Formal Groups and Invariant Differentials of Elliptic Curves

TL;DR

This paper derives a power series expansion for the invariant differential of elliptic curves over rationals and explores congruence relations related to Frobenius traces, advancing understanding of elliptic curve invariants.

Contribution

It introduces explicit power series expansions for invariant differentials and establishes new congruence relations for Frobenius traces of elliptic curves over ield.

Findings

Power series expansion of ield invariant differential derived

New congruence relations for Frobenius trace established

Enhanced understanding of elliptic curve invariants over ield

Abstract

In this paper, we find a power series expansion of the invariant differential $\omega_E$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we introduce several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.