Forthcoming events in this series


Thu, 28 Apr 2022

16:00 - 17:00
L4

A modular construction of unramified p-extensions of Q(N^{1/p})

Jaclyn Lang
( Temple University )
Abstract

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.

Thu, 03 Mar 2022

16:00 - 17:00
L4

Density of rational points on del Pezzo surfaces of degree 1

Rosa Winter
(King's College London)
Abstract

Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology?


Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d >= 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general.


I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.

Thu, 24 Feb 2022

16:00 - 17:00
L4

Euler characteristics and epsilon constants of curves over finite fields - some wild stuff

Bernhard Koeck
(University of Southampton)
Abstract

Let X be a smooth projective curve over a finite field equipped with an action of a finite group G. I’ll first briefly introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result will be a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic if the projection X  X/G is at most weakly ramified. This generalises a theorem of Chinburg for the tamely ramified case. I’ll end with some speculations in the arbitrarily wildly ramified case. This is joint work with Helena Fischbacher-Weitz and with Adriano Marmora.

Thu, 17 Feb 2022

16:00 - 17:00
L5

Values of the Ramanujan tau-function

Vandita Patel
(University of Manchester)
Abstract

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

Thu, 12 Mar 2020

16:00 - 17:00
L5

Growth in soluble linear groups over finite fields

Brendan Murphy
(University of Bristol)
Abstract

In joint work with James Wheeler, we show that if a subset $A$ of $GL_n(\mathbb{F}_q)$ is a $K$-approximate group and the group $G$ it generates is soluble, then there are subgroups $U$ and $S$ of $G$ and a constant $k$ depending only on $n$ such that:

$A$ quickly generates $U$: $U\subseteq A^k$,
$S$ contains a large proportion of $A$: $|A^k\cap S| \gg K^{-k}|A|, and
$S/U$ is nilpotent.

Briefly: approximate soluble linear groups over any finite field are (almost) finite by nilpotent.

The proof uses a sum-product theorem and exponential sum estimates, as well as some representation theory, but the presentation will be mostly self-contained.

Thu, 05 Mar 2020
16:00
L6

Dynamical systems for arithmetic schemes

Christopher Deninger
(University of Muenster)
Abstract

We construct a functor from arithmetic schemes (and dominant morphisms) to dynamical systems which allows to recover the Hasse-Weil zeta function of a scheme as a Ruelle type zeta function of the corresponding dynamical system. We state some further properties of this correspondence and explain the relation to the work of Kucharczyk and Scholze who realize the Galois groups of fields containing all roots of unity as (etale) fundamental groups of certain topological spaces. We also explain the main reason why our dynamical systems are not yet the right ones and in what regard they need to be refined.
 

Thu, 27 Feb 2020
16:00
L6

Apéry series and Mellin transforms of solutions of differential equations

Spencer Bloch
(University of Chicago)
Abstract


One can study periods of algebraic varieties by a process of "fibering out" in which the variety is fibred over a punctured curve $f:X->U$. I will explain this process and how it leads to the classical Picard Fuchs (or Gauss-Manin) differential equations. Periods are computed by integrating solutions of Picard Fuchs over suitable closed paths on $U$. One can also couple (i.e.tensor) the Picard Fuchs connection to given connections on $U$. For example, $t^s$ with $t$ a unit on $U$ and $s$ a parameter is a solution of the connection on $\mathscr{O}_U$ given by $\nabla(1) = sdt/t$. Our "periods" become integrals over suitable closed chains on $U$ of $f(t)t^sdt/t$. Golyshev called the resulting functions of $s$ "motivic Gamma functions". 
Golyshev and Zagier studied certain special Picard Fuchs equations for their proof of the Gamma conjecture in mirror symmetry in the case of Picard rank 1. They write down a generating series, the Apéry series, the knowledge of the first few terms of which implied the gamma conjecture. We show their Apéry series is the Taylor series of a product of the motivic Gamma function times an elementary function of $s$. In particular, the coefficients of the Apéry series are periods up to inverting $2\pi i$. We relate these periods to periods of the limiting mixed Hodge structure at a point of maximal unipotent monodromy. This is joint work with M. Vlasenko. 
 

Thu, 20 Feb 2020

16:00 - 17:00
L5

Analytic rank of automorphic L-functions

Hung Bui
(University of Manchester)
Abstract

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-called analytic rank, which is the order of vanishing of the associated L-functions at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe". This is joint work with Kyle Pratt and Alexandru Zaharescu.

Thu, 13 Feb 2020
16:00
L5

Symmetric power functoriality for modular forms

James Newton
(KCL)
Abstract

Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym^r from automorphic representations of GL(2) to automorphic representations of GL(r+1). I will discuss some joint work with Jack Thorne on the symmetric power lifting for holomorphic modular forms.

Thu, 06 Feb 2020

16:00 - 17:00
L5

The Riemann zeta function in short intervals

Adam Harper
(University of Warwick)
Abstract

I will describe some new-ish results on the average and maximum size of the Riemann zeta function in a "typical" interval of length 1 on the critical line. A (hopefully) interesting feature of the proofs is that they reduce the problem for the zeta function to an analogous problem for a random model, which can then be solved using various probabilistic techniques.

Thu, 30 Jan 2020
16:00
L5

The p-part of BSD for residually reducible elliptic curves of rank one

Giada Grossi
(UCL)
Abstract

Let E be an elliptic curve over the rationals and p a prime such that E admits a rational p-isogeny satisfying some assumptions. In a joint work with J. Lee and C. Skinner, we prove the anticyclotomic Iwasawa main conjecture for E/K for some suitable quadratic imaginary field K. I will explain our strategy and how this, combined with complex and p-adic Gross-Zagier formulae, allows us to prove that if E has rank one, then the p-part of the Birch and Swinnerton-Dyer formula for E/Q holds true.
 

Thu, 23 Jan 2020

16:00 - 17:00
L5

Efficient congruence and discrete restriction for (x,x^3)

Kevin Hughes
(University of Bristol)
Abstract

We will outline the main features of Wooley's efficient congruencing method for the parabola. Then we will go on to prove new bounds for discrete restriction to the curve (x,x^3). The latter is joint work with Trevor Wooley (Purdue).

Thu, 05 Dec 2019
16:00
L6

On the negative Pell equation

Stephanie Chan
(UCL)
Abstract

Stevenhagen conjectured that the density of d such that the negative Pell equation x^2-dy^2=-1 is solvable over the integers is 58.1% (to the nearest tenth of a percent), in the set of positive squarefree integers having no prime factors congruent to 3 modulo 4. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we use a recent breakthrough of Smith to prove that the infimum of this density is at least 53.8%, improving previous results of Fouvry and Klüners, by studying the distribution of the 8-rank of narrow class groups of quadratic number fields.

Thu, 28 Nov 2019
16:00
L6

Propagating algebraicity of automorphic representations via functoriality

Wushi Goldring
(Stockholm University)
Abstract

My talk will have two protagonists: (1) Automorphic representations which -- let's be honest -- are very complicated and mysterious, but also (2) Involutions  (=automorphisms of order at most 2) of connected reductive groups -- these are very concrete and can often be represented by diagonal matrices with entries 1,-1 or i, -i. The goal is to explain how difficult questions about (1) can be reduced to relatively easy, concrete questions about (2).
Automorphic representations are representation-theoretic generalizations of modular forms. Like modular forms, automorphic representations are initially defined analytically. But unlike modular forms -- where we have a reinterpretation in terms of algebraic geometry -- for most automorphic representations we currently only have a (real) analytic definition. The Langlands Program predicts that a wide class of automorphic representations admit the same algebraic properties which have been known to hold for modular forms since the 1960's and 70's. In particular, certain complex numbers "Hecke eigenvalues" attached to these automorphic representations are conjectured to be algebraic numbers. This remains open in many cases (especially those cases of interest in number theory and algebraic geometry), in particular for Maass forms -- functions on the upper half-plane which are a non-holomorphic variant of modular forms.
I will explain how elementary structure theory of reductive groups over the complex numbers provides new insight into the above algebraicity conjectures; in particular we deduce that the Hecke eigenvalues are algebraic for an infinite class of examples where this was not previously known. 
After applying a bunch of "big, old theorems" (in particular Langlands' own archimedean correspondence), it all comes down to studying how involutions of a connected, reductive group vary under group homomorphisms. Here I will write down the key examples explicitly using matrices.

Thu, 21 Nov 2019
16:00
L6

The Weyl subconvex exponent for Dirichlet L-functions.

Ian Petrow
(UCL)
Abstract

In the 1920s Weyl proved the first non-trivial estimate for the Riemann zeta function on the critical line: \zeta(1/2+it) << (1+|t|)^{1/6+\epsilon}. The analogous bound for a Dirichlet L-function L(1/2,\chi) of conductor q as q tends to infinity is still unknown in full generality. In a breakthrough around 2000, Conrey and Iwaniec proved the analogue of the Weyl bound for L(1/2,\chi) when \chi is assumed to be quadratic of conductor q.  Building on the work of Conrey and Iwaniec, we show (joint work with Matt Young) that the Weyl bound for L(1/2,\chi) holds for all primitive Dirichlet characters \chi. The extension to all moduli q is based on aLindelöf-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when q^*|d, where q^* is the least positive integer such that q^2|(q^*)^3.

Thu, 14 Nov 2019
16:00
L6

Propinquity of divisors

Ben Green
(Oxford)
Abstract

Let n be a random integer (sampled from {1,..,X} for some large X). It is a classical fact that, typically, n will have around (log n)^{log 2} divisors. Must some of these be close together? Hooley's Delta function Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the number of divisors of n in I. I will report on joint work with Kevin Ford and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is about (log log n)^c for some c = 0.353.... given by an equation involving an exotic recurrence relation, and then prove (in some sense) half of this conjecture, establishing that Delta(n) is at least this big almost surely.

Thu, 07 Nov 2019
16:00
L6

Number fields with prescribed norms

Rachel Newton
(Reading)
Abstract

Let G be a finite abelian group, let k be a number field, and let x be an element of k. We count Galois extensions K/k with Galois group G such that x is a norm from K/k. In particular, we show that such extensions always exist. This is joint work with Christopher Frei and Daniel Loughran.

Thu, 31 Oct 2019
16:00
L6

Mordell-Weil groups as Galois modules

Christian Wuthrich
(Nottingham)
Abstract

Let $E/k$ be an elliptic curve over a number field and $K/k$ a Galois extension with group $G$. What can we say about $E(K)$ as a Galois module? Not just what complex representations appear, but its structure as a $\mathbb{Z}[G]$-module. We will look at some examples with small $G$.

Thu, 24 Oct 2019

16:00 - 17:00
L6

L-functions of Kloosterman sums

Javier Fresán
(Ecole Polytechnique)
Abstract

Guided by the analogy with certain moments of the Bessel function that appear as Feynman integrals, Broadhurst and Roberts recently studied a family of L-functions built up by assembling symmetric power moments of Kloosterman sums over finite fields. I will prove that these L-functions arise from potentially automorphic motives over the field of rational numbers, and hence admit a meromorphic continuation to the complex plane that satisfies the expected functional equation. If time permits, I will identify the periods of the corresponding motives with the Bessel moments and make a few comments about the special values of the L-functions. This is a joint work with Claude Sabbah and Jeng-Daw Yu.

Thu, 17 Oct 2019
16:00
L6

One-level density of Dirichlet L-functions

Kyle Pratt
(Oxford)
Abstract

I will discuss work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.
 

Thu, 20 Jun 2019

16:00 - 17:00
L6

Explicit Non-Abelian Chabauty via Motivic Periods

David Corwin
(UC Berkeley)
Abstract

We report on a line of work initiated by Dan-Cohen and Wewers and continued by Dan-Cohen and the speaker to explicitly compute the zero loci arising in Kim's non-abelian Chabauty's method. We explain how this works, an important step of which is to compute bases of a certain motivic Hopf algebra in low degrees. We will summarize recent work by Dan-Cohen and the speaker, extending previous computations to $\mathbb{Z}[1/3]$ and proposing a general algorithm for solving the unit equation. Many of the methods in the more recent work are inspired by recent ideas of Francis Brown. Finally, we indicate future work, in which we hope to use elliptic motivic periods to explicitly compute points on punctured elliptic curves and beyond.

Thu, 13 Jun 2019

16:00 - 17:00
L6

Arakelov theory on degenerating curves

Gerd Faltings
(University of Bonn and MPIM)
Abstract

We compute the asymptotics of Arakelov functions if smooth curves degenerate to semistable singular curves. The motivation was to determine whether the delta function defines a metric on the boundary of moduli space. In fact things are slightly more complicated. The main result states that the asymptotics is mostly governed by the graph associated to the degeneration, with some subleties. The topic has been also treated by R. deJong and my student R. Wilms.

Thu, 06 Jun 2019

16:00 - 17:00
L6

A non-abelian algebraic criterion for good reduction of curves

Valentina DiProietto
(University of Exeter)
Abstract


For a family of proper hyperbolic complex curves $f: X \longrightarrow \Delta^*$ over a puntured disc $\Delta^*$ with semistable reduction at the center, Oda proved, with transcendental methods, that the outer monodromy action of $\pi_1(\Delta^*) \cong \mathbb{Z}$ on the classical unipotent fundamental group of the generic fiber of $f$ is trivial if and only if $f$ has good reduction at the center. In this talk I explain a joint work with B. Chiarellotto and A. Shiho in which we give a purely algebraic proof of Oda's result.

Thu, 30 May 2019

16:00 - 17:00
L6

Fourier expansions at cusps and the Manin constant of elliptic curves

Michalis Neururer
(TU Darmstadt)
Abstract

I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.

Thu, 23 May 2019

16:00 - 17:00
L6

The Sum-Product Phenomenon

George Shakan
(Oxford University)
Abstract

In 1983, Erdos and Szemerédi conjectured that for any finite subset of the integers, either the sumset or the product set has nearly quadratic growth. Applications include incidence geometry, exponential sums, compressed image sensing, computer science, and elsewhere. We discuss recent progress towards the main conjecture and related questions.