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

Abstract

We prove a formula for the coefficients of a weight [Formula: see text] Cohen–Eisenstein series of square-free level [Formula: see text]. This formula generalizes a result of Gross, and in particular, it proves a conjecture of Quattrini. Let [Formula: see text] be an odd prime number. For any elliptic curve [Formula: see text] defined over [Formula: see text] of rank zero and square-free conductor [Formula: see text], if [Formula: see text], under certain conditions on the Shafarevich–Tate group [Formula: see text], we show that [Formula: see text] divides [Formula: see text] if and only if [Formula: see text] divides the class number [Formula: see text] of [Formula: see text]