Effectivity in Faltings' Theorem.

Abstract: In joint work with Brian Lawrence we show that, assuming standard motivic conjectures (Fontaine-Mazur, Grothendieck-Serre, Hodge, Tate), there is a finite-time algorithm that, on input $(K,C)$ with $K$ a number field and $C/K$ a smooth projective hyperbolic curve, outputs $C(K)$. On the other hand, in certain cases (i.e. after restricting the inputs $(K,C)$ --- e.g. so that $K/\mathbb Q$ is totally real and of odd degree) there is an unconditional finite-time algorithm to compute $(K,C)\mapsto C(K)$, using potential modularity theorems. I will discuss these two results, focusing in the latter case on how to unconditionally compute the $K$-rational points on the curves $C_a : x^6 + 4y^3 = a^2$ (i.e. $a\in K^\times$ fixed) when $K/\mathbb Q$ is totally real of odd degree.

(The talk will cover Chapters 7, 9, and 11 of my thesis, available on e.g. my website.)

algebraic geometrynumber theory

Audience: researchers in the topic

MAGIC (Michigan - Arithmetic Geometry Initiative - Columbia)

Series comments: Description: Research seminar in arithmetic geometry

(Zoom password = order of the alternating group on six letters)