Friday 19th April 2024: University of Bristol

Jonathan Bober (Bristol)

Title: Murmurations

Abstract: I will survey the phenomenon first discovered by He, Lee, Oliver, and Pozdnyakov (arXiv:2204.10140) in the context of elliptic curves, described as "murmurations". We now see that this type of phenomenon seems to be present in many families of L-functions; there are low-order biases in the coefficients of L-functions which fluctuate regularly as a function of the index of the coefficient divided by the conductor of the L-function. In a few instances these phenomena can be proved.

My own work in this area is joint with Andrew Booker, David Lowry-Duda, and Min Lee, though that might play a relatively small role in this talk.

Ofir Gorodetsky (Oxford)

Title: On an Erdős-Kac-type conjecture of Elliott, and shifted primes

Abstract: In this talk we will review the Erdős-Kac Theorem on the number of prime factors of a random integer. We will explain how this classical result and its generalizations are proved nowadays. We will then explain how, in recent work with Lasse Grimmelt, we proved an analogous result for a certain distribution on shifted primes, thereby solving a conjecture of Elliott from 2014.

Time permitting, we will also review Billingsley's Theorem on the shape of the largest prime factors of a random integer and say what our methods tell about similar questions in Elliott's set-up.

Based on joint work with Lasse Grimmelt.

Jenny Roberts (Bristol)

Title: Newform Eisenstein congruences of local origin

Abstract: The theory of Eisenstein congruences dates back to Ramanujan’s surprising discovery that the Fourier coefficients of the discriminant function are congruent to the 11th power divisor sum modulo 691. This observation can be explained via the congruence of two modular forms of weight 12 and level 1; the discriminant function and the Eisenstein series, E_{12}. Eisenstein congruences were later used by Ribet in his proof of the converse to Herbrand's theorem. 

In this talk, I will first discuss the steps Ribet used in his proof and then compare these steps to my recent work, joint with Dan Fretwell, on congruences between Eisenstein series and newforms of weight k > 2, squarefree level and non-trivial character. 

Netan Dogra (KCL)

Title: On the Zilber--Pink conjecture for a product of curves 

Abstract:  Let X be a curve of genus g>1 over the complex numbers. What is the Zariski closure, inside X^n, of the set of n-tuples of points (z_i) for which there exists a non-constant function f on X with divisor supported on {z_i}? This question can be viewed as a special case of the Zilber--Pink conjecture, which is a broad generalisation of the Andre--Oort conjecture. In this talk I will describe new results which answer this question for some (X,n). This is joint work with Arnab Saha (IIT Gandhinagar).