Number Theory Seminar
The seminars are held on Mondays from 3pm to 4pm.
Regular seminars have been suspended in light of the ongoing COVID19 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.
202021 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 padic 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 padics 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 coordinates 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 étaleBrauer 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 ℓadic 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 ℓadic cohomology group of B. Over C our pairing is in fact Qvalued, 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 Let S be a set of rational primes and consider the set of all elliptic curves over 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 ThueMahler 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 Sunit. A theorem of BennettRechnitzer shows that the problem of computing all elliptic curves over of conductor N reduces to solving a number of ThueMahler equations. To resolve all such equations, there exists a practical method of Tzanakisde Weger using bounds for linear forms in padic 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 Dmodules on rigid analytic spaces ArdakovWadsley introduced padic Dcapmodules on rigid analytic spaces in order to study padic representations geometrically, in analogy to the theory of BeilinsonBernstein 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 Dcapmodules, 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 Buyukbudok (Dublin) PerrinRiou style critical padic Lfunctions I will report on joint work with Denis Benois, where we gave a PerrinRioustyle construction of Bellaïche's padic Lfunction (as well as its improvements) at a $\theta$critical point on the eigencurve. Besides the interpolation of the BeilinsonKato elements about this point, the key input to prove the interpolative properties is a new "eigenspacetransition via differentiation" principle. 
7th December 
Jessica Fintzen (Cambridge) 