Gan--Gross--Prasad cycles and derivatives of p-adic L-functions

2024/10/10 by Daniel Disegni, Wei Zhang, Disegni, Daniel +1 · 1 citation
Mathematics · #advanced mathematical theories #Advanced Algebra and Geometry #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2410.08401

Abstract

We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let Π be a conjugate-selfdual cuspidal automorphic representation of GL_n x GL_n+1 over a CM field, which is algebraic of minimal regular weight at infinity. We first show the rationality of twists of the ratio of L-values of Π appearing in the GGP conjectures. Then, when Π is p-ordinary at a prime p, we construct a cyclotomic p-adic L-function L_p(M_Π) interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of L_p(M_Π) to the p-adic heights of Selmer classes arising from arithmetic diagonal cycles on unitary Shimura varieties. We deduce applications to the p-adic Beilinson-Bloch-Kato conjecture for the motive attached to Π. All proofs are based on some relative-trace formulas in p-adic coefficients.

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• Central values of Asai L-functions and twisted Gan--Gross--Prasad conjecture

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