Thirteenth Algorithmic Number Theory Symposium ANTS-XIII University of Wisconsin, Madison July 16 – 20, 2018 Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII) July 16 – 20, 2018

Curves with many points over number fields

Noam Elkies

Abstract: Fix g ≥ 2, and let C be a curve of genus g over a number field K. Then C(K) is finite (Faltings), but the upper bound on #C(K) is not uniform even for fixed g and K. Caporaso, Harris, and Mazur proved, assuming the Bombieri-Lang conjectures on V(K) for varieties V of general type, that #C(K) is bounded above, and that limsup_C #C(K) is bounded even when K varies; but these upper bounds are ineffective already for g=2 and K=Q. One then naturally seeks examples of genus-g curves, both individually and in infinite families, with many rational points. We review various techniques and records from the past few decades, and report on some recent results for g=2 and g=3 that use the arithmetic and geometry of special K3 surfaces.

Files available: slides