizontal Iwasawa-theoretic framework to study their zeros. By expressing central $L$-values via Birch--Stevens-type modular-symbol expansions and encoding them into horizontal measures, the authors prove strong non-vanishing results, including quantitative lower bounds and simultaneous non-vanishing for multiple curves. A key innovation is a horizontal analogue of Weierstrass preparation supported by a Fourier-analytic structure on non-noetherian horizontal Iwasawa algebras, together with a global-to-local norm-relations machinery. Conditional on mod $p$ Kurihara conjectures (and benefiting from recent progress) they obtain pervasive non-vanishing results for twists of general order, with explicit density and growth-parameter statements, providing new evidence toward conjectures of Goldfeld and related Diophantine stability phenomena.

Abstract

We define new objects called 'horizontal $p$-adic $L$-functions' associated to $L$-values of twists of elliptic curves over $\mathbb{Q}$ by characters of $p$-power order and conductor prime to $p$. We study the fundamental properties of these objects and obtain applications to non-vanishing of finite order twists of central $L$-values, making progress toward conjectures of Goldfeld and David--Fearnley--Kisilevsky. For general elliptic curves $E$ over $\mathbb{Q}$ we obtain strong quantitative lower bounds on the number of non-vanishing central $L$-values of twists by Dirichlet characters of fixed order $d\equiv 2 \mod 4$ greater than two. We also obtain non-vanishing results for general $d$, including $d = 2$, under mild assumptions. In particular, for elliptic curves with $E[2](\mathbb{Q}) = 0$ we improve on the previously best known lower bounds on the number of non-vanishing $L$-values of quadratic twists due to Ono. Finally, we obtain results on simultaneous non-vanishing of twists of an arbitrary number of elliptic curves with applications to Diophantine stability.

Horizontal $p$-adic $L$-functions

TL;DR

The paper constructs horizontal -adic -functions attached to twists of elliptic curves by -power order characters and develops a horizontal Iwasawa-theoretic framework to study their zeros. By expressing central -values via Birch--Stevens-type modular-symbol expansions and encoding them into horizontal measures, the authors prove strong non-vanishing results, including quantitative lower bounds and simultaneous non-vanishing for multiple curves. A key innovation is a horizontal analogue of Weierstrass preparation supported by a Fourier-analytic structure on non-noetherian horizontal Iwasawa algebras, together with a global-to-local norm-relations machinery. Conditional on mod Kurihara conjectures (and benefiting from recent progress) they obtain pervasive non-vanishing results for twists of general order, with explicit density and growth-parameter statements, providing new evidence toward conjectures of Goldfeld and related Diophantine stability phenomena.

Abstract

We define new objects called 'horizontal -adic -functions' associated to -values of twists of elliptic curves over by characters of -power order and conductor prime to . We study the fundamental properties of these objects and obtain applications to non-vanishing of finite order twists of central -values, making progress toward conjectures of Goldfeld and David--Fearnley--Kisilevsky. For general elliptic curves over we obtain strong quantitative lower bounds on the number of non-vanishing central -values of twists by Dirichlet characters of fixed order greater than two. We also obtain non-vanishing results for general , including , under mild assumptions. In particular, for elliptic curves with we improve on the previously best known lower bounds on the number of non-vanishing -values of quadratic twists due to Ono. Finally, we obtain results on simultaneous non-vanishing of twists of an arbitrary number of elliptic curves with applications to Diophantine stability.
Paper Structure (36 sections, 64 theorems, 258 equations)

This paper contains 36 sections, 64 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Non-vanishing Results
  3. Method of Proof
  4. Previous Results
  5. Notation
  6. Horizontal p-adic Measures
  7. Horizontal Iwasawa Algebras
  8. Constructing measures from norm relations
  9. Ring Theoretic Properties of Pro- p Iwasawa Algebras
  10. Twisting of measures
  11. Fourier Theory of Pro- p Abelian Groups
  12. Maximum modulus principle for finite group rings
  13. Structure theorem for character zeroes
  14. Non-vanishing of maximal order characters
  15. Horizontal Amice Transform
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $E/\mathbb{Q}$ be an elliptic curve and let $d\geq 2$ be an integer such that one of the following holds: Then there exists a constant $\alpha=\alpha_{E,d}>0$ such that

Theorems & Definitions (150)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.4
  • Theorem 1.6
  • ...and 140 more