Binary theta functions and Borcherds products

TL;DR

This paper demonstrates that certain binary theta functions can be expressed as Borcherds products, revealing their zeros only at quadratic irrationalities and expanding understanding of their structure in number theory.

Contribution

It shows that all weight 1 binary theta functions are Borcherds products, providing new infinite product expansions for these functions.

Findings

Binary theta functions have infinite product expansions as Borcherds products.

All weight 1 binary theta functions are Borcherds products.

Zeros of these functions occur only at quadratic irrationalities.

Abstract

We obtain infinite product expansions in the sense of Borcherds for theta functions associated with certain positive-definite binary quadratic and binary hermitian forms. Among other things, we show that every weight 1 binary theta function is a Borcherds product. In particular, binary theta functions have zeros only at quadratic irrationalities.