[Paper Review] On a conjecture of Perrin-Riou

Kâzım Büyükboduk|arXiv (Cornell University)|Nov 19, 2015

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper proves Perrin-Riou's conjecture on the non-vanishing of the $p$-adic Beilinson-Kato class for elliptic curves over $\mathbb{Q}$ with ordinary or supersingular reduction at $p$, using a unified approach based on $\Lambda$-adic Kolyvagin systems. The result extends prior work by Bertolini-Darmon and Venerucci and establishes the non-vanishing under mild hypotheses, advancing the $p$-adic Birch and Swinnerton-Dyer conjecture.

ABSTRACT

Our goal in this article is to give a proof of Perrin-Riou's conjecture (under reasonably mild hypotheses) on the non-vanishing of the $p$-adic Beilinson-Kato class associated to an elliptic curve $E_{/\mathbb{Q}}$, when $E$ has ordinary (i.e., good ordinary or multiplicative) or supersingular reduction at $p$. This generalizes the forthcoming work of Bertolini and Darmon for a good ordinary prime $p$ and of Venerucci for a split multiplicative prime $p$. Our method is based on the general theory of $\Lambda$-adic Kolyvagin systems, as developed by the author previously (and enhanced slightly here) and it applies equally well to treat all these cases simultaneously.

Motivation & Objective

• To prove Perrin-Riou's conjecture on the non-vanishing of the $p$-adic Beilinson-Kato class for elliptic curves over $\mathbb{Q}$ with ordinary or supersingular reduction at $p$.
• To generalize previous results by Bertolini and Darmon (good ordinary case) and Venerucci (split multiplicative case) into a single, unified framework.
• To establish the non-vanishing of the $p$-adic $L$-function special $L$-values via $\Lambda$-adic Kolyvagin systems under mild hypotheses.
• To extend the applicability of $\Lambda$-adic Kolyvagin systems to include both ordinary and supersingular reduction cases simultaneously.

Proposed method

• Utilizes the general theory of $\Lambda$-adic Kolyvagin systems, previously developed by the author, as the foundational framework.
• Applies a refined version of the $\Lambda$-adic Kolyvagin system machinery, slightly enhanced for this work.
• Establishes the non-vanishing of the $p$-adic Beilinson-Kato class by analyzing its image in the $\Lambda$-adic Selmer group.
• Uses the structure of the Iwasawa module and the control theorem to relate the $p$-adic $L$-values to the Kolyvagin system construction.
• Applies the theory of Euler systems and $\Lambda$-modules to deduce non-vanishing from the non-triviality of the system.
• Treats both ordinary and supersingular cases uniformly by leveraging the $\Lambda$-adic framework, avoiding case-by-case analysis.

Experimental results

Research questions

• RQ1 Does the $p$-adic Beilinson-Kato class associated to an elliptic curve over $\mathbb{Q}$ with ordinary or supersingular reduction at $p$ remain non-zero in the $\Lambda$-adic Selmer group?
• RQ2 Can the non-vanishing of the $p$-adic $L$-value special $L$-values be deduced uniformly across ordinary and supersingular reduction types?
• RQ3 To what extent can the $\Lambda$-adic Kolyvagin system method be extended to cover both ordinary and supersingular cases simultaneously?
• RQ4 Is there a unified framework that generalizes the results of Bertolini-Darmon and Venerucci under a common set of mild hypotheses?
• RQ5 How does the structure of the $\Lambda$-module control the non-vanishing of the $p$-adic $L$-function in the supersingular case?

Key findings

• The $p$-adic Beilinson-Kato class is non-vanishing for elliptic curves over $\mathbb{Q}$ with ordinary or supersingular reduction at $p$, under reasonably mild hypotheses.
• The proof unifies the treatment of ordinary and supersingular cases via the $\Lambda$-adic Kolyvagin system framework.
• The non-vanishing result extends prior work of Bertolini and Darmon for good ordinary primes and of Venerucci for split multiplicative primes.
• The method relies on the enhanced theory of $\Lambda$-adic Kolyvagin systems, which allows for a coherent treatment across different reduction types.
• The result provides strong evidence toward the $p$-adic Birch and Swinnerton-Dyer conjecture in the supersingular case.
• The framework demonstrates the robustness of $\Lambda$-adic systems in handling diverse $p$-adic $L$-function behaviors.