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

الأصل · EN

Let E be an elliptic curve over a finite field k, and ℓ a prime number different from the characteristic of k. In this paper we consider the problem of finding the structure of the Tate module Tℓ(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 application to the study of prime-splitting in torsion fields of elliptic curves over number fields.