Elliptic curves, modular forms, and sums of Hurwitz class numbers

TL;DR

This paper explores the sums of Hurwitz class numbers with additional congruence conditions, using elliptic curves, modular forms, and trace formulas, providing new methods and partial results for specific moduli.

Contribution

It introduces three novel approaches to evaluate Hurwitz class number sums with congruence restrictions, extending understanding for moduli 2, 3, 4, and partial results for 5 and 7.

Findings

Explicit formulas for sums with m=2,3,4

Partial results and conjectures for m=5,7

Connections between Hurwitz sums and modular forms

Abstract

Let H(N) denote the Hurwitz class number. It is known that if $p$ is a prime, then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper, we investigate the behavior of this sum with the additional condition $r\equiv c\pmod m$. Three different methods will be explored for determining the values of such sums. First, we will count isomorphism classes of elliptic curves over finite fields. Second, we will express the sums as coefficients of modular forms. Third, we will manipulate the Eichler-Selberg trace for ula for Hecke operators to obtain Hurwitz class number relations. The cases $m=2,3$ and 4 are treated in full. Partial results, as well as several conjectures, are given for $m=5$ and 7.