On the Surjectivity of Mod ` Representations Associated to Elliptic Curves
نویسنده
چکیده
Let E be an elliptic curve over the rationals that does not have complex multiplication. For each prime `, the action of the absolute Galois group on the `-torsion points of E can be given in terms of a Galois representation ρE,` : Gal(Q/Q) → GL2(F`). An important theorem of Serre says that ρE,` is surjective for all sufficiently large `. In this paper, we describe an algorithm based on Serre’s proof that can quickly determine the finite set of primes ` for which ρE,` is not surjective. We will also give some improved bounds for Serre’s theorem.
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