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.

Accepted for publication by International Journal of Number Theory, 12 pages

Cited by in corpus (7)

• Computing images of Galois representations attached to elliptic curves
• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Possible indices for the Galois image of elliptic curves over Q
• Computing endomorphism rings and Frobenius matrices of Drinfeld modules
• On the symplectic type of isomorphims of the p-torsion of elliptic curves
• Endomorphism rings of reductions of Drinfeld modules
• On the degree of the $p$-torsion field of elliptic curves over ${\mathbb{Q}}_{\ell }$ for $\ell \ne p$