Classical and Modern Approaches to Diophantine Problems

20th July 2022, Department of Mathematics, The University of Manchester, UK

In this one day meeting, we hope to bring together active researchers in the field of explicit resolution to Diophantine equations and related topics. We expect this topic to be interesting to academics and PGR students in Number Theory and related areas.
Three talks will be held, with a post-talk reception and a social dinner in the evening.

Registration: There is no registration fee, but to estimate the number of participants and to register for dinner, please fill out the following survey: link

Location: University Place_5.206, The University of Manchester, UK.
All activities will be held in person.

Speakers: Samir Siksek (Warwick), Lassina Dembele (KCL), Vandita Patel (Manchester).

Queries: Please contact Vandita Patel at vandita.patel@manchester.ac.uk.

Title and Abstracts:

Lassina Dembele: Explicit inertial Langlands correspondence for GL_2 and arithmetic applications

Abstract: In this talk we will describe an algorithm for computing automorphic types for GL_2. Then, we will give several arithmetic applications. (This is joint work with Nuno Freitas and John Voight.)

Vandita Patel: Values of the Ramanujan tau function

Abstract: The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett, Adela Gherga and Samir Siksek.

Samir Siksek: Efficient resolution of Thue–Mahler equations

Abstract: A Thue–Mahler equation has the form F(X,Y)=p_1^{m_1} \cdots p_r^{m_r} where F is an irreducible homogeneous binary form of degree at least 3 with integer coefficients, and p_1,..,p_r are primes. A standard algorithm due to Tzanakis and de Weger solves Thue–Mahler equations when the degree and the number of primes is small. We give lattice-based sieving techniques that are capable of handling large Thue–Mahler equations. This is joint work with Adela Gherga.

This meeting is generously supported by an LMS Scheme 9 Grant and by the Department of Mathematics, Manchester.