ERLASS VI

Program

• 10:00-10:30, Tea and coffee

• 10:30-11:20, Kyle Pratt (Oxford)

Title: Power savings for counting solutions to polynomial-factorial equations

Abstract: Let $P$ be a polynomial with integer coefficients and degree at least two. I recently studied, joint with Hung Bui and Alexandru Zaharescu, the number of integer solutions to the polynomial-factorial equation $n! = P(x)$. The ABC conjecture implies there are only finitely many solutions to this equation, but we are interested in unconditional results. There are trivially $\leq N$ solutions with $n\leq N$, and we proved there are actually $\leq c(P) N^{33/34}$ solutions, where $c(P)$ is a positive constant depending only on $P$. The previous best result was that the number of solutions is $o(N)$. I will discuss some history of the problem and discuss some of the techniques (Diophantine and Pad\'e approximation) used in the proof.

• 11:30-12:20, Félicien Comtat (Queen Mary)

Title: The Kuznetsov formula for GSp(4)

Abstract: In this talk, I will present my work on the Kuznetsov formula for GSp(4). The latter relates Whittaker coefficients of Maass forms on GSp(4) for a certain congruence subgroup to sums of generalised Kloosterman sums. In the first part of the talk, I will give an overview of how the Kuznetsov formula for GSp(4) can be proved by integrating a pre-trace formula against a character of the unipotent subgroup. I will then present an application to equidistribution of Satake parameters of GSp(4) Maass forms with respect to the Sato-Tate measure as the level tends to infinity, providing some evidence towards the Generalised Ramanujan Conjecture in this setting.

• 14:30-15:20, Sadiah Zahoor (Sheffield)

Title: Congruences between modular forms on integer weight and half-integer weight

Abstract: The theory of half-integral weight modular forms may be adopted to prove 'congruences' between Selmer type groups. William McGraw and Ken Ono used this approach by passing from 'congruences' between modular forms of integer weight to congruences between modular forms of half-integer weight. For example, recall the famous 'congruence modulo 11' between the normalised Discriminant function ‘Delta’ of weight 12 and the newform f of weight 2 attached to Elliptic curve of conductor 11. Using Shimura's correspondence and Kohnen’s isomorphism, which connects modular forms of weight 2k for a positive integer k with half-integer modular forms of weight k+(1/2), our congruence descends to a congruence modulo 11 between half integer modular forms of weight 3/2 and 13/2.

The talk shall begin with a brief introduction to modular forms of integer and half-integer weight leading to statement and overview of the main result I have been

working on. I will also give an overview of current progress and generalisation of Theorem of McGraw and Ono to Hilbert modular forms of integer and half-integer weight.

• 15:30-16:00, Tea and coffee

• 16:00-16:50, Sam Chow (Warwick)

Title: Galois groups of random polynomials