# Mathematical Sciences Research Institute

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# Seminar

DDC - Diophantine Problems: On Hilbert's tenth problem in two variables November 30, 2020 (09:00 AM PST - 10:00 AM PST)
Parent Program: Decidability, definability and computability in number theory: Part 1 - Virtual Semester MSRI: Online/Virtual
Speaker(s) Levent Alpoge (Columbia University)
Description

This seminar will focus on Diophantine problems in a broad sense, with a view towards (but not limited to) interactions between Number Theory and Logic. Particular attention will be given to topics with the potential of further developments in the context of this MSRI scientific program. This will provide an opportunity for researchers to update on new results, techniques and some of the main problems of the field.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Video

#### On Hilbert's Tenth Problem In Two Variables

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

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 (i.e. genus > 1) curve, outputs C(K). The algorithm has the property that, if it terminates, the output is unconditionally correct --- one uses the conjectures to show that it always terminates in finite time.

On the other hand, in certain cases (i.e. after imposing conditions on K and C) there is an unconditional finite-time algorithm to compute (K,C)\mapsto C(K), using potential modularity theorems instead. Example: given K totally real of odd degree and a\in K^\times, one can effectively compute C_a(K) where C_a : x^6 + 4y^3 = a^2.

I will focus on the first of these two results but will try to mention at least the ideas that go into the second if time permits.