On sparsity of positive-definite automorphic forms within a family

TL;DR

This paper investigates the sparsity of positive-definite automorphic forms within various families, showing that most forms are not positive-definite under certain averaging conditions, extending classical results about polynomial zeros.

Contribution

It generalizes Baker and Montgomery's result to automorphic forms, providing an axiomatic framework for their non-positive-definiteness in diverse families.

Findings

Most automorphic forms in studied families are not positive-definite.

The results apply to holomorphic cusp forms, Hilbert class characters, and elliptic curves.

Automorphic forms rarely satisfy positive-definiteness under the given conditions.

Abstract

It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. On generalizing their result, we formulate an axiomatic result about sets of automorphic forms satisfying certain averages when suitably ordered, which ensures that almost all forms are not positive-definite within such sets. We then apply the result to various families, including the family of holomorphic cusp forms, the family of the Hilbert class characters of imaginary quadratic fields, and the family of elliptic curves.