Number Theory Seminar

I will show how to find uniform finiteness results for rational
points on curves in terms of the Laplace spectrum of an associated graph.
The method is to bound the gonality of a curve (minimal degree of a map
onto P^1) by the "stable gonality" of an associated stable reduction
graph, and then to bound this stable gonality of the graph (some kind of
minimal degree of a map to a tree) in terms of spectral data. The latter
bound is a graph theoretical analogue of a famous inequality of Li and Yau
in differential geometry. Applications include a lower bound on the
gonality of Drinfeld modular curves that is linear in the genus, and the
proof of a conjecture of Papikian on he degree of the modular
parametrisation of elliptic curves over function fields. We have also
studied stable gonality as graph invariants, including their behaviour for
Ramanujan graphs and in some random graph models. (Joint work with
Fumiharu Kato and Janne Kool.)