A note on the Fourier coefficients of a Cohen-Eisenstein series

TL;DR

This paper derives a formula for the Fourier coefficients of a weight 3/2 Cohen-Eisenstein series of square-free level, generalizing previous results and confirming a conjecture related to class numbers and elliptic curves.

Contribution

It provides a generalized formula for Cohen-Eisenstein series coefficients, extending Gross's work and proving Quattrini's conjecture about divisibility relations involving class numbers and elliptic curve groups.

Findings

The formula generalizes Gross's result for square-free levels.

It establishes a divisibility criterion linking class numbers and elliptic curve group orders.

Under certain conditions, it confirms the conjecture relating Shafarevich-Tate groups and class numbers.

Abstract

We prove a formula for the coefficients of a weight $3/2$ Cohen-Eisenstein series of square-free level $N$. This formula generalizes a result of Gross and in particular, it proves a conjecture of Quattrini. Let $l$ be an odd prime number. For any elliptic curve $E$ defined over $\mathbb{Q}$ of rank zero and square-free conductor $N$, if $l \mid |E(\mathbb{Q})|$, under certain conditions on the Shafarevich-Tate group $III_D$, we show that $l$ divides $|III_D|$ if and only if $l$ divides the class number $h(-D)$ of $\mathbb{Q}(\sqrt{-D}).$