Alex Torzewski

Lawrence-Venkatesh bounds for curves in families

We outline how the method of Lawrence-Venkatesh to bound rational points via p-adic period mappings can be used to study curves in families. This leads to upper bounds on the number of rational points on curves of genus > 1 depending only on the reduction modulo a well chosen prime and the primes of bad reduction. This was first shown by Faltings as a consequence of the Mordell and Shafarevich Conjectures.