Eta quotients, Eisenstein series and Elliptic Curves

TL;DR

This paper expresses all newforms of specific weight and levels as combinations of eta quotients and Eisenstein series, and derives generating functions and congruences for elliptic curve point counts over finite fields.

Contribution

It provides explicit representations of newforms in terms of eta quotients and Eisenstein series, and develops generating functions for elliptic curve point counts over finite fields.

Findings

Explicit expressions for newforms as eta quotients and Eisenstein series.

Generating functions for the order of elliptic curve points over finite fields.

Congruence relations for elliptic curve point counts.

Abstract

We express all the newforms of weight $2$ and levels $30$, $33$, $35$, $38$, $40$, $42$, $44$, $45$ as linear combinations of eta quotients and Eisenstein series, and list their corresponding strong Weil curves. Let $p$ denote a prime and $E (\zz_p)$ denote the the group of algebraic points of an elliptic curve $E$ over $\zz_p$. We give a generating function for the order of $E (\zz_p)$ for certain strong Weil curves in terms of eta quotients and Eisenstein series. We then use our generating functions to deduce congruence relations for the order of $E (\zz_p)$ for those strong Weil curves.