# 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 Vladimir Dokchitser (UCL) 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