Binary quadratic forms of odd class number

2024/07/31 by Akbary, Amir, Totani, Yash
#11E25 #11E45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.00184

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 ((-D)/(.)) 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.

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