Integral Tate modules and splitting of primes 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.