London Number Theory Seminar

UCL King's Imperial College

The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL. Here is our diversity policy.

IMPORTANT: the organisers have decided to cancel the last two seminars and study groups (18th and 25th March) because of the ongoing COVID-19 situation.

This term (Spring 2020), the seminar will be hosted by UCL, and will be on Wednesdays at 4pm in room 505 of the maths department (25 Gordon St), starting on 15th Jan. The one exception: on 18th March it will be in 26 Bedford Way, room LG04. The organisers are Chris Birkbeck and Peter Humphries. Up to date details are here. The seminar will be preceded by tea/coffee in the 6th floor common room.

15 Jan 2020 - Matthew Bisatt (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: Gamma Functions, Monodromy, and Apéry Constants.
1) Recall of the theory of periods for local systems on curves.
2) Definition (V. Golyshev) of motivic gamma functions as Mellin transforms of period integrals.
3) Main theorem (joint with M. Vlasenko)
4) Application to the gamma conjecture in mirror symmetry (work of Golyshev + Zagier).

29 Jan 2020- Giada Grossi (UCL)
Title: The p-part 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 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.

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 non-square integer. When f has degree up to 4 (or when f is a product of integer linear polynomials) it has been shown that the Brauer-Manin obstruction is the only one to the Hasse principle. This is the result of decades of investigations by Swinnerton-Dyer, Colliot-Thélène, 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 L-functions
Abstract: Distribution of values of L-functions on the critical line, or more generally central values in families of L-functions, 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 sub-power rate of growth. Soundararajan's method of resonators and its recent improvement due to Bondarenko-Seip are flexible first moment methods that unconditionally show that zeta(1/2+it), or central values of other degree one L-functions, achieve very large values.
In this talk, we address large central values L(1/2, f x chi) of a fixed GL(2) L-function twisted by Dirichlet characters chi to a large prime modulus q. We show that many of these twisted L-functions 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, ready-to-use 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 L-functions with Blomer, Fouvry, Kowalski, Michel, and Sawin.

04 Mar 2020 - Asbjørn Nordentoft (Copenhagen)
Title: The distribution of modular symbols and additive twists of L-functions
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 L-functions (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)
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).

25 Mar 2020 - Chris Lazda (Warwick University)
Title: A Neron-Ogg-Shafarevich criterion for K3 surfaces
Abstract: The naive analogue of the Néron-Ogg-Shafarevich 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 semi-stable 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.

The seminar will be preceded by various study groups.

A list of previous seminar talks is here.

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