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).

## Bilu-Parent update

**Tagged**algebraic geometry, Bilu, elliptic curves, Galois representations, modular curves, number theory, Parent, serre