Forthcoming events in this series


Tue, 26 Mar 2024
16:00
Quillen Room

Global Galois representations with prescribed local monodromy

Lambert A'Campo
(MPIM Bonn)
Abstract

The compatibility of local and global Langlands correspondences is a central problem in algebraic number theory. A possible approach to resolving it relies on the existence of global Galois representations with prescribed local monodromy.  I will provide a partial solution by relating the question to its topological analogue. Both the topological and arithmetic version can be solved using the same family of projective hypersurfaces, which was first studied by Dwork.

Thu, 07 Mar 2024
16:00
Lecture Room 4

Unitary Friedberg–Jacquet periods and anticyclotomic p-adic L-functions

Andrew Graham
(MPIM Bonn)
Abstract
I will describe the construction of a “square root” anticyclotomic p-adic L-function for symplectic type automorphic representations of the unitary group U(1, 2n-1). This can be seen as a higher dimensional generalisation of the work of Bertolini–Darmon–Prasanna, and one of the main ingredients is the p-adic iteration of Maass–Shimura operators in higher degrees of coherent cohomology. If time permits, I will describe the expected relation with Euler systems outside the region of interpolation.
Thu, 29 Feb 2024
16:00
Lecture Room 4

A new approach to modularity

Andrew Wiles
(University of Oxford)
Abstract

In the 1960's Langlands proposed a generalisation of Class Field Theory. I will review this and describe a new approach using the trace formua as well as some analytic arguments reminiscent of those used in the classical case. In more concrete terms the problem is to prove general modularity theorems, and I will explain the progress I have made on this problem.

Thu, 22 Feb 2024
16:00
Lecture Room 4

Tangent spaces of Schubert varieties

Rong Zhou
(University of Cambridge)
Abstract

Schubert varieties in (twisted) affine Grassmannians and their singularities are of interest to arithmetic geometers because they model the étale local structure of the special fiber of Shimura varieties. In this talk, I will discuss a proof of a conjecture of Haines-Richarz classifying the smooth locus of Schubert varieties, generalizing a classical result of Evens-Mirkovic. The main input is to obtain a lower bound for the tangent space at a point of the Schubert variety which arises from considering certain smooth curves passing through it. In the second part of the talk, I will explain how in many cases, we can prove this bound is actually sharp, and discuss some applications to Shimura varieties. This is based on joint work with Pappas and Kisin-Pappas.

Thu, 15 Feb 2024
16:00
Lecture Room 4, Mathematical Institute

Strong Bounds for 3-Progressions

Zander Kelley
(UIUC)
Abstract
Suppose you have a set $A$ of integers from $\{1, 2,\ldots, N\}$ that contains at least $N / C$ elements.
Then for large enough $N$, must $A$ contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?
In 1953, Roth showed that this is indeed the case when $C \approx \log \log N$, while Behrend in 1946 showed that $C$ can be at most $2^{\sqrt{\log N}}$ by giving an explicit construction of a large set with no 3-term progressions.
Since then, the problem has been a cornerstone of the area of additive combinatorics.
Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on $C$ to $C = (\log N)^{1 + c}$, for some constant $c > 0$.
This talk will describe our work which shows that the same holds when $C \approx 2^{(\log N)^{1/12}}$, thus getting closer to Behrend's construction.
Based on a joint work with Raghu Meka.
Thu, 15 Feb 2024
15:00
Lecture Room 4, Mathematical Institute

Goldbach beyond the square-root barrier

Jared Duker Lichtman
(Stanford)
Abstract

We show the primes have level of distribution 66/107 using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level 3/5 of Maynard. We also extend this level to 5/8, assuming Selberg's eigenvalue conjecture. As a result, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the 'square-root barrier', and leads to the greatest improvement on the problem since Bombieri--Davenport from 1966.

Thu, 08 Feb 2024
16:00
Lecture Room 4, Mathematical Institute

Inhomogeneous Kaufman measures and diophantine approximation

Sam Chow
(Dept. Mathematics, University of Warwick)
Abstract

Kaufman constructed a family of Fourier-decaying measures on the set of badly approximable numbers. Pollington and Velani used these to show that Littlewood’s conjecture holds for a full-dimensional set of pairs of badly approximable numbers. We construct analogous measures that have implications for inhomogeneous diophantine approximation. In joint work with Agamemnon Zafeiropoulos and Evgeniy Zorin, our idea is to shift the continued fraction and Ostrowski expansions simultaneously.

Thu, 01 Feb 2024
16:00
Lecture Room 4, Mathematical Institute

Tame Triple Product Periods

Alice Pozzi
(University of Bristol )
Abstract

A recent conjecture proposed by Harris and Venkatesh relates the action of derived Hecke operators on the space of weight one modular forms to certain Stark units. In this talk, I will explain how this can be rephrased as a conjecture about "tame" analogues of triple product periods for a triple of mod p eigenforms of weights (2,1,1). I will then present an elliptic counterpart to this conjecture relating a tame triple product period to a regulator for global points of elliptic curves in rank 2. This conjecture can be proved in some special cases for CM weight 1 forms, with techniques resonating with the so-called Jochnowitz congruences. This is joint work in preparation with Henri Darmon. 

Thu, 18 Jan 2024
16:00
Lecture Room 4, Mathematical Institute

Traces of random matrices over F_q, and short character sums

Ofir Gorodetsky
(University of Oxford)
Abstract
Let g be a matrix chosen uniformly at random from the GL_n(F_q), where F_q is the field with q elements. We consider two questions:
1. For fixed k and growing n, how fast does Tr(g^k) converge to the uniform distribution on F_q?
2. How large can k be taken, as a function of n, while still ensuring that Tr(g^k) converges to the uniform distribution on F_q?
We will answer these two questions (as well as various variants) optimally. The questions turn out to be strongly related to the study of particular character sums in function fields.
Based on joint works with Brad Rodgers (arXiv:1909.03666) and Valeriya Kovaleva (arXiv:2307.01344).
Thu, 30 Nov 2023
16:00
L5

Computing p-adic heights on hyperelliptic curves

Stevan Gajović
(Charles University Prague)
Abstract

In this talk, we present an algorithm to compute p-adic heights on hyperelliptic curves with good reduction. Our algorithm improves a previous algorithm of Balakrishnan and Besser by being considerably simpler and faster and allowing even degree models. We discuss two applications of our work: to apply the quadratic Chabauty method for rational and integral points on hyperelliptic curves and to test the p-adic Birch and Swinnerton-Dyer conjecture in examples numerically. This is joint work with Steffen Müller.

Thu, 23 Nov 2023
16:00
L5

Anticyclotomic p-adic L-functions for U(n) x U(n+1)  

Xenia Dimitrakopoulou
(University of Warwick)
Abstract

I will report on current work in progress on the construction of anticyclotomic p-adic L-functions for Rankin--Selberg products. I will explain how by p-adically interpolating the branching law for the spherical pair (U(n)xU(n+1), U(n)) we can construct a p-adic L-function attached to cohomological automorphic representations of U(n) x U(n+1), including anticyclotomic variation. Due to the recent proof of the unitary Gan--Gross--Prasad conjecture, this p-adic L-function interpolates the square root of the central L-value. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.

Thu, 16 Nov 2023
16:00
L5

90 years of pointwise ergodic theory

Ben Krause
(University of Bristol)
Abstract

This talk will cover the greatest hits of pointwise ergodic theory, beginning with Birkhoff's theorem, then Bourgain's work, and finishing with more modern directions.

Thu, 02 Nov 2023
16:00
L5

Partition regularity of Pythagorean pairs

Joel Moreira
(University of Warwick)
Abstract

Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x2+y2=z2)? This is one of the simplest questions in arithmetic Ramsey theory which is still open. I will present a recent partial result, showing that in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x2+y2=z2 for some integer z which may be in a different cell. 

The proof consists, after some initial maneuvers inspired by ergodic theory, in controlling the behavior of completely multiplicative functions along certain quadratic polynomials. Considering separately aperiodic and "pretentious" functions, the last major ingredient is a concentration estimate for functions in the latter class when evaluated along sums of two squares.

The talk is based on joint work with Frantzikinakis and Klurman.

Thu, 26 Oct 2023
16:00
L5

The sum-product problem for integers with few prime factors (joint work with Hanson, Rudnev, Zhelezov)

Ilya Shkredov
(LIMS)
Abstract

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbf{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper we show that this maximum is at least of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\varepsilon}$ for some $\varepsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. Our proof consists of three parts: combinatorial, analytical and number theoretical.

 

Thu, 19 Oct 2023
16:00
L5

Siegel modular forms and algebraic cycles

Aleksander Horawa
(Oxford University)
Abstract

(Joint work with Kartik Prasanna)

Siegel modular forms are higher-dimensional analogues of modular forms. While each rational elliptic curve corresponds to a single holomorphic modular form, each abelian surface is expected to correspond to a pair of Siegel modular forms: a holomorphic and a generic one. We propose a conjecture that explains the appearance of these two forms (in the cohomology of vector bundles on Siegel modular threefolds) in terms of certain higher algebraic cycles on the self-product of the abelian surface. We then prove three results:
(1) The conjecture is implied by Beilinson's conjecture on special values of L-functions. Amongst others, this uses a recent analytic result of Radzwill-Yang about non-vanishing of twists of L-functions for GL(4).
(2) The conjecture holds for abelian surfaces associated with elliptic curves over real quadratic fields.
(3) The conjecture implies a conjecture of Prasanna-Venkatesh for abelian surfaces associated with elliptic curves over imaginary quadratic fields.

Thu, 12 Oct 2023
16:00
L5

Moments of families of quadratic L-functions over function fields via homotopy theory

Dan Petersen
(Stockholm University)
Abstract

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

Thu, 22 Jun 2023
16:00
L5

Anticyclotomic Euler systems and Kolyvagins' methods

Christopher Skinner
(Princeton University)
Abstract

I will explain a formalism for anticyclotomic Euler systems for a large class of Galois representations and explain how to prove analogs of Kolyvagins' celebrated "rank one" results. A novelty of this approach lies in the use of primes that split in the CM field. This is joint work with Dimitar Jetchev and Jan Nekovar. I will also describe some higher-dimensional examples of such Euler systems.

Thu, 15 Jun 2023
16:00
L5

Computations, heuristics and analytic number theory

Andrew Granville
(Université de Montréal)
Abstract

Abstract. I will talk about projects in which we combine heuristics with computational data to develop a theory in problems where it was previously hard to be confident of the guesses that there are in the literature.

 

1/ "Speculations about the number of primes in fast growing sequences". Starting from studying the distribution of primes in sequences like $2^n-3$, Jon Grantham and I have been developing a heuristic to guess at the frequency of prime values in arbitrary linear recurrence sequences in the integers, backed by calculations.

 

If there is enough time I will then talk about:

 

2/ "The spectrum of the $k$th roots of unity for $k>2$, and beyond".  There are many questions in analytic number theory which revolve around the "spectrum", the possible mean values of multiplicative functions supported on the $k$th roots of unity. Twenty years ago Soundararajan and I determined the spectrum when $k=2$, and gave some weak partial results for $k>2$, the various complex spectra.  Kevin Church and I have been tweaking MATLAB's package on differential delay equations to help us to develop a heuristic theory of these spectra for $k>2$, allowing us to (reasonably?) guess at the answers to some of the central questions.

Tue, 13 Jun 2023
16:00
L5

Revisiting the Euler system for imaginary quadratic fields

Christopher Skinner
(Princeton University)
Abstract

I will explain how to construct an Euler system for imaginary quadratic fields using Eisenstein series and their cohomology classes. This illustrates a template for a construction that should yield many new Euler systems.

Thu, 08 Jun 2023
16:00
L5

The elliptic Gamma function and Stark units for complex cubic fields

Luis Garcia
(University College London)
Abstract

The elliptic Gamma function — a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function — is a meromorphic special function in several variables that mathematical physicists have shown to satisfy modular functional equations under SL(3,Z). In this talk I will present evidence (numerical and theoretical) that this function often takes algebraic values that satisfy explicit reciprocity laws and that are related to derivatives of Hecke L-functions at s=0. Thus this function conjecturally allows to extend the theory of complex multiplication to complex cubic fields as envisioned by Hilbert's 12th problem. This is joint work with Nicolas Bergeron and Pierre Charollois.

Thu, 01 Jun 2023
16:00
L5

An Euler system for the symmetric square of a modular form

Christopher Skinner
(Princeton University)
Abstract

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an example of a larger class of similarly constructed Euler systems.  This is a report on joint work with Marco Sangiovanni Vincentelli.

Thu, 25 May 2023
16:00
L5

Balanced triple product p-adic L-functions and classical weight one forms

Luca Dall'Ava
(Università degli Studi di Milano)
Abstract

The main object of study of the talk is the balanced triple product p-adic L-function; this is a p-adic L-function associated with a triple of families of (quaternionic) modular forms. The first instances of these functions appear in the works of Darmon-Lauder-Rotger, Hsieh, and Greenberg-Seveso. They have proved to be effective tools in studying cases of the p-adic equivariant Birch & Swinnerton-Dyer conjecture. With this aim in mind, we discuss the construction of a new p-adic L-function, extending Hsieh's construction, and allowing classical weight one modular forms in the chosen families. Such improvement does not come for free, as it coincides with the increased dimension of certain Hecke-eigenspaces of quaternionic modular forms with non-Eichler level structure; we discuss how to deal with the problems arising in this more general setting. One of the key ingredients of the construction is a p-adic extension of the Jacquet-Langlands correspondence addressing these more general quaternionic modular forms. This is joint work in progress with Aleksander Horawa.

Thu, 18 May 2023
16:00
L5

Rational points on Erdős-Selfridge curves

Kyle Pratt
(University of Oxford)
Abstract

Many problems in number theory are equivalent to determining all of the rational points on some curve or family of curves. In general, finding all the rational points on any given curve is a challenging (even unsolved!) problem. 

The focus of this talk is rational points on so-called Erdős-Selfridge curves. A deep conjecture of Sander, still unproven in many cases, predicts all of the rational points on these curves. 

I will describe work-in-progress proving new cases of Sander's conjecture, and sketch some ideas in the proof. The core of the proof is a `mass increment argument,' which is loosely inspired by various increment arguments in additive combinatorics. The main ingredients are a mixture of combinatorial ideas and quantitative estimates in Diophantine geometry.

Thu, 11 May 2023
16:00
L5

Parity of ranks of abelian surfaces

Celine Maistret
(University of Bristol)
Abstract
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 sign of the functional equation of the L-series 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.
In both cases, we prove analogous unconditional results for Selmer groups.