Binary quadratic forms of odd class number

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 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.