Integral Tate modules and splitting of primes in torsion fields of elliptic curves

TL;DR

This paper investigates the structure of Tate modules of elliptic curves over finite fields as Galois representations, providing explicit procedures and applications to prime-splitting in torsion fields over number fields.

Contribution

It introduces an explicit method to determine Tate module structures from characteristic polynomials and j-invariants, linking local and global properties of elliptic curves.

Findings

Explicit procedure for Tate module structure determination

Connection between characteristic polynomial, j-invariant, and Galois representations

Application to prime-splitting in torsion fields of elliptic curves

Abstract

Let $E$ be an elliptic curve over a finite field $k$, and $\ell$ a prime number different from the characteristic of $k$. In this paper we consider the problem of finding the structure of the Tate module $T_\ell(E)$ as an integral Galois representations of $k$. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial $f_E(x)$ and the $j$-invariant $j_E$ of $E$. Hilbert Class Polynomials of imaginary quadratic orders play here an important role. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields.