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Number Theory Seminar

The seminars are held on Mondays from 3pm to 4pm.

Regular seminars have been suspended in light of the ongoing COVID-19 pandemic. We will be running seminars on Zoom during this time.

See [ this link ] for guidance/suggestions for using Zoom.

If you'd like to attend a seminar but don't have the meeting link, please email one of the organisers in advance.

2020-21 Term 2

Organisers: Sam Chow, Chris Lazda and Chris Williams

11th January

Valentijn Karemaker (Utrecht)

Mass formulae for supersingular abelian threefolds

Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field k of characteristic p. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic principally polarised supersingular abelian threefold over k of characteristic >2 has automorphism group Z/2Z.

This is joint work with F. Yobuko and C.-F. Yu.

18th January

Tim Browning (IST Austria)

Rational points on Grassmannians: freeness and equidistribution
The distribution of rational points on Grassmannians is equivalent to the distribution of
lattices in a real vector space. A precise asymptotic formula for this distribution
was worked out by Wolfgang Schmidt in the 60s. Peyre has recently put forward the notion
of "free" rational points as a (potential) means of circumventing certain counter-examples
in the Manin conjecture for Fano varieties. We will discuss this in the context of
Grassmannians and show how it can be recast as an equidistribution problem about
certain tensor product lattices. This is joint work with Tal Horesh and Florian Wilsch.

25th January

Alex Bartel (Glasgow)

Statistics of Mordell-Weil groups as Galois modules

Let G be a finite group, let E/Q be an elliptic curve, and fix a finite-dimensional Q[G]-module V. Let F/Q run over all Galois extensions whose Galois group is isomorphic to G (together with a fixed such isomorphism for each F) and such that E(F) tensor Q is isomorphic to V as a G-module. Then what does E(F) itself look like "on average" in this family? I will report on joint work with Adam Morgan, in which we consider a particular special case of this general question. We propose a heuristics that predicts a precise answer in that case, and make some progress towards proving it. Our heuristics turns out to be an elliptic curve analogue of Stevenhagen's heuristic on the solubility of negative Pell equations.

1st February

Tom Bloom (Cambridge)

8th Feburary

Jan Vonk (Leiden)

15th February

Katharina Hubner (Heidelberg)

22nd February

Sally Gilles (Imperial)

1st March

8th March

Luis Garcia (UCL)

15th March

Kirsti Biggs (Gothenburg)

2020-21 Term 1

Organisers: Sam Chow, Chris Lazda and Chris Williams

5th October

Dan Fretwell (Bristol)

(Real Quadratic) Arthurian Tales

In recent years, there has been a lot of interest in explicitly identifying the global Arthur parameters attached to certain automorphic forms. In particular, Chenevier and Lannes were able to completely identify and prove the full lists of Arthur parameters in the case of level 1, trivial weight automorphic forms for defintiely orthogonal groups of ranks 8,16 and 24 (not a simple task!).

One finds interesting modular forms hidden in these parameters (e.g. Delta and a handful of special Siegel modular forms of genus 2). Comparing Arthur parameters mod 0 proves/reproves various Eisenstein congruences for these special modular forms, e.g. the famous 691 congruence of Ramanujan and, more importantly, an example of a genus 2 Eisenstein congruence predicted by Harder (which, up to then, had not been proved for even a modular form!).

In this talk I will discuss recent work with Neil Dummigan on extending the above to definite orthogonal groups over certain real quadratic fields and try to tell the analogous Arthurian tales (mysteries included).

12th October

Valeriya Kovaleva (Oxford)

On the distribution of equivalence classes of p-adic quadratic forms

Some questions about quadratic forms can be reduced to a question about their canonical form, or equivalence class. In the statistical sense this means that one may use the distribution of equivalence classes to compute the proportion of quadratic forms with a certain property. In this talk we will show how to derive the probability that a random quadratic form over p-adics lies in an equivalence class, and give examples of applications.

19th October

Simon Myerson (Warwick)

Sifting rational points on elliptic curves

This is work in progress with Katharina Müller and Subham Bhakta. We discuss the problem of counting rational points on elliptic curves with bounded height and co-ordinates which are restricted in some way. We relate this to work of Loughran and Smeets on counting the varieties in a family which have a rational point.

26th October

Francesca Balestrieri (American University of Paris)

Strong approximation for homogeneous spaces of linear algebraic groups

Building on work by Yang Cao, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over a number field k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some natural compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent. (This latter result is somewhat related to the Inverse Galois Problem.)

2nd November

Damian Rossler (Oxford)

A generalization of Beilinson's geometric height pairing

In the first section of his seminal paper on height pairings, Beilinson constructed an ℓ-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the function field of a curve over an algebraically closed field, and asked about an generalization to higher dimensional bases. In this paper we answer Beilinson's question by constructing a pairing for varieties defined over the function field of a smooth variety B over an algebraically closed field, with values in the second ℓ-adic cohomology group of B. Over C our pairing is in fact Q-valued, and in general we speculate about its geometric origin. This is joint work with Tamás Szamuely.

 

9th November

Adela Gherga (Warwick)

Implementing Algorithms to Compute Elliptic Curves Over Q

Let S be a set of rational primes and consider the set of all elliptic curves over Q having good reduction outside S and bounded conductor N. Currently, using modular forms, all such curves have been determined for N less than 500000, the bulk of this work being attributed to Cremona.

Early attempts to tabulate all such curves often relied on reducing the problem to one of solving a number of certain integral binary forms called Thue-Mahler equations. These are Diophantine equations of the form F(x,y) = u, where F is a given binary form of degree at least 3 and u is an S-unit.

A theorem of Bennett-Rechnitzer shows that the problem of computing all elliptic curves over Q of conductor N reduces to solving a number of Thue-Mahler equations. To resolve all such equations, there exists a practical method of Tzanakis-de Weger using bounds for linear forms in p-adic logarithms and various reduction techniques. In this talk, we describe our refined implementation of this method and discuss the key steps used in our algorithm.

16th November

Peter Varju (Cambridge)

The mixing time of the ax+b Markov chain

Chung, Diaconis and Graham studied the Markov chain on Z/qZ with transitions x -> 2x+B_n, where B_n is an independent sequence of random variables uniformly distributed in {-1,0,1}. They showed that the chain is approximately uniformly distributed after c_1 log_2(q) steps for almost all q, where c_1 is a constant slightly larger than 1. They asked whether it is possible to reduce the value of c_1 to 1. This was shown not to be possible by Hildebrand who showed that the chain is far from uniformly distributed after c_2 log_2(q) steps for any q, where c_2 is some constant with 1<c_2<c_1. I will talk about a joint work with Sean Eberhard, in which we determine the best possible value for the constant. This result is based on the large sieve inequality. If 2 in the definition of the chain is replaced by 10, then this argument does not work any longer, and we deduce a weaker result using Gallagher's larger sieve.

23rd November

Andreas Bode (Lyon)

Bornological D-modules on rigid analytic spaces

Ardakov-Wadsley introduced p-adic D-cap-modules on rigid analytic spaces in order to study p-adic representations geometrically, in analogy to the theory of Beilinson-Bernstein localization over the complex numbers. In this talk, we report on an ongoing project to extend their framework to the (derived) category of complete bornological D-cap-modules, which allows us to define analogues of the usual six operations. We then consider a subcategory playing the role of D^b_coh(D) and prove a number of stability results.

30th November

Kazim Buyukboduk (Dublin)

Perrin-Riou style critical p-adic L-functions

I will report on joint work with Denis Benois, where we gave a Perrin-Riou-style construction of Bellaïche's p-adic L-function (as well as its improvements) at a $\theta$-critical point on the eigencurve. Besides the interpolation of the Beilinson-Kato elements about this point, the key input to prove the interpolative properties is a new "eigenspace-transition via differentiation" principle.

7th December

Jessica Fintzen (Cambridge)

Representations of p-adic groups and applications

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress.
I will also outline how new results about the representation theory of p-adic groups can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p, which is joint work with Sug Woo Shin. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.