Rational points on X_0^+ (p^r)

By Yuri F. Bilu, Pierre Parent, M. Rebolledo. 2011

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We show how the recent isogeny bounds due to E. Gaudron and G. Remond allow to obtain the triviality of X_0^+ (p^r)(Q), for r>1 and p a prime exceeding 2.10^{11}. This includes the case of the curves X_split (p). We then prove, with the help of computer calculations, that the same holds true for p in the range 10 < p < 10^{14}, p\neq 13. The combination of those results completes the qualitative study of such sets of rational points undertook in previous papers, with the exception of p=13.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f2fc76b25f9aacbd531fd99ab36222d21f33f2c