Number theory seminar in memory of F. Momose
Rational points of $X_0^+ (p^r )$ for all primes.
Presenters
Pierre Parent
Abstract
In a series of papers (Compositio Math. 1984, J. Fac. Sci. Univ. Tokyo 1986, Nagoya Math. J. 2002), and motivated by some expected consequences on the arithmetic of elliptic curves, F. Momose tackled the question of rational points on the quotient modular curves $X_0^+ (p^r )$, for $r > 1$ and $p$ a prime number. In joint works with Yu. Bilu we proved recently, by adding analytic tools to Momose’s ideas, that, as desired, those sets of rational points are only made of CM points, at least for $p$ larger than some very large bound. Finally, the case of (not so) small primes was settled by the same authors and M. Rebolledo, by using still another algorithmic approach, thereby completing a positive answer to Momose’s question (to the single exception of $p^r = 132$). We will try to sketch those proofs in our talk.
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