Upcoming talks
Previous Seminars
In the spring term (2021) Luis Garcia and I are organizing the London Number Theory Seminar hosted at UCL. The list of speakers, dates and zoom details can be found here
In the spring term (2020) Peter Humphries and I are organizing the London Number Theory Seminar hosted at UCL. The list of speakers, dates and provisional details are as follows:
The seminar is held on Wednesdays at 4pm in the Mathematics Department (25 Gordon St) Room 505, with the exception of the 18th of March, where it will take place in 26 Bedford Way, room LG04. Immediately prior to the seminar there is tea and coffee available in the 6th floor common room.

15 Jan 2020  Matthew Bisatt (University of Bristol)

Title: Tame torsion of Jacobians and the tame inverse Galois problem

Abstract: Fix positive integers g and m. Does there exist a genus g curve, defined over the rationals, such that the mod m representation of its Jacobian is everywhere tamely ramified? I will give an affirmative answer to this question when m is squarefree via the theory of hyperelliptic Mumford curves. I will also and give an application of this to a variant of the inverse Galois problem. This is joint work with Tim Dokchitser.


22 Jan 2020  Spencer Bloch (University of Chicago)

Title: Motivic Gamma functions

Abstract:

Recall of the theory of periods for local systems on curves.

Definition (V. Golyshev) of motivic gamma functions as Mellin transforms of period integrals.

Main theorem (joint with M. Vlasenko)

Application to the gamma conjecture in mirror symmetry (work of Golyshev + Zagier).



29 Jan 2020 Giada Grossi (UCL)

Title: The ppart of BSD for residually reducible elliptic curves of rank one

Abstract: Let E be an elliptic curve over the rationals and p a prime such that E admits a rational pisogeny 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 padic GrossZagier formulae, allows us to prove that if E has rank one, then the ppart of the Birch and SwinnertonDyer formula for E/Q holds true.


05 Feb 2020  Efthymios Sofos (University of Glasgow)

Title: Rational points on Châtelet surfaces

Abstract: This talk is on ongoing joint work with Alexei Skorobogatov.
Châtelet surfaces of degree d are surfaces of the form x^2−ay^2=f(t), where f is a fixed integer polynomial of even degree d and a is a fixed nonsquare integer. When f has degree up to 4 (or when f is a product of integer linear polynomials) it has been shown that the BrauerManin obstruction is the only one to the Hasse principle. This is the result of decades of investigations by SwinnertonDyer, ColliotThélène, Skorobogatov, Browning and Matthiesen, among others.
Going beyond degree 4 for polynomials of general type has been a very popular question which has seen no progress in the last decades. We use techniques from analytic number theory, related to equidistribution of the Möbius function, to prove that for 100% of all polynomials f (ordered by the size of the coefficients) gives Châtelet surfaces that satisfy the Hasse principle.


12 Feb 2020  Sandro Bettin (Università degli studi di Genova)

Title: The value distribution of quantum modular forms

Abstract: In a joint work with Sary Drappeau, we obtain results on the value distribution of quantum modular forms. As particular examples we consider the distribution of modular symbols and the Estermann function at the central point.


19 Feb 2020  Sarah Peluse (Oxford University)

Title: Bounds in the polynomial Szemerédi theorem

Abstract: Let P_1,...,P_m be polynomials with integer coefficients and zero constant term. Bergelson and Leibman’s polynomial generalization of Szemerédi’s theorem states that any subset A of {1,...,N} that contains no nontrivial progressions x,x+P_1(y),...,x+P_m(y) must satisfy A=o(N). In contrast to Szemerédi's theorem, quantitative bounds for Bergelson and Leibman's theorem (i.e., explicit bounds for this o(N) term) are not known except in very few special cases. In this talk, I will discuss recent progress on this problem.


26 Feb 2020  Djordje Milicevic (Bryn Mawr/Max Planck)

Title: Extreme values of twisted Lfunctions

Abstract: Distribution of values of Lfunctions on the critical line, or more generally central values in families of Lfunctions, has striking arithmetic implications. One aspect of this problem are upper bounds and the rate of extremal growth. The Lindelof Hypothesis states that zeta(1/2+it)<<(1+t)^eps for every eps>0 ; however neither this statement nor the celebrated Riemann Hypothesis (which implies it) by themselves do not provide even a conjecture for the precise extremal subpower rate of growth. Soundararajan's method of resonators and its recent improvement due to BondarenkoSeip are flexible first moment methods that unconditionally show that zeta(1/2+it), or central values of other degree one Lfunctions, achieve very large values.
In this talk, we address large central values L(1/2, f x chi) of a fixed GL(2) Lfunction twisted by Dirichlet characters chi to a large prime modulus q. We show that many of these twisted Lfunctions achieve very high central values, not only in modulus but in arbitrary angular sectors modulo pi*Z, and that in fact given any two modular forms f and g, the product L(1/2, f x chi) * L(1/2, g x chi) achieves very high values. To obtain these results, we develop a flexible, readytouse variant of Soundararajan's method that uses only a limited amount of information about the arithmetic coefficients in the family. In turn, these conditions involve small moments of various combinations of Hecke eigenvalues over primes, for which we develop the corresponding Prime Number Theorems using functorial lifts of GL(2) forms.
This is part of joint work on moments of twisted Lfunctions with Blomer, Fouvry, Kowalski, Michel, and Sawin.


04 Mar 2020  Asbjørn Nordentoft (Copenhagen)

Title: The distribution of modular symbols and additive twists of Lfunctions

Abstract: Recently Mazur and Rubin, motivated by questions in Diophantine stability, put forth some conjectures regarding the distribution of modular symbols, one of which predicts asymptotic Gaussian behavior. An average version of this conjecture was settled by Petridis and Risager using automorphic methods. Modular symbols are certain line integrals associated to weight two cusp forms and we will in this talk discuss generalizations of the result of Petridis and Risager to higher weight cusp forms. In particular we will explain how to generalize the automorphic methods to show that central values of additive twists of cuspidal Lfunctions (of arbitrary even weight) are also asymptotically Gaussian.


11 Mar 2020  Javier Fresán (École polytechnique)

Title: Irregular Hodge filtration and eigenvalues of Frobenius

Abstract: The de Rham cohomology of a connection of exponential type on an algebraic variety carries a filtration, indexed by rational numbers, that generalises the usual Hodge filtration on the cohomology with constant coefficients. I will explain a few results and conjectures relating this filtration to exponential sums over finite fields.


18 Mar 2020  Tiago da Fonseca (Oxford University) CANCELLED

Title: On Fourier coefficients of Poincaré series

Abstract: Poincaré series are among the first examples of holomorphic and weakly holomorphic modular forms. They are useful in many analytical questions, but their Fourier coefficients seem hard to grasp algebraically. In this talk, I will discuss the arithmetic nature of Fourier coefficients of Poincaré series by characterizing them as cohomological invariants (periods).

Note: The talk will take place in 26 Bedford Way, room LG04


25 Mar 2020  Chris Lazda (Warwick University) CANCELLED

Title: A NeronOggShafarevich criterion for K3 surfaces

Abstract: The naive analogue of the NéronOggShafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semistable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if is second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke.

For more information you can sign up to one the London mailing list available here (where you can also find information about previous talks).
In the spring term (2019) Alex Torzewski and I are organizing the London Number Theory Seminar hosted at UCL. The list of speakers, dates and provisional details are as follows:
The seminar is held on Wednesdays at 4pm in the Mathematics Department (25 Gordon St) Room 505. Immediately prior to the seminar there is tea and coffee available in the 6th floor common room. Earlier in the afternoon there are the algebraic study groups on "Derived Deformation Theory" and "Derived Hecke Algebras" held in Room 505 as well as a to be announced analytic study group.

09 Jan 2019  Adam Morgan (Glasgow)

Title: Parity of Selmer ranks in quadratic twist families.

Abstract: We study the parity of 2Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we prove results about the proportion of twists having odd (resp. even) 2Selmer rank. This generalises work of Klagsbrun–Mazur– Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square.


16 Jan 2019  Helene Esnault (Freie Universität Berlin)

Title: Vanishing theorems for étale sheaves

Abstract: The talk is based on two results: Scholze’s Artin type vanishing theorem for the projective space, which I proved without perfectoid geometry (which implies in particular that it holds in positive characteristic), and a rigidity theorem for subloci of the ladic character variety stable under the Galois group over a number field (joint work in progress with Moritz Kerz).

Note: This seminal will take place in room 500.


23 Jan 2019  Adam Logan

Title: Automorphism groups of K3 surfaces over nonclosed fields

Abstract: Using the Torelli theorem for K3 surfaces of PyatetskiiShapiro and Shafarevich one can describe the automorphism group of a K3 surface over ${\mathbb C}$ up to finite error as the quotient of the orthogonal group of its Picard lattice by the subgroup generated by reflections in classes of square $2$. We will give a similar description valid over an arbitrary field in which the reflection group is replaced by a certain subgroup. We will then illustrate this description by giving several examples of interesting behaviour of the automorphism group, and by showing that the automorphism groups of two families of K3 surfaces that arise from Diophantine problems are finite. This is joint work with Martin Bright and Ronald van Luijk (University of Leiden).


30 Jan 2019  Jan Kohlhaase (Universität DuisburgEssen)

Title: Fourier analysis on universal formal covers

Abstract: The padic Fourier transform of Schneider and Teitelbaum has complicated integrality properties which have not yet been fully understood. I will report on an approach to this problem relying on the universal formal cover of a pdivisible group as introduced by Scholze and Weinstein. This has applications to the representation theory of padic division algebras.


06 Feb 2019  Mladen Dimitrov (Université de Lille)

Title: padic Lfunctions of Hilbert cusp forms and the trivial zero conjecture

Abstract: In a joint work with Daniel Barrera and Andrei Jorza, we prove a strong form of the trivial zero conjecture at the central point for the padic Lfunction of a noncritically refined cohomological cuspidal automorphic representation of GL(2) over a totally real field, which is Iwahori spherical at places above p. We will focus on the novelty of our approach in the case of a multiple trivial zero, where in order to compute higher order derivatives of the padic Lfunction, we study the variation of the root number in partial finite slope families and establish the vanishing of many Taylor coefficients of the padic Lfunction of the family.

Note: This seminal will take place in room 500.


13 Feb 2019  Yiannis Petridis (UCL)

Title: Symmetries and spaces

Abstract: It is a long established idea in mathematics that in order to understand space we need to study its symmetries. This is the centrepoint of the Erlangen program, which, published by Felix Klein in 1872 in Vergleichende Betrachtungen über neuere geometrische Forschungen, is a method of characterizing geometries based on group theory. In a group we can multiply, while on a space we can integrate. I will explore the link between the two starting with the mathematics of the seventeenth century and leading to the arithmetic of elliptic curves.

Note: Inaugural lecture


20 Feb 2019  Pankaj Vishe (Durham)

Title: Rational points over global fields and applications.

Abstract: We present analytic methods for counting rational points on varieties defined over global fields. The main ingredient is obtaining a version of HardyLittlewood circle method which incorporates elements of Kloosterman refinement in new settings.


27 Feb 2019  Martin Gallauer (Oxford)

Title: How many real ArtinTate motives are there?

Abstract: The goals of my talk are 1) to place this question within the framework
of tensortriangular geometry, and 2) to report on joint work with Paul
Balmer (UCLA) which provides an answer to the question in this
framework.


06 Mar 2019  Edgar Assing (Bristol)

Title: The supnorm problem over number fields.

Abstract: In this talk we study the supnorm of automorphic forms over number fields. This topic sits on the intersection of Quantum chaos, harmonic analysis and number theory and has seen a lot progress lately. We will discuss some of the recent result in the rank one setting.

Note: This seminal will take place in room 500.


13 Mar 2019  Jan Vonk (Oxford)

Title: Singular moduli for real quadratic fields and padic mock modular forms

Abstract: The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct padic analogues of singular moduli through classes of rigid meromorphic cocycles. I will discuss padic counterparts for our proposed RM invariants of classical relations between singular moduli and the theory of weak harmonic Maass forms.


20 Mar 2019  Alice Pozzi (UCL)

Title: The eigencurve at Eisenstein weight one points

Abstract: Coleman and Mazur constructed the eigencurve, a rigid analytic space classifying padic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is better understood at points corresponding to cuspforms of weight greater than 1, while the weight one case is far more intricate. In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. We focus on the unusual phenomenon of cuspidal Hida families specializing to Eisenstein series at weight one. We discuss the relation between the geometry of the eigencurve and the GrossStark Conjecture.
