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# 2020-21

The seminars are held on Mondays from 3pm to 4pm (or on Tuesday if Monday is a bank holiday).

2020-21 Term 3

Organisers: Sam Chow, Chris Lazda and Chris Williams

 26th April Oscar Rivero (Warwick) Eisenstein congruences and Euler systems Let f be a cuspidal eigenform of weight two, and let p be a prime at which f is congruent to an Eisenstein series. Beilinson constructed a class arising from the cup-product of two Siegel units and proved a relationship with the first derivative $L$ at the near central point s=0 of the L-series of f. In this talk, I will motivate the study of congruences between modular forms at the level of cohomology classes, and will report on a joint work with Victor Rotger where we prove two congruence formulas relating the motivic part of $L$ modulo p and $L$ modulo p with circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson--Kato elements and, most crucially, the work of Fukaya--Kato. 4th May (Tuesday) Josha Box (Warwick) Computing models for quotients of modular curves In Sage or Magma, you can ask for the defining equations of modular curves of the form $X_0(N)$ and it will give you the answer. For more general modular curves, however, an algorithm to determine such models did not exist until recently. I will describe such an algorithm based on the existing method for $X_0(N)$ . This builds on earlier work of John Cremona 10th May Beth Romano (Oxford) Depth in the local Langlands correspondence The local Langlands correspondence is a kaleidoscope of conjectures relating representations of p-adic groups, local Galois theory, and the theory of complex Lie groups. I'll discuss how the notion of depth appears on both sides of the correspondence. The positive-depth part of the correspondence becomes mysterious for small residue characteristic, and I'll talk about how a construction of Reeder--Yu uses geometric invariant theory to shed light on this area. Finally, I'll talk about my recent results that build on Reeder--Yu to give new positive-depth representations for certain exceptional groups. 17th May Jon Chapman (Manchester) Partition and density regularity for diagonal Diophantine systems A system of equations is called partition regular if every finite colouring of the positive integers produces monochromatic solutions to the system. A system is called density regular if it has solutions over every set of integers with positive upper density. A classical result of Rado characterises all partition regular linear systems, whilst Szemerédi’s theorem classifies all density regular linear systems. In this talk, I will report on recent developments on these topics for non-linear systems. I will also show how techniques from analytic number theory and additive combinatorics can be used to classify partition and density regularity for sufficiently non-singular systems of diagonal polynomial equations. 24th May Ashwin Iyengar (KCL) The Iwasawa main conjecture and the extended eigencurve I will explain how one can formulate the Iwasawa main conjecture for p-adic families of modular forms. I will focus specifically on a certain piece of the “extended eigencurve”, which is a mixed characteristic adic space where the p-adic families live. 1st June (Tuesday) Daniel Loughran (Bath) Probabilistic Arithmetic Geometry A theorem of Erdos-Kac states that the number of prime divisors of an integer behaves like a normal distribution (once suitably renormalised). In this talk I shall explain a version of this result for integer points on varieties. This is joint work with Efthymios Sofos and Daniel El-Baz. 7th June Ross Paterson (Glasgow) Statistics for Elliptic Curves over Galois Extensions As E varies among elliptic curves defined over the rational numbers, a theorem of Bhargava and Shankar shows that the average rank of the Mordell--Weil group E(Q) is bounded. If we now fix a number field K, is the same true of E(K)? Moreover, if K/F is a Galois extension then how does the Galois group act on E(K) "on average"? This talk will report on recent progress on these questions: answering the first in the affirmative for certain choices of K, and, after a more precise formulation, offering a partial answer to the second. 14th June Natalie Evans (KCL) Correlations of almost primes The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$. 21st June Shuntaro Yamagishi (Utrecht) Solving polynomial equations in many variables in primes Solving polynomial equations in primes is a fundamental problem in number theory. For example, the twin prime conjecture can be phrased as the statement that the equation $x_1 - x_2 - 2 = 0$ has infinitely many solutions in primes. Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a degree $d$ homogeneous form. In 2014, Cook and Magyar proved the existence of prime solutions to the equation $F(x_1, \ldots, x_n) = 0$ under certain assumptions on $F$. In particular, their result requires the number of variables $n$ to be an exponential tower in $d$. I will talk about a result related to this work of Cook and Magyar improving on the number of variables required. 28th June Hanneke Wiersema (KCL) Minimal weights of mod p Galois representations The strong form of Serre's conjecture states that every two-dimensional continuous, odd, irreducible mod p representation of the absolute Galois group of Q arises from a modular form of a specific minimal weight, level and character. In this talk we show the minimal weight is equal to a notion of minimal weight inspired by work of Buzzard, Diamond and Jarvis. Moreover, using the Breuil-Mézard conjecture we give a third interpretation of this minimal weight as the smallest k>1 such that the representation has a crystalline lift of Hodge-Tate type (0, k-1). After discussing the interplay between these three weight characterisations in the more general setting of Galois representations over totally real fields, we investigate its consequences for generalised Serre conjectures.

### 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) Sets without square differences How dense can a set of integers be if no two distinct elements have a square difference? This question was first asked by Lovasz, and answered in a qualitative sense independently by Sarkozy and Furstenberg in the 1970s, who showed that such a set must have zero upper density. The best known upper bound previously came from an elaborate Fourier analytic argument due to Pintz, Steiger, and Szemeredi, in 1988. In joint work with James Maynard, we present the first improvement on this bound. Our method, although still relying on Fourier analysis and the circle method, is more direct, and the key new innovation is a new bound for the number of additive relations between rationals of small denominator, which is of independent interest, and we hope has many other applications to circle method problems. 8th Feburary Jan Vonk (Leiden) Diagonal restrictions of p-adic families of modular forms The theory of complex multiplication occupies an important place in number theory, an early manifestation of which was the use of special values of the j-function in explicit class field theory of imaginary quadratic fields, and the works of Eisenstein, Kronecker, Weber, Hilbert, and many others. In the early 20th century, Hecke studied the diagonal restrictions of Eisenstein series over real quadratic fields, which later lead to highly influential developments in the theory of complex multiplication initiated by Gross and Zagier in their famous work on Heegner points on elliptic curves. In this talk, we will explore what happens when we replace the imaginary quadratic fields in CM theory with real quadratic fields, and propose a framework for a tentative 'RM theory', based on the notion of rigid meromorphic cocycles, studied in joint work with Henri Darmon. I will discuss recent progress obtained in various joint works with Henri Darmon, Alice Pozzi, Yingkun Li. 15th February Katharina Hubner (Heidelberg) The tame site For a scheme of characteristic p > 0 (or mixed characteristic) etale cohomology with p-torsion coefficients does not behave very well: Smooth base change, cohomological purity, homotopy invariance, just to name a few, only hold for coefficients prime to the characteristic. The reason for this failure is the existence of wild ramification. This talk presents a modification of the etale topology that does not admit for wild ramification, called the tame site. For coefficients away from the characteristic the etale and tame cohomology groups are isomorphic and for p-torsion coefficients they are better behaved than the etale cohomology groups. 22nd February Sally Gilles (Imperial) Syntomic cohomology and period morphisms In 2017, Colmez and Niziol proved a comparison theorem between arithmetic p-adic nearby cycles and syntomic cohomology sheaves. To prove it, they gave a local construction using (\phi, \Gamma)-modules theory which allows to reduce the period isomorphism to a comparison theorem between cohomologies of Lie algebras. I will explain the geometric version of this local construction and how to globalize it to get a new period isomorphism. In particular, the explicit description of this new isomorphism can be used to compare previous constructions of period morphisms and prove they are equal. 1st March Vladimir Dokchitser (UCL) Hasse-Weil, Tate-Shafarevich and Birch-Swinnerton-Dyer Both the Birch-Swinnerton-Dyer conjecture and the Shafarevich-Tate conjecture provide ways of studying rational points on elliptic curves. Curiously, some basic properties of L-functions translate into out-of-reach statements about rational points, and vice versa. I will discuss several unexplained consequences that rational points, Selmer groups and L-functions imply about each other. 8th March Luis Garcia (UCL) Eisenstein classes and hyperplane complements In recent years several authors (Sczech, Nori, Hill, Charollois-Dasgupta-Greenberg, Beilinson-Kings-Levin) have defined and studied certain group cocycles ("Eisenstein cocycles") in the cohomology of arithmetic groups. I will discuss how these constructions can be understood in terms of equivariant cohomology and characteristic classes. This point of view relates the cocycles to the theta correspondence and leads to generalisations relating the homology of arithmetic groups to algebraic objects such as meromorphic differentials on hyperplane complements. I will discuss these generalisations and an application to critical values of L-functions (joint with Nicolas Bergeron and Pierre Charollois). 15th March Kirsti Biggs (Chalmers/Gothenburg) Ellipsephic efficient congruencing for the Vinogradov system An ellipsephic set consists of natural numbers with digital restrictions in a given base. Such sets have a fractal structure and can be viewed as p-adic analogues of real Cantor sets. Using Wooley's nested efficient congruencing method, we bound the number of ellipsephic solutions to the Vinogradov system of general degree k; that is, the system of diagonal equations x_1^j + ... + x_s^j = y_1^j + ... + y_s^j for j from 1 to k. In this talk, I will focus on the key step in the proof, which uses an additive property of our chosen ellipsephic sets to improve on certain congruence conditions at a low cost. I will also touch briefly on connections to harmonic analysis.

### 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