- Monday 12 May, Seminar
Time and Room: 12 – 1, MS.04
Speaker: David Loeffler (Imperial)
Title: Computing automorphic forms for definite unitary
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,
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
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
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
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,
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
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.
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