Tag Archives: Bilu

Speaking of Emmanuel Kowalski

How can I have forgotten to put his blog on my blogroll?  Well, it’s up there now — a great place for thoughtful posts on number theory both contemporary and historical, not to mention engaging diversions on mysterious symbols on slide rules and the important question of whether Grothendieck appears in the movie of Zazie dans le métro.

In Emmanuel’s most recent post, he reports on something I too should have mentioned; that the beautiful result of Bilu and Parent about rational points of X^split(p), whose original version fell prey to a subtle error, has apparently been corrected, and the original result is now once again independent of GRH.

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Bilu-Parent update

The result of Yuri Bilu and Pierre Parent that I blogged about last summer has appeared in a new, modified version on the arXiv. The authors discovered a mistake in the earlier version — their theorem on rational points on X^split(p) is now conditional on GRH, while they get an unconditional version for points on X^split(p^2). The dependence on GRH (Proposition 5.2 in the new version) is via explicit Chebotarev bounds; under GRH one has that if E/Q is a non-CM elliptic curve whose mod-p Galois representation lands in the normalizer of a split Cartan, then p << log (N_E)^(1+eps). The idea is that when E is not CM, one can find a nonzero Fourier coefficient a_l with l at most (log N_E)^(2+eps), which is required to reduce to 0 mod p; this immediately implies the desired bound on p. In the old version, the unconditional weaker bound p << (height(j(E)))^2, due to Masser, Wustholtz, and Pellarin, was sufficient; in the present version, it’s this bound that gives you control of X^split(p^2)(Q).

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Bilu-Parent: Serre’s uniformity in the split Cartan case

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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