London Number Theory Seminar

The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL. Here is our diversity policy.

This term (Summer 2020), the seminar will be hosted by Kings College, virtually. The organiser is James Newton. The talks will be online. The meeting link is https://zoom.us/j/91638038954 and the password was distributed on the mailing lists. A reminder that there are two mailing lists -- london-number-theory-seminar, a low traffic list for people interested in the seminar, and london-number-theorists, for number theorists living in London who might be interested in things like discussions on what topics to run future study groups on and so on.

The talks start on Wed 22nd April and finish on 24th June. An up to date list of titles and abstracts is here and occasionally I copy the contents of that page to the below.

22/4/20 Tiago Jardim Da Fonseca (Oxford)
Title: On Fourier coefficients of Poincaré series
Abstract: Poincaré series are among the first examples of holomorphic and weakly holomorphic modular forms. They are useful in many analytical questions, but their Fourier coefficients seem hard to grasp algebraically. In this talk, I will discuss the arithmetic nature of Fourier coefficients of Poincaré series by characterizing them as cohomological invariants (periods).

29/4/20 Ila Varma (Toronto)
Title: Malle's Conjecture for octic D4-fields.
Abstract: We consider the family of normal octic fields with Galois group D4, ordered by their discriminants. In forthcoming joint work with Arul Shankar, we verify the strong form of Malle's conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove this and related results.

6/5/20 Chris Lazda (Warwick)
Title: A Néron–Ogg–Shafarevich criterion for K3 surfaces
Abstract: The naive analogue of the Néron–Ogg–Shafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semi-stable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if is second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke.

13/5/20 Chantal David (Concordia)
Title: Non-vanishing cubic Dirichlet L-functions at $s = 1/2$
Abstract: Joint work with A. Florea and M. Lalin. A famous conjecture of Chowla predicts that $L(1/2,\chi) \not= 0$ for Dirichlet L-functions attached to primitive characters χ. It was conjectured first in the case where χ is a quadratic character, which is the most studied case. For quadratic Dirichlet L-functions, Soundararajan then proved that at least 87.5% of the quadratic Dirichlet L-functions do not vanish at $s = 1/2$, by computing the first two mollified moments. Under GRH, there are slightly stronger results by Ozlek and Snyder obtained by computing the one-level density.

We consider in this talk cubic Dirichlet L-functions. There are few papers in literature about Dirichlet cubic L-functions, compared to the abundance of papers on Dirichlet qua- dratic L-functions, as this family is more difficult, in part because of the cubic Gauss sums. The first moment for $L(1/2,\chi)$ where $\chi$ is a primitive cubic character was computed by Baier and Young over $\mathbb{Q}$ (the non-Kummer case), by Luo over $\mathbb{Q}(\sqrt{−3})$ (the Kummer case), and by David, Florea and Lalin over function fields, in both the Kummer and non-Kummer case. Bounding the second moment, those authors could obtain lower bounds for the number of non-vanishing cubic twists, but not a positive proportion. Moreover, for the case of Dirichlet cubic L-functions, computing the one-level density under the GRH also gives lower bounds which are weaker than any positive proportion.

We prove in this talk that there is a positive proportion of cubic Dirichlet L-functions non-vanishing at $s = 1/2$ over function fields. This can be achieved by using the recent breakthrough work on sharp upper bounds for moments of Soundararajan and Harper. There is nothing special about function fields in our proof, and our results would transfer over number fields (but we would need to assume GRH in this case).

20/5/20 Rong Zhou (Imperial)
Title: TBA
Abstract: TBA

27/5/20 Matteo Tamiozzo (Imperial)
Title: TBA
Abstract: TBA

03/06/20 Yunqing Tang (Paris-Saclay)
Title: TBA
Abstract: TBA

10/6/20 Vesselin Dimitrov (Toronto)
Title: TBA
Abstract: TBA

17/6/20 TBA
Title: TBA
Abstract: TBA

24/6/20 TBA
Title: TBA
Abstract: TBA

The seminar will be preceded by various study groups.

A list of previous seminar talks is here.

There are two mailing lists for number theory in London: