Integral Tate modules and splitting of primes in torsion fields of elliptic curves
Let [Formula: see text] be an elliptic curve over a finite field [Formula: see text], and [Formula: see text] a prime number different from the characteristic of [Formula: see text]. In this paper, we consider the problem of finding the structure of the Tate module [Formula: see text] as an integral Galois representations of [Formula: see text]. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial [Formula: see text] and the [Formula: see text]-invariant [Formula: see text] of [Formula: see text]. Hilbert Class Polynomials of imaginary quadratic orders play an important role here. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields.