# 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 (MS.03 if Tuesday). 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 3

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

 25th April Akshat Mudgal (Oxford) On sums of kth powers of arbitrary sets of integers An important problem in analytic number theory is Waring's problem, which concerns the number of ways in which a given natural number can be represented as a sum of $s$ k^th powers, for some fixed natural numbers s,k. Most modern approaches to this problem often end up analysing solutions to equations of the form $x_1^k + ... + x_s^k = x_{s+1}^k + \cdots + x_{2s}^k$, with the variables $x_1, ..., x_{2s}$ lying in some interval ${1,2, ..., N}$. In this talk, we study a more general setting, and so, we consider additive equations of the shape $f(x_1) + ... + f(x_s) = f(x_{s+1}) + ... + f(x_{2s})$, where f is a convex function with suitable properties and $x_1, ..., x_{2s}$ lie in some arbitrary finite set of integers. We will survey some recent results on this type of problem as well as highlight the various connections between this question and some other well-known conjectures in additive combinatorics. In person TUESDAY 3rd May, MS.03 broadcast Alexei Skorobogatov (Imperial) Reduction of Kummer surfaces modulo 2 If a Kummer surface has good reduction, then it comes from an abelian surface with good reduction. The converse holds if the residual characteristic is not 2. In a joint work with Chris Lazda we obtain a necessary and sufficient condition for good reduction of Kummer surfaces attached to abelian surfaces with good, non-supersingular reduction in residual characteristic 2. When these conditions are satisfied we construct an explicit smooth model. Teams 9th May James Newton (Oxford) Modularity over CM fields Since the seminal works of Wiles and Taylor-Wiles, robust methods were developed to prove the modularity of 'polarised' Galois representations. These include, for example, those coming from elliptic curves defined over totally real number fields. Over the last 10 years, new developments in the Taylor-Wiles method (Calegari, Geraghty) and the geometry of Shimura varieties (Caraiani, Scholze) have broadened the scope of these methods. One application is the recent work of Allen, Khare and Thorne, who prove modularity of a positive proportion of elliptic curves defined over a fixed imaginary quadratic field. I'll review some of these developments and work in progress with Caraiani which has further applications to modularity of elliptic curves over imaginary quadratic fields. In person 16th May Amina Abdurrahman (Princeton) Square roots of symplectic L-functions and Reidemeister torsion In the 70s Deligne gave a topological formula for the local epsilon factors attached to an orthogonal representation. We consider the case of a symplectic representation and present a conjecture giving a topological formula for a finer invariant, the square class of its central value. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds, and give a sketch of the proofs. This is joint work with Akshay Venkatesh. Teams 23rd May Lola Thompson (Utrecht) Summing $\mu(n)$: an even faster elementary algorithm We present a new elementary algorithm for computing $M(x) = \sum_{n \le x} \mu(x)$, where $\mu(n)$ is the Moebius function. Our new algorithm takes time $O(x^{3/5} (\log x)^{\epsilon+3/5})$ and space $O(x^{3/10}(\log x)^{13/10})$, which improves upon existing combinatorial algorithms. While there is an analytic algorithm due to Lagarias and Odlyzko with computations based on Riemann zeta integrals that only takes time $O(x^{\epsilon+1/2})$, our algorithm has the advantage of being easier to implement. The new approach roughly amounts to analyzing the difference between a model that we obtain via Diophantine approximation and reality, and showing that it has a simple description in terms of congruence classes and segments. This simple description allows us to compute the difference quickly by means of a table lookup. This talk is based on joint work with Harald Andrés Helfgott. In person 6th June Min Lee (Bristol) Non-vanishing of symmetric cube L-functions The non-vanishing of L-series at the centre of the critical strip has long been a subject of great interest. An important example of the significance of non-vanishing is in the case of an L-series corresponding to a modular form of weight 2, where the non-vanishing at the central point has been shown to be equivalent to the finiteness of the group of rational points of the associated elliptic curve. In the case of higher rank L-functions whose Euler product has an even degree, such connections between non-vanishing at the central point and the finiteness of certain groups are believed to be true, but the relations remain purely conjectural. The symmetric cube L-series plays a role in one of these conjectures. Ginzburg, Jiang and Rallis (2001) proved that the non-vanishing at the central point of the critical strip of the symmetric cube L-series of any GL(2) automorphic form is equivalent to the non-vanishing of a certain triple product integral. The main purpose of this talk is to use this equivalence to prove that there are infinitely many Maass–Hecke cuspforms over the imaginary quadratic field of discriminant -3 such that the central values of their symmetric cube L-functions do not vanish. This is joint work with Jeff Hoffstein and Junehyuk Jung. In person TUESDAY 7th June, MS.03 broadcast Tess Anderson (Purdue) Let's count things: Arithmetic statistics meets Fourier analysis Arithmetic statistics is an area devoted to counting a wide range of objects of algebraic interest, such as polynomials, fields, and elliptic curves. Fueled by the interplay of analysis and number theory, we'll count polynomials and number fields, which though basic objects of study in number theory, are quite difficult to actually count. How often does a random polynomial fail to have full Galois group? How many number fields of a given degree and bounded discriminant are there? We will address both of these questions today. Teams 13th June Francesco Lemma (IMJ-PRG) Tempered currents and higher regulators of Siegel sixfolds Beilinson conjectures relate non-critical special values of motivic L-functions to higher regulators from motivic cohomology to Deligne cohomology. We will present a new description of the Deligne cohomology of a smooth quasi-projective complex analytic variety in terms of tempered currents, which is particularly well adapted to the computation of higher regulators of Shimura varieties. As an application, we will compute the image of a class in the motivic cohomology of a Siegel sixfold in terms of a non-critical value of the degree 8 L-function attached to certain cuspidal automorphic representations of GSp(6). This is a joint work with Cauchi and Rodrigues Jacinto. In person TUESDAY 14 June, MS.03 Rob Kurinczuk (Sheffield) Representations of p-adic groups over Z[1/p] and moduli of Langlands parameters In recent joint work with Dat, Helm, and Moss, we constructed moduli spaces of Langlands parameters over Z[1/p] and studied their geometry. We expect this geometry is reflected in the representation theory of the p-adic group. Our main conjecture “local Langlands in families” relates the GIT quotient of the moduli space of Langlands parameters, the centre of the category of representations of the p-adic group, and the endomorphisms of a "Gelfand-Graev representation". I will explain how after inverting all "non-banal" primes we can compute the centre of the category of representations for any p-adic group, and using this (and the local Langlands correspondence of Arthur, Mok and others) prove the conjecture for classical groups after inverting these primes. In person 27th June Stevan Gajović (Groningen / Max Planck) Quadratic Chabauty for integral points and p-adic heights on even degree hyperelliptic curves In this talk, we show how to construct a locally analytic function that we use to compute integral points on certain even degree hyperelliptic curves whose Jacobian has Mordell–Weil rank over Q equal to the genus. We use p-adic (Coleman–Gross) heights to construct such a function. We explain an algorithm to compute p-adic heights on even degree hyperelliptic curves, more precisely, its local component above p. We briefly discuss how to extend this approach to compute integral points in number fields. This is joint work with Steffen Müller. In person

### 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 (Kevin) Kwan (Columbia) Spectral Moment Formulae for Rankin–Selberg L-functions Central values of L-functions are of great arithmetic interest. One way to access their statistics is via moment calculations. Moments of L-functions have been at the center stage of analytic number theory for decades. There are lots of spectacular advances and applications in this direction. In this talk, I will give an overview of the progress, techniques in this area, as well as discuss some of my recent and on-going work. Teams 7th February Marco D'Addezio (IMJ-PRG) Boundedness of the p-primary torsion of the Brauer group of an abelian variety I will present a new finiteness result for the Brauer group of abelian varieties over finitely generated fields of positive characteristic. More precisely, I will explain how to prove in this case that the transcendental Brauer group has finite exponent. The proof uses the crystalline Tate conjecture, proven by de Jong, and an ad-hoc comparison between the fppf cohomology of Z_p(1) and the crystalline cohomology over imperfect fields. In the end, I will also explain why certain p-divisible p-torsion classes of the Brauer group over the algebraic closure (which are not in the transcendental Brauer group by the main theorem) give an obstruction for the surjectivity of the specialisation morphism of the Néron–Severi group. Teams 14th February Chris Birkbeck (UCL) Formalising modular forms, Eisenstein series and the modularity conjecture in Lean I’ll discuss some recent work on defining modular forms and Eisenstein series in Lean. This is an interactive theorem prover which has recently attracted mathematicians and computer scientist working together to create a unified digitised library of mathematics. In my talk I will explain what Lean is and explain the process of taking basic definitions/examples of modular forms and formalising them. In person 21st February Han Yu (Cambridge) Radial projections of carpets and arithmetic of numbers with restricted digits Given a self-similar or self-affine carpet on $\mathbb{R}^2$, one can construct Bernoulli probability measures. Let $\mu$ be one of them. Consider the radial projections $R_a(\mu)$ with center $a\in\mathbb{R}^2$. Initiated by Marstrand in 1954, a great amount of attention has been attracted in considering regularities of $R_a(\mu)$. A very challenging problem is to understand when is $R_a(\mu)$ absolutely continuous or even continuous. In this talk, I will present new results, for example, I will show that for a class of carpet measures $\mu$, $R_a(\mu)$ is absolutely continuous for all $a$. This improves earlier results that $R_a(\mu)$ is absolutely continuous for 'most of' $a$ in terms of the Lebesgue measure. We can then discuss various applications of those new results. For example, the arithmetic product set of numbers with digits $\{1,\dots, 10^{7000}\}$ with base $10^{9000}$ contains intervals and the arithmetic division set has a positive Lebesgue measure. A crucial ingredient of the proof is a $l^1$-Fourier method developed by D. Allen, S. Chow, P. Varju and myself which will also be discussed in detail. In person 28th February Celine Maistret (Bristol) Parity of ranks of abelian surfaces and product of elliptic curves Let K be a number field and A/K an abelian surface. By the Mordell–Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the global root number of A/K determines the parity of the rank of A/K. Assuming finiteness of the Shafarevich–Tate group, we prove the parity conjecture for principally polarized abelian surfaces under suitable local constraints. Using a similar approach, we show that for two elliptic curves E_1 and E_2 over K with isomorphic 2-torsion, the parity conjecture is true for E_1 if and only if it is true for E_2. This latter result allows us to complete the proof of the p-parity conjecture for elliptic curves over totally real fields. Teams 7th March V. Vinay Kumaraswamy (Tata Institute of Fundamental Research, Mumbai) Quantitative Diophantine approximation for generic ternary diagonal forms In this talk, I will discuss the problem of finding ‘small’ solutions to Diophantine inequalities involving diagonal ternary quadratic forms. We study this problem on average over a one-parameter family of forms. Building on work by Bourgain and Schindler, I will present new results examining this problem over the primes, as well as a generalisation to inhomogeneous forms. This is joint work with Anish Ghosh. *This talk is not hybrid, but is online only: our usual room (B3.02) is unavailable. Fully online* 14th March Alice Pozzi (Imperial) Rigid meromorphic cocycles and p-adic variations of modular forms A rigid meromorphic cocycle is a class in the first cohomology of the group SL(2,Z[1/p]) acting on the non-zero rigid meromorphic functions on the Drinfeld p-adic upper half plane by Mobius transformation. Rigid meromorphic cocycles can be evaluated at points of "real multiplication", and their values conjecturally lie in composita of abelian extensions of real quadratic fields, suggesting striking analogies with the classical theory of complex multiplication. In this talk, we discuss the proof of this conjecture for a special class of rigid meromorphic cocycles. Our proof connects the values of rigid meromorphic cocycles to the study of certain p-adic variations of Hilbert modular forms. This is joint work with Henri Darmon and Jan Vonk. Teams

### 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