Galois Representations Attached to Elliptic Curves: around Serre's Uniformity Question

Serre’s uniformity question asks whether there exists a bound N > 0 such that, for every non-CM elliptic curve E over Q and every prime p > N , the residual Galois representation ρE,p : Gal(Q/Q) → Aut(E[p]) is surjective. The work of many authors has shown that, for p > 37, this representation is either surjective or has image contained in the normaliser of a non-split Cartan subgroup C+ns(p). Zywina has further proved that, whenever ρE,p is not surjective for p > 37, its image can be either C+ns(p) or an index-3 subgroup of it. Recently, Le Fourn and Lemos showed that the index-3 case cannot arise for p > 1.4 · 107. We show that the same statement holds for any prime larger than 37. We then further explore effectivisation of Serre’s open image theorem. Given an elliptic curve E defined over Q without complex multiplication, we provide an explicit sharp bound on the index of the image of the adelic representation ρE . In particular, we show that [GL2(Z) : Im ρE ] is bounded above by h(j(E))3+o(1) for h(j(E)) tending to infinity, where the constants are explicit. We also classify the possible (conjecturally non-existent) images of the representations ρE,pn whenever Im ρE,p is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo