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 ` 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`(E) as an integral Galois representations of k. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial fE(x) and the j-invariant jE of E. Hilbert Class Polynomials of imaginary quadratic orders play here an important role. We give a global appli-cation to the study of prime-splitting in torsion fields of elliptic curves over number fields. 1