Special values of triple-product p-adic L-functions and non-crystalline diagonal classes

Francesca Gatti, Xavier Guitart, Marc Masdeu, Victor Rotger

Research output: Contribution to journalArticleResearchpeer-review

Abstract

The main purpose of this note is to understand the arithmetic encoded in the special value of the p-adic L-function Lgp(f, g, h) associated to a triple of modular forms (f, g, h) of weights (2, 1, 1), in the case where the classical L-function (Formula presented) (which typically has sign +1) does not van-ish at its central critical point s = 1. When f corresponds to an elliptic curve E/Q and the classical L-function vanishes, the Elliptic Stark Conjecture of Darmon–Lauder–Rotger predicts that Lgp(f, g, h)(2, 1, 1) is either 0 (when the order of vanishing of the complex L-function is > 2) or related to logarithms of global points on E and a certain Gross–Stark unit associated to g (when the order of vanishing is exactly 2). We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value Lgp(f, g, h)(2, 1, 1) in the case where (Formula presented).

Original languageEnglish
Pages (from-to)809-834
Number of pages26
JournalJournal de Theorie des Nombres de Bordeaux
Volume33
Issue number3.1
DOIs
Publication statusPublished - 2021

Keywords

  • Elliptic curves
  • P-adic L-functions
  • Selmer groups

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