Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

 

Past events in this series


Thu, 12 Jun 2025
16:00
Lecture Room 4

The exceptional zero conjecture for GL(3)

Chris Williams
(University of Nottingham)
Abstract

The BSD conjecture predicts that a rational elliptic curve $E$ has infinitely many points if and only if its $L$-function vanishes at $s=1$.

There are $p$-adic versions of similar phenomena. If $E$ is $p$-ordinary, there is, for example, a $p$-adic analytic analogue $L_p(E,s)$ of the $L$-function, and if $E$ has good reduction, then it has infinitely many rational points iff $L_p(E,1) = 0$. However if $E$ has split multiplicative reduction at $p$ - that is, if $E/\mathbf{Q}_p$ admits a Tate uniformisation $\mathbf{C}_p^{\times}/q^{\mathbf{Z}}$ - then $L_p(E,1) = 0$ for trivial reasons, regardless of $L(E,1)$; it has an 'exceptional zero'. Mazur--Tate--Teitelbaum's exceptional zero conjecture, proved by Greenberg--Stevens in '93, states that in this case the first derivative $L_p'(E,1)$ is much more interesting: it satisfies $L_p'(E,1) = \mathrm{log}(q)/\mathrm{ord}(q) \times L(E,1)/(\mathrm{period})$. In particular, it should vanish iff $L(E,1) = 0$ iff $E(\mathbf{Q})$ is infinite; and even better, it has a beautiful and surprising connection to the Tate period $q$, via the 'L-invariant' $\mathrm{log}(q)/\mathrm{ord}(q)$.

In this talk I will discuss exceptional zero phenomena and L-invariants, and a generalisation of the exceptional zero conjecture to automorphic representations of GL(3). This is joint work in progress with Daniel Barrera and Andrew Graham.

Thu, 19 Jun 2025
16:00
Lecture Room 4

TBA

Hanneke Wiersema
(University of Cambridge)