A p-adic arithmetic inner product formula

• 成果类型: Article
• 署名作者: Disegni, Daniel; Liu, Yifeng
• 署名单位: Ben-Gurion University of the Negev; Centre National de la Recherche Scientifique (CNRS); Aix-Marseille Universite; Zhejiang University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01243-7
• 发表日期: 2024
• 页码: 219-371
• 关键词: semi-stable reduction; local-global compatibility; gross-zagier formula; etale tate twists; l-derivatives; crystalline cohomology; theta correspondence; eisenstein series; vanishing cycles; heegner points

摘要

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.