# 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 2021-22. Speakers have been given the choice of in-person or remote talks. Remote talks will be held on Microsoft Teams.

In-person 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.

### 2021-22 Term 2

Organisers: Sam Chow, Pak-Hin Lee and Chris Williams

 10th January Jaclyn Lang (Temple) A modular construction of unramified p-extensions of Q(N1/p) In his 1976 proof of the converse of Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-p extensions of the p-th 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 degree-p 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 boundingthe variance of some arithmetic sequences in arithmetic progressions. The main distinctionis between “deterministic” sequences (e.g. the constant function 1) and “pseudorandom” ones (like the prime numbers) or between additive type sequences (such as the prime factors counting functions) and multiplicative type ones (e.g. the smooth numbers). In particular, we will state the results we obtained, building on work of Harper and Soundararajan, about the variance of α-fold divisor functions, for complex parameters α, explaining why they sort of interpolate between the aforementioned functions as α approaches 1. 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 Chung-Hang 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)

### 2021-22 Term 1

Organisers: Sam Chow, Chris Lazda and Chris Williams

 4th October Rob Rockwood (Warwick) Bloch-Kato for finite slope Siegel modular forms I’ll discuss extending results of Loeffler-Zerbes on the analytic rank 0 Bloch-Kato conjecture for GSp4 and GSp(4) x GL(2) to include the small slope case. In person 11th October Lennart Gehrmann (Duisburg-Essen) Plectic Stark-Heegner 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 Mordell-Weil group of elliptic curves of rank 1. The work of Bertolini, Darmon and their schools has shown that p-adic 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 so-called Stark-Heegner points completely control the Mordell-Weil group of elliptic curves of rank 1. In this talk I will report on a plectic generalization of Stark-Heegner points. Inspired by Nekovar and Scholl's conjectures, these points are expected to control Mordell-Weil groups of higher rank elliptic curves. If time permits I sketch a proof that higher order derivatives ofanticyclotomic p-adic L-functions are computed by plectic Stark-Heegner points. 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 Hardy-Littlewood 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 Lang--Vojta conjecture, unit equations in families, the Frey--Mazur conjecture and questions in the Chabauty--Coleman method. Teams 8th November Eva Viehmann (TU Munich) Harder-Narasimhan-strata in the $B_\mathrm{dR}^+$-Grassmannian We establish a Harder-Narasimhan formalism for modifications of $G$-bundles on the Fargues-Fontaine curve. The semi-stable 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 Harder-Narasimhan 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 non-discrete 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 non-trivial defect is one of the main obstacles to long-standing 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 non-degenerate 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) Log-motivic cohomology and a deformational semistable $p$-adic Hodge conjectureLet $k$ be a perfect field of characteristic $p>0$ and let $X$ be a proper scheme over $W(k)$ with semistable reduction. I shall define a logarithmic version of motivic cohomology for the special fibre $X_k$, and relate it to logarithmic Milnor $K$-theory and logarithmic Hyodo-Kato Hodge-Witt cohomology. The bidegree $(2r,r)$ log-motivic cohomology group can be seen as a log-Chow group $\mathrm{CH}^r_{\log}(X_k)$; when $r=1$ we recover the log-Picard group $\mathrm{Pic}^{\log}(X_k)$. Then, by gluing a logarithmic variant of the Suslin-Voevodsky motivic complex to a log-syntomic complex along the logarithmic Hyodo-Kato Hodge-Witt sheaf, I will prove that an element of the $\mathrm{CH}^r_{\log}(X_k)$ formally lifts to the continuous log-Chow group of $X$ if and only if it is “Hodge” (i.e. its log-crystalline Chern class lands in the $r$-th step of the Hodge filtration of the generic fibre of X under the Hyodo-Kato isomorphism). This simultaneously generalises the semistable $p$-adic Lefschetz $(1,1)$ theorem of Yamashita (which is the case $r=1$), and the deformational p-adic Hodge conjecture of Bloch-Esnault-Kerz (which is the case of good reduction). This is joint work with Andreas Langer. 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 Deligne-Lusztig varieties. The result for the basic locus is then deduced from this using the Rapoport-Zink uniformization. This is joint work with X. He and Y. Zhu. In person