Yuri Bilu and Pierre Parent posted a beautiful paper on the arXiv last week, settling part of a very old problem about the mod-p Galois representations attached to elliptic curves over Q.

If E is an elliptic curve over Q, the action of Galois on the p-torsion points of E yields a Galois representation

rho_{E,p}: Gal(Q) -> GL_2(F_p).

A famous theorem of Serre tells us that if E does not have complex multiplication, then rho_{E,p} is surjective for p large enough. But what “large enough” means depends, a priori, on E.

In practice, one seldom comes across an elliptic curve without CM such that rho_{E,p} is non-surjective. Thus the conjecture, originally due to Serre and now very widely believed, that “large enough” *need not* depend on the elliptic curve; that is, there is some absolute constant P such that rho_{E,p} is surjective for all non-CM elliptic curves over Q and all p > P.

More number theory below the fold:

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