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- Title
A p-adic arithmetic inner product formula.
- Authors
Disegni, Daniel; Liu, Yifeng
- 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 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 π ; 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.
- Subjects
ARITHMETIC; PRIME numbers; UNITARY groups
- Publication
Inventiones Mathematicae, 2024, Vol 236, Issue 1, p219
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-024-01243-7