Binary quadratic forms of odd class number

TL;DR

The paper studies representations of integers by primitive positive-definite binary quadratic forms of discriminant $-D$ when the class number $h(-D)$ is odd, establishing a modular-forms-based formula that expresses the representation numbers $a(n,Q)$ as a combination of a divisor-sum term and cusp-form coefficients $t_r(n)$ from cusp forms of weight 1, level $D$, and a Kronecker-type character. It then shows that the associated cusp forms $F_{D,r}(z)$ have eta-quotient representations only in the exceptional case $D=23$, providing a theta-function-based generalization of van der Blij's 1952 result for discriminant $-23$. The work further classifies eta quotients of prime level $D$ that arise as half the difference of two theta functions, introducing Schoeneberg pairs as the relevant mechanism, and establishing a complete eta-quotient classification in this setting. Throughout, the paper leverages the Eholzer–Skoruppa product-expansion framework and modular-forms techniques to draw precise connections between representation numbers, theta functions, and eta quotients, with computational illustrations for class numbers one, three, and five. The results yield a structured, modular-theoretic approach to calculating representation numbers and clarify when eta-quotient representations can occur.

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 $D$.