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Triple product $p$-adic $L$-functions for finite slope families and a $p$-adic Gross-Zagier formula

Ting-Han Huang

Abstract

In this paper, we generalize two results of H. Darmon and V. Rotger on triple product $p$-adic $L$-functions associated with Hida families to finite slope families. We first prove a $p$-adic Gross-Zagier formula, then demonstrate an application to a special case of the equivariant Birch and Swinnerton-Dyer conjecture for supersingular elliptic curves.

Triple product $p$-adic $L$-functions for finite slope families and a $p$-adic Gross-Zagier formula

Abstract

In this paper, we generalize two results of H. Darmon and V. Rotger on triple product -adic -functions associated with Hida families to finite slope families. We first prove a -adic Gross-Zagier formula, then demonstrate an application to a special case of the equivariant Birch and Swinnerton-Dyer conjecture for supersingular elliptic curves.
Paper Structure (24 sections, 29 theorems, 145 equations)

This paper contains 24 sections, 29 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Triple product -adic -functions
  3. The modular sheaf and the de Rham sheaf
  4. The $q$-expansions.
  5. The operators $U$ and $V$.
  6. $p$-adic iterations of the Gauss--Manin connection.
  7. The overconvergent projections.
  8. Definition of the triple product -adic -function
  9. Triple product $p$-adic $L$-functions for finite slope families.
  10. The -adic Gross--Zagier formula
  11. Special values at balanced classical weights
  12. The -adic Abel--Jacobi maps and the generalized diagonal cycles
  13. Kuga--Sato varieties and the generalized diagonal cycles.
  14. Explicit computations of the Abel--Jacobi maps.
  15. Proof of the -adic Gross--Zagier formula
  16. ...and 9 more sections

Key Result

Theorem 1.1

Given $(x, y, z) \in \Sigma_{\mathop{\mathrm{\operatorname{bal}}}\nolimits}$. Let $c := (x+ y+ z- 2)/2$ and write $x = y +z -2t$ with $t \in \mathbb{Z}_{> 0}$. Then where the Euler factors are given by

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2: DR2
  • Theorem 1.3: DR and reciprocity_balanced
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 60 more