## 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).

## Two idle questions about modular curves

This post is an math-blogging experiment in writing down small questions that have occurred to me, and which I haven’t thought about seriously — thus it is highly possible they are poorly formed, or that the answers are obvious.

1. Let f be a cuspform on S_2(Gamma_0(N)) such that A_f has dimension greater than 1. Then the map X_0(N) -> A_f factors through X_0(N)/W, where W is some group of Atkin-Lehner involutions which act as +1 on A_f. Do we know an example of such an f where the map X_0(N)/W -> A_f is not a closed embedding? What if dim A_f is greater than 2? (In some sense, a map from a curve to a three-fold should be less likely to intersect itself “by chance” than a map from a curve to a surface.)
2. In the original proof of Fermat’s Last Theorem, Mazur’s theorem on rational isogenies of elliptic curves over Q was used in a critical way; when E is your Frey curve, you prove that E is modular, then derive a contradiction from the fact that E[p] is an _irreducible_ modular mod p Galois representation with very little ramification. Nowadays, can one write down a proof of Fermat that doesn’t pass through Mazur’s theorem?

## 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:

## Are there elliptic curves with small rank?

Usually when people muse about distribution of ranks of elliptic curves, they’re wondering how large the rank of an elliptic curve can be. But as Tim Dokchitser pointed out to me, there are questions in the opposite direction which are equally natural, and equally mysterious. For instance: we do not know that, for every number field K, there exists an elliptic curve E/K whose Mordell-Weil rank is less than 100. Isn’t that strange?

Along similar lines, Bjorn Poonen asks: is it the case that, for every number field K, there exists an elliptic curve E/Q such that E(Q) and E(K) have the same positive rank? Again, we don’t have a clue. See Bjorn’s expository article “Undecidability in Number Theory” (.pdf link) to find out what this has to do with Hilbert’s 10th problem.

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