
Thirteenth Algorithmic Number Theory Symposium ANTSXIII
University of Wisconsin, Madison
July 16 – 20, 2018
Thirteenth Algorithmic Number Theory Symposium (ANTSXIII)
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 BombieriLang 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 genusg 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
© 20172018 Jennifer Paulhus (with thanks to Kiran S. Kedlaya, and by extension Pierrick Gaudry and Emmanuel ThomÃ©)