[Colloquium] Abhijit Mudigonda MS Presentation/May 6, 2024

[Megan Woodward]

This is an announcement of Abhijit Mudigonda's MS Presentation
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Candidate: Abhijit Mudigonda

Date: Monday, May 06, 2024

Time: 10:30 am CT

Location: JCL 257

Title: Hilbert Modular Forms

Abstract: The absolute Galois group of the rational numbers - the group of "reasonable" automorphisms of the algebraic numbers - is the central object of study in algebraic number theory. This group can be accessed via its representations, called Galois representations. Many important objects coming from geometry, like elliptic curves, have Galois representations which capture important properties of the object. As such, a productive method for studying objects from geometry is to look at their Galois representations. Conjecturally, these Galois representations also can be obtained from highly symmetric complex analytic functions on the upper half plane known as modular forms. For example, the core of Wiles' proof of Fermat's last theorem was proving that for every elliptic curve defined over the rationals there is a modular form which has the same Galois representation. One major advantage with working with modular forms instead of geometric objects is that modular forms are much more amenable to explicit computation, and today there are many computational packages for constructing and manipulating modular forms. However, the computational infrastructure for working with Hilbert modular forms - a generalization of modular forms to totally real fields (instead of the rationals) - is substantially more limited. In particular, "partial weight 1" and "nonparitious" Hilbert modular forms are relatively poorly understood both computationally and theoretically, as their Galois representations do not (directly) come from geometry. In this presentation, I'll discuss ongoing work to develop an open-source computational package for constructing and manipulating Hilbert modular forms as well as the applications of this package to constructing interesting geometric objects. Time permitting, I'll also discuss potential applications to the ECM factorization algorithm.


Advisors: Francesco Calegari and Janos Simon

Committee Members: Francesco Calegari, Janos Simon, and Alexander Razborov










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