Binary quadratic forms of odd class number

TL;DR

This paper relates the number of representations by binary quadratic forms with odd class number to cusp forms and eta quotients, generalizing classical results and classifying special eta quotients.

Contribution

It expresses representation counts as rational linear combinations involving cusp form Fourier coefficients and classifies eta quotients related to these forms.

Findings

Representation counts are linked to cusp form coefficients.

Eta quotient representations occur only for D=23.

Classified eta quotients as differences of theta functions.

Abstract

Let $-D$ be a fundamental discriminant. We express the number of representations of an integer by a positive definite binary quadratic form of discriminant $-D$ with an odd class number $h(-D)$ as a rational linear expression involving the Kronecker symbol $\left(\frac{-D}{.}\right)$ and the Fourier coefficients of certain cusp forms. We prove these cusp forms have eta quotient representations only if $D=23$. This provides, using theta functions, a generalization of a result of F. van der Blij from 1952 for binary quadratic forms of discriminant $-23$ to the case of forms of discriminant $-D$ with odd $h(-D)$. We also classify all the eta quotients of prime level $D$ which are half the difference of two theta functions of level $D$.