A $p$ -adic arithmetic inner product formula

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-02-23 DOI:10.1007/s00222-024-01243-7

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Abstract

Fix a prime number $p$ and let $E/F$ be a CM extension of number fields in which $p$ splits relatively. Let $\pi $ be an automorphic representation of a quasi-split unitary group of even rank with respect to $E/F$ such that $\pi $ is ordinary above $p$ with respect to the Siegel parabolic subgroup. We construct the cyclotomic $p$ -adic $L$ -function of $\pi $ , and a certain generating series of Selmer classes of special cycles on Shimura varieties. We show, under some conditions, that if the vanishing order of the $p$ -adic $L$ -function is 1, then our generating series is modular and yields explicit nonzero classes (called Selmer theta lifts) in the Selmer group of the Galois representation of $E$ associated with $\pi $ ; in particular, the rank of this Selmer group is at least 1. In fact, we prove a precise formula relating the $p$ -adic heights of Selmer theta lifts 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.

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p$ 的算术内积公式

Abstract Fix a prime number$p$ and let $E/F$ be a CM extension of number fields in which $p$ splits relatively.让$\pi $是一个关于$E/F$的偶数阶的准分裂单元群的自变量表示，使得$\pi $在关于西格尔抛物面子群的$p$之上是普通的。我们构造了 $\pi $的cyclotomic $p$ -adic $L$-function，以及Shimura varieties上特殊循环的Selmer类的某个产生数列。我们在一些条件下证明了，如果 $p$ -adic $L$ -function 的消失阶是 1，那么我们的产生数列就是模数化的，并在与$\pi $相关联的 $E$ 的伽罗瓦表示的塞尔玛群中产生明确的非零类（称为塞尔玛θ提升）；特别是，这个塞尔玛群的秩至少是 1。事实上，我们证明了一个精确的公式，这个公式将塞尔默θ提升的 $p$ -adic 高度与 $p$ -adic $L$ -function 的导数联系起来。与 Perrin-Riou 的 Gross-Zagier 公式的 $p$ -adic 类似，我们的公式是李超和第二作者最近建立的算术内积公式的 $p$ -adic 类似。

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