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TWISTED TRIPLE PRODUCT $\text{p}$-ADIC L-FUNCTIONS AND HIRZEBRUCH–ZAGIER CYCLES

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  20 February 2019

Iván Blanco-Chacón  and
Michele Fornea
Iván Blanco-Chacón
Affiliation:
University College Dublin, Ireland (ivnblanco@gmail.com)
Michele Fornea
Affiliation:
McGill University, Montreal, Canada (michele.fornea@mail.mcgill.ca)
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Abstract

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross–Zagier formula relating the values of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.

Keywords

twisted triple product L-functionssyntomic Abel–Jacobi mapHirzebruch–Zagier cycles

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11G35: Varieties over global fields
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 1947 - 1992
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Copyright
© Cambridge University Press 2019

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