Number Theory Seminar
The seminars are held on Mondays from 3pm to 4pm (or on Tuesday if Monday is a bank holiday).
We are running a hybrid seminar series in 202122. Speakers have been given the choice of inperson or remote talks. Remote talks will be held on Microsoft Teams.
Inperson seminars at held in room B3.02 of the Zeeman building. If you'd like to attend a seminar but don't have the meeting link, please email one of the organisers in advance.
202122 Term 2
Organisers: Sam Chow, PakHin Lee and Chris Williams
10th January 
Jaclyn Lang (Temple) A modular construction of unramified pextensions of Q(N^{1/p})
In his 1976 proof of the converse of Herbrand’s theorem, Ribet used Eisensteincuspidal congruences to produce unramified degreep extensions of the pth cyclotomic field when p is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degreep extensions of Q(N^{1/p}) when N is a prime that is congruent to 1 mod p. This answers a question posed on Frank Calegari’s blog.

In person 
17th January 
Daniele Mastrostefano (Warwick) The variance of generalized divisor functions and other sequences in arithmetic progressions In this talk we will overview old and new results on the problem of lower bounding 
Teams 
24th January 
Ayesha Hussain (Bristol) The Distribution of Character Sums Over the past few decades, there has been a lot of interest in partial sums of Dirichlet characters. Montgomery and Vaughan showed that these character sums remain a constant size on average and, as a result, a lot of work has been done on the distribution of the maximum. In this talk, we will investigate the distribution of these character sums themselves, with the main goal being to describe the limiting distribution as the prime modulus approaches infinity. This is motivated by Kowalski and Sawin’s work on Kloosterman paths. 
In person 
31st January 
ChungHang Kwan (Columbia)  Teams 
7th February  Marco D'Addezio (Jussieu)  Teams 
14th February  Chris Birkbeck (UCL)  In person 
21st February 
Han Yu (Cambridge)  In person 
28th February 
Celine Maistret (Bristol)  TBC 
7th March 
V. Vinay Kumaraswamy (TATA Institute) *This talk is not hybrid, but is online only: our usual room (B3.02) is unavailable. 
Fully online* 
14th March 
Alice Pozzi (Imperial) 
In person (TBC)

202122 Term 1
Organisers: Sam Chow, Chris Lazda and Chris Williams
4th October 
Rob Rockwood (Warwick) BlochKato for finite slope Siegel modular forms I’ll discuss extending results of LoefflerZerbes on the analytic rank 0 BlochKato conjecture for GSp4 and GSp(4) x GL(2) to include the small slope case. 
In person 
11th October 
Lennart Gehrmann (DuisburgEssen) Plectic StarkHeegner points Heegner points play an important role in our understanding of the arithmetic of modular elliptic curves. These points, that arise from CM points on Shimura curves, control the MordellWeil group of elliptic curves of rank 1. The work of Bertolini, Darmon and their schools has shown that padic methods can be successfully employed to generalize the definition of Heegner points to quadratic extensions that are not necessarily CM. Numerical evidence strongly supports the belief that these socalled StarkHeegner points completely control the MordellWeil group of elliptic curves of rank 1. In this talk I will report on a plectic generalization of StarkHeegner points. Inspired by Nekovar and Scholl's conjectures, these points are expected to control MordellWeil groups of higher rank elliptic curves. If time permits I sketch a proof that higher order derivatives of This is joint work with Michele Fornea. 
Teams 
18th October 
Kaisa Matomaki (Turku) Almost primes in almost all very short intervals By probabilistic models one expects that, as soon as $h \to \infty$ with $X \to \infty$, short intervals of the type $(x h \log X, x]$ contain primes for almost all $x \in (X/2, X]$. However, this is far from being established. In the talk I discuss related questions and in particular describe how to prove the above claim when one is satisfied with finding $P_2$numbers (numbers that have at most two prime factors) instead of primes. 
Teams 
25th October 
Ben Green (Oxford) Quadratic forms in 8 prime variables I will discuss a recent paper of mine, the aim of which is to count the number of prime solutions to Q(p_1,..,p_8) = N, for a fixed quadratic form Q and varying N. The traditional approach to problems of this type, the HardyLittlewood circle method, does not quite suffice. The main new idea is to involve the Weil representation of the symplectic groups Sp_8(Z/qZ). I will explain what this is, and what it has to do with the original problem. I hope to make the talk accessible to a fairly general audience. 
In person only, 4pm, Humanities H0.56 
1st November 
Netan Dogra (KCL) $p$adic integrals and correlated points on families of curves In this talk, we explain how studying the common zeroes of Coleman integrals in families has applications to studying the LangVojta conjecture, unit equations in families, the FreyMazur conjecture and questions in the ChabautyColeman method. 
Teams 
8th November 
Eva Viehmann (TU Munich) HarderNarasimhanstrata in the $B_\mathrm{dR}^+$Grassmannian We establish a HarderNarasimhan formalism for modifications of $G$bundles on the FarguesFontaine curve. The semistable stratum of the associated stratification of the $B_{\mathrm{dR}}^+$Grassmannian coincides with the weakly admissible locus. When restricted to minuscule affine Schubert cells, it corresponds to the HarderNarasimhan stratification of Dat, Orlik and Rapoport. I will also explain the relation to the Newton stratification as well as some geometric properties of the strata. This is joint work with K.H. Nguyen. 
Teams 
15th November 
Vaidehee Thatte (KCL) Arbitrary Valuation Rings and Wild Ramification Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow nondiscrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a nontrivial defect is one of the main obstacles to longstanding problems, such as obtaining resolution of singularities in positive characteristic. Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups. 
In person 
22nd November 
Victor Beresnevich (York) Rational points near manifolds, Khintchine's theorem and Diophantine exponents I will talk about recent progress in estimating the number of rational points lying at a small distance from a given nondegenerate submanifold of $\mathbb{R}^n$ and the implications it has for problems in Diophantine approximation, in particular, for establishing Khintchine's theorem for manifolds and certain Diophantine exponents. This is a joint work with Lei Yang. 
Teams 
29th November 
Oli Gregory (Exeter) Logmotivic cohomology and a deformational semistable $p$adic Hodge conjecture 
In person 
6th December 
Rong Zhou (Cambridge)
Components in the basic locus of Shimura varieties
The basic locus of Shimura varieties is the generalization of the supersingular locus in the modular curve and provides us with an interesting class of cycles in the special fiber of Shimura varieties. In this talk, we give a description of the set of irreducible components in the basic locus of Hodge type Shimura varieties in terms of class sets for an inner form of the structure group, generalizing a classical result of Deuring and Serre. A key input for our approach is an analysis of certain twisted orbital integrals using techniques from local harmonic analysis in order to understand the geometry of affine DeligneLusztig varieties. The result for the basic locus is then deduced from this using the RapoportZink uniformization. This is joint work with X. He and Y. Zhu. 
In person
