Hirzebruch-Zagier cycles in $p$-adic families and adjoint $L$-values

TL;DR

This paper constructs $p$-adic families of Hirzebruch-Zagier cycles on Hilbert modular varieties and uses them to geometrically realize multivariable $p$-adic adjoint $L$-functions associated with Hida families.

Contribution

It introduces a novel $p$-adic family framework for Hirzebruch-Zagier cycles and connects these to the geometric construction of multivariable $p$-adic $L$-functions.

Findings

Hirzebruch-Zagier cycles can be organized into $p$-adic families.

A geometric construction of multivariable $p$-adic adjoint $L$-functions is achieved.

The approach links cycles on Hilbert modular varieties to special values of $L$-functions.

Abstract

Let $E/F$ be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch-Zagier cycles arising from the associated Hilbert modular varieties can be put in $p$-adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable $p$-adic adjoint $L$-function twisted by the Hecke character of $E/F$, attached to Hida families of Hilbert modular forms over $F$.