The modularity of Siegel's zeta functions

TL;DR

This paper proves Siegel's conjecture on the modularity of his zeta functions associated with indefinite quadratic forms by employing a Weil-type converse theorem for Maass forms, revealing their connection to holomorphic modular forms.

Contribution

The paper establishes the modularity of Siegel's zeta functions using a modern converse theorem, confirming Siegel's original plan and linking these functions to holomorphic modular forms.

Findings

Siegel's zeta functions are shown to be modular forms.

Half of Siegel's zeta functions correspond to holomorphic modular forms.

The proof employs a Weil-type converse theorem for Maass forms.

Abstract

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel's original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that "half" of Siegel's zeta functions correspond to holomorphic modular forms.