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A $p$-adic arithmetic inner product formula

Daniel Disegni, Yifeng Liu

Abstract

Fix a prime number $p$ and let $E/F$ be a CM extension of number fields in which $p$ splits relatively. Let $π$ be an automorphic representation of a quasi-split unitary group of even rank with respect to $E/F$ such that $π$ is ordinary above $p$ with respect to the Siegel parabolic subgroup. We construct the cyclotomic $p$-adic $L$-function of $π$, and show, under certain conditions, that if its order of vanishing at the trivial character is $1$, then the rank of the Selmer group of the Galois representation of $E$ associated with $π$ is at least $1$. Furthermore, under a certain modularity hypothesis, we use special cycles on unitary Shimura varieties to construct some explicit elements in the Selmer group called Selmer theta lifts; and we prove a precise formula relating their $p$-adic heights to the derivative of the $p$-adic $L$-function. In parallel to Perrin-Riou's $p$-adic analogue of the Gross--Zagier formula, our formula is the $p$-adic analogue of the arithmetic inner product formula recently established by Chao~Li and the second author.

A $p$-adic arithmetic inner product formula

Abstract

Fix a prime number and let be a CM extension of number fields in which splits relatively. Let be an automorphic representation of a quasi-split unitary group of even rank with respect to such that is ordinary above with respect to the Siegel parabolic subgroup. We construct the cyclotomic -adic -function of , and show, under certain conditions, that if its order of vanishing at the trivial character is , then the rank of the Selmer group of the Galois representation of associated with is at least . Furthermore, under a certain modularity hypothesis, we use special cycles on unitary Shimura varieties to construct some explicit elements in the Selmer group called Selmer theta lifts; and we prove a precise formula relating their -adic heights to the derivative of the -adic -function. In parallel to Perrin-Riou's -adic analogue of the Gross--Zagier formula, our formula is the -adic analogue of the arithmetic inner product formula recently established by Chao~Li and the second author.
Paper Structure (44 sections, 88 theorems, 513 equations)

This paper contains 44 sections, 88 theorems, 513 equations.

Table of Contents

  1. Introduction
  2. Cyclotomic $p$-adic $L$-function
  3. Modularity of generating functions in Selmer groups
  4. A $p$-adic arithmetic inner product formula
  5. Application to symmetric power of elliptic curves
  6. Structure and strategy
  7. Notation and conventions
  8. Siegel hermitian varieties
  9. Running notation
  10. Siegel hermitian varieties and line bundles of automorphy
  11. Relation with PEL type moduli spaces
  12. Cyclotomic $p$-adic $L$-function
  13. Doubling space and degenerate principal series
  14. Siegel hermitian Eisenstein series
  15. Relevant representations
  16. ...and 29 more sections

Key Result

Theorem 1.4

Under the above setup, suppose that $\pi_v$ is Panchishkin unramified for every $v\in\mathtt{V}_F^{(p)}$. For every finite set $\lozenge$ of places of $\mathbb{Q}$ containing $\{\infty,p\}$ such that $\pi_v$ is unramified for every $v\in\mathtt{V}_F^\mathrm{fin}\setminus\mathtt{V}_F^{(\lozenge)}$, t where

Theorems & Definitions (224)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.7: Theorem \ref{['th:modularity']}
  • Theorem 1.8: $p$-adic arithmetic inner product formula, Theorem \ref{['th:aipf']}
  • Corollary 1.9
  • Remark 1.10
  • Remark 1.11
  • ...and 214 more