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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2 thoughts on “Bilu-Parent update

  1. As I mentioned to Parent, the “epsilon” in (log N)^{2+epsilon} can be removed: it’s mentioned by Serre in the notes to his Chebotarev paper in his collected works (where he says one can use the ell-adic trick of Faltings), and it’s also well known in the analytic theory of L-functions (where the trick is to use a smooth enough “explicit formula” to get rid of the log log factor which otherwise occurs).

  2. P.S. I forgot to add: unfortunately, removing this epsilon doesn’t help in any serious way for the main result…

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