Title: 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/Q is totally real and of odd degree) there is an unconditional finite-time algorithm to compute (K,C)→ 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 Ca : x6 + 4y3 = a2 (i.e. a in Kx fixed) when K/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.)