On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

Abstract

Abstract. Let E be an elliptic curve defined over Q, of conductor N and without complex multiplication.

Almost all elliptic curves are Serre curves

“…In particular, this implies that any fixed non-CM elliptic curve E has only finitely many exceptional primes, since any such exceptional prime must divide m E . One might wonder how the integer m E (chosen minimally so that (1) still holds) depends on the curve E. Various results exist which bound the largest possible exceptional prime for E. For example, Mazur [17] proves that if E is semistable, then no prime N ≥ 11 can be exceptional for E. Other authors have bounded the largest possible exceptional prime in terms of invariants of the elliptic curve, such as the height [16] and conductor ( [21], [15], and [2]). …”

A bound for the torsion conductor of a non-CM elliptic curve

“…The work of Parent [8] represents further progress towards resolution of the split Cartan case, while the work of Chen [2] shows that in the non-split case, new ideas are needed. Other authors have bounded the largest prime p satisfying (3) in terms of invariants of the elliptic curve ( [11], [4], [3], and [6]). …”

“…The torsion conductor m E should not be confused with the number [3], which has the useful property that, for any integer n,…”

Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik?s problem

“…1. For the first statement of part 1 we refer the reader to [Se68] or [acC4,Appendix]. Now for k coprime to A(E) let us show that Q(ζ k ) is the maximal abelian extension contained in…”