Number Theory
Workshop

All talks in Room G.209 - Alan Turing Building

15:30 Rebecca Bellovin
University of Glasgow
The eigencurve at the boundary
Roughly speaking, the eigencurve is a moduli space of modular forms.  It carries a natural family of Galois representations.  I will discuss the behavior of these Galois representations at the boundary of the eigencurve.

16:10 Lasse Grimmelt
University of Oxford
Twisted correlations of the divisor function via discrete averages of Poincare series
The spectral theory of automorphic forms finds remarkable applications in analytic number theory. Notably, it is utilised in results concerning the distribution of primes in large arithmetic progressions and in questions on variants of the fourth moment of the zeta function. Traditionally, these problems are addressed by reducing them to sums of Kloosterman sums, followed by either the use of existing black-box results or by-hand application of spectral theory through Kuznetsov's formula.

In this presentation, based on joint work with Jori Merikoski, I will introduce an alternative approach that entirely circumvents the need for Kloosterman sums. This approach offers increased flexibility compared to existing black-box methods. It is inspired by Bruggeman-Motohashi's work on the fourth moment of the zeta function. As an application, I will present a novel result on correlations of the divisor function in arithmetic progressions and moments of Dirichlet L-functions.

16:50 Alexei Skorobogatov
Imperial College London
Hasse principle for intersections of two quadrics via Kummer surfaces
Smooth intersections of two quadrics in the projective space of dimension at least 5 over a number field are expected to satisfy the Hasse principle. This was proved by Wittenberg in his thesis, conditionally on the finiteness of the Tate-Shafarevich groups of elliptic curves and Schinzel's Hypothesis. In a joint work with Adam Morgan, we remove the dependence on Schinzel's Hypothesis, while assuming the finiteness of the Tate-Shafarevich groups of Jacobians of genus 2 curves. The proof proceeds by proving the Hasse principle for certain Kummer surfaces attached to such Jacobians and deducing the Hasse principle for generic smooth intersections of two quadrics in the projective space of dimension 4, which is known to imply the Hasse principle for intersections of quadrics in higher dimension.

11:30 Steve Lester
King's College London
The distribution of the Riemann zeta-function at its relative maxima
On the critical line, the modulus of Riemann zeta-function has exactly one relative maximum between consecutive zeros assuming the Riemann hypothesis. In this talk I will discuss the distribution of values of the Riemann zeta-function at these relative maxima and give an application to counting the number of solutions  T < t ≤ 2T to the equation |ζ(½+it)| = a where a > 0 is a real number. This is joint work with Micah Milinovich.

13:45 Vaidehee Thatte
King's College London
Generalized ramification theory and its applications
Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. In the classical setup, the degree of a finite Galois extension of "nice" fields splits up neatly into the product of two well-understood numbers (ramification index and inertia degree) that encode how the base field changes. In the general case, however, a third factor called the defect (or ramification deficiency) can pop up. The defect is a mysterious phenomenon and the main obstruction to several long-standing open problems, such as obtaining resolution of singularities. The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many adjustments" fails when the defect is non-trivial. I will discuss my previous and ongoing work in arithmetic algebraic geometry, number theory, and valuation theory that allows us to understand and treat the defect.

14:25 Min Lee
University of Bristol
Murmurations of automorphic forms in archimedean families
In April 2022, He, Lee, Oliver and Pozdnyakov made an interesting discovery using machine learning: a surprising correlation between the root numbers of elliptic curves and the coefficients of their L-functions. They coined this correlation 'murmurations of elliptic curves'. Naturally, one might wonder whether we can identify a common thread of 'murmurations' in other families of L-functions. In this talk, I will introduce joint works with Jonathan Bober, Andrew R. Booker, David Lowry-Duda, Andrei Seymour-Howell and Nina Zubrilina, demonstrating murmurations in holomorphic modular forms and Maass forms in archimedean families.

11:30 Robin Visser
University of Warwick
Abelian surfaces with good reduction away from 2

11:45 Rafail Psyroukis
Durham University
A Dirichlet Series attached to orthogonal modular forms

12:00 Sam Streeter
University of Bristol
Whose Line Is It Anyway?

12:15 David Angdinata
London School of Geometry and Number Theory
Twisted L-values of elliptic curves

14:00 Elyes Boughattas
University of Bath
Arithmetic of conic bundles

14:15 Aled Williams
London School of Economics
Considering a Classical Upper Bound on the Frobenius Number

14:30 Thomas Karam
University of Oxford
Fourier analysis modulo p on the Boolean cube

14:45 Pedro Cazorla Garcia
University of Manchester
Perfect codes and Diophantine equations