Likely and unlikely ramification of algebraic points on curves

Abstract

This paper initiates the study of arithmetic properties of closed points of arbitrary degree on curves. We show that a pointless curve of rank $0$ can have only finitely many cyclic degree $\ell$ points for all sufficiently large primes $\ell$. More generally, we show that the residue fields of points on curves of genus at least $2$ and rank $0$ are often biased away from total ramification. The proof technique illuminates a natural intermediate conjecture between the Manin--Mumford conjecture and the Zilber--Pink conjecture for self-products of abelian varieties. This intermediate conjecture implies analogous results for other splitting profiles.

arXiv comment35 pages