On Zeta Functions and Families of Siegel Modular Forms

TL;DR

This paper explores $p$-adic families of zeta functions and Siegel modular forms, analyzing their $L$-functions, critical values, and establishing higher genus Rankin's lemma, with conjectures on liftings and constructions of $p$-adic families.

Contribution

It introduces new $p$-adic constructions of Siegel modular forms and formulates a conjecture on liftings between symplectic groups, advancing understanding of their $L$-functions.

Findings

Description of $L$-functions in terms of motivic $L$-functions.

Establishment of higher genus Rankin's lemma.

Construction methods for $p$-adic families of Siegel modular forms.

Abstract

Let $p$ be a prime, and let $\Gamma=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modular forms are given using Ikeda-Miyawaki constructions.