Week 5: Monday 12 – Friday 16 May

- Monday 12 May,
*Seminar*

Time and Room: 12 – 1, MS.04

Speaker:**David Loeffler**(Imperial)

Title: Computing automorphic forms for definite unitary groups

Abstract: "The problem of computing modular forms, which are the automorphic forms for the group GL(2), has been much studied, and there are now efficient algorithms for calculating the full space of modular forms of a given weight and level. But for automorphic forms on more general groups less is known. I shall outline the definition of automorphic forms on general reductive groups, and describe a particular class of groups (those that are compact at infinity) for which explicit calculations are possible. Then I will present an algorithm and some examples of the results for one such group, the definite unitary group U(3)."

**Tuesday 13 May – Wednesday 14 May,***Short Course*

Speaker:**Bill Hart**(Warwick)

Title: Introduction to Theta Functions

Times and Rooms: Tuesday 10 – 11, MS.04

Tuesday 12 – 1, MS.04

Wednesday 10 – 11, B3.03

Wednesday 12 – 1, B1.01

Abstract: Historically theta functions were defined as holomorphic functions with quasi-periodicity and were important in the study of the Jacobi elliptic functions. Riemann related the Jacobi theta function with zeta, establishing a functional equation for the latter. Hecke synthesized this approach in associating an integer weight modular form with an $L$-series. However, unlike the integral weight modular forms studied by Hecke, theta functions are modular forms of half integer weight. Weil promulgated the idea that the classical theta function of Jacobi should be viewed as an automorphic form on a 2-fold cover of the complex upper half plane. Shimura later clarified the link with $L$-series. Theta functions have been generalised to higher dimensions and their theory is important to the study of partitions, $L$-series, moduli spaces, abelian varieties, quadratic forms, moonshine, string theory, modular equations and a panoply of other problem areas. We will provide an all too brief account of some of the classical theory and modern interpretation and applications of theta functions.

**Thursday 15 May,***Seminar*

Time and Room: 12 – 1, MS.04

Speaker:**Thotsaphon Thongjunthug**(Warwick)

Title: Computing a lower bound for the canonical height on ellliptic curves over number fields

Abstract: Computing a lower bound for the canonical height is one of crucial steps in determining a Mordell--Weil basis for an elliptic curve. In this seminar I will explain my recent work on extending an algorithm of Cremona and Siksek (2006) to compute such lower bound for any elliptic curves defined over more general number fields $K$. The talk will mainly focus on the case when $K$ is a totally real number field, where the algorithm is now successfully extended, together with some examples. Some current progress on the case when $K$ has a complex embedding will be also discussed.