## Upcoming seminars

**Abstract:**We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form of degree k and one general form of degree d, where d is smaller than k. By employing a hybrid method that combines ideas from the study of general forms with techniques adapted to the diagonal case, we are able to obtain bounds that grow exponentially in d but only quadratically in k, reflecting the growth rates typically obtained for both problems separately. Time permitting, we may also discuss some of the most interesting generalisations of our approach. This is joint work with Scott T. Parsell.

## Past seminars

18.11.2020 - Tim Browning (IST Austria): Poor bounds for a rich problem

Abstract: Thanks to Neron heights and the Mordell-Weil theorem we can count rational points of bounded height on elliptic curves. Falting's theorem tells us that there are only finitely many rational points (of any height) on hyperelliptic curves of genus at least 2. Unfortunately, these estimates depend intimately on the individual curve and it is hard to apply them to study the corresponding problem for surfaces that are fibred into elliptic or hyperelliptic curves. In this talk I shall discuss an alternative approach that uses sieves, though the bounds are far from optimal! This is joint work with Dante Bonolis.

04.11.2020 - John Christian Ottem (University of Oslo): Enriques surface fibrations of even index

Abstract: I will explain a geometric construction of an Enriques surface fibration over P1 of even index. This answers a question of Colliot-Thèlene and Voisin, and provides new counterexamples to the integral Hodge conjecture. This is joint work with Fumiaki Suzuki.

28.10.2020 - Scott Ahlgren (University of Illinois, Urbana-Champaign): OBS! 14.30-15.30

Title: Congruences for the partition function

21.10.2020 - Peter Sarnak (Princeton University/Institute for Advanced Study): Applications of points on subvarieties of tori

Abstract: The intersection of the division group of a finitely generated subgroup of a torus with an algebraic subvariety has been understood for some time (Lang, Laurent,...). After a brief review of some of the tools in the analysis and their recent extensions (André-Oort conjectures), we give some old and new applications; in particular to the additive structure of the spectra of metric graphs and crystalline measures.

Joint work with P. Kurasov.

07.10.2020 - Jack Shotton (University of Durham): Moduli of local Galois representations and representation theory

24.09.2020 - Jan Gerken (Chalmers/GU): Single-valued maps at genus zero and one

Abstract: The single-valued map for for multiple zeta values (MZVs) due to Francis Brown and Oliver Schnetz is an intriguing algebra homomorphism for MZVs. Surprisingly, it appears in the leading contribution to scattering amplitudes in string theory, relating open- and closed-string amplitudes. Since MZVs are periods of configuration spaces of punctured genus-zero Riemann surfaces, a natural next step is the generalization of the single-valued map to genus-one surfaces. In string theory, these correspond to the subleading contributions to the scattering amplitudes. Using the structures provided by string theory, we propose a genus-one generalization of the single-valued map for MZVs which acts on the level of generating functions of genus-one periods.

Abstract: Tensor hierarchy algebras constitute a new class of non-contragredient Lie superalgebras, whose finite-dimensional members are the simple Lie superalgebras of Cartan type in Kac’s classification. They have proven useful in describing gauge structures in physical models related to string theory. I will review their construction by generators and relations and some of the remarkable features they exhibit.

*L*-functions

Abstract: In an unpublished paper from 2007, Conrey discovered certain ‘reciprocity relations’ satisfied by twisted moments of Dirichlet L-functions, linking the arithmetics of the finite fields F_p, F_q for two different primes p,q (as is the case with quadratic reciprocity). In this talk I will discuss a generalization to twisted moments of twisted modular L-functions. This will lead to a discussion of the notion of quantum modular forms due to Zagier, and in particular we will explain that additive twists of modular L-functions define examples of quantum modular forms.

Abstract: Hurwitz class numbers, class numbers of imaginary quadratic fields, and partition counts are among the most classic quantities in number theory, and for each of them their factorizations, i.e. divisibilities, are celebrated open questions. In the case of class numbers the Cohen-Lehnstra Heuristics provides predictions of of statistical nature. In the case of partition counts, Ramanujan congruences opened the door to a whole new research area in~1920.

We survey recent progress on divisibilities of class numbers and partition counts on arithmetic progressions. These result rely on a two new methods exploiting the finer structure of Fourier coefficients of real-analytic and meromorphic modular forms.

The project on class numbers is partially based on joint work with Olivia Beckwith and Olav Richter. The project on partition counts is partially based on joint work with Olivia Beckwith and Scott Ahlgren.

Abstract: There are many difficult conjectures about automorphic representations, many of which seem to be out of reach at the moment. It has therefore become increasingly popular to study instead families of automorphic representations and their statistical properties, which allows for additional analytic techniques to be used.

In my talk I want to discuss the distribution of Hecke eigenvalues or, in other words, Satake parameters in the family of spherical unramified automorphic representations of split classical groups. We obtain an effective distribution of the Satake parameters, when we order the family according to the size of analytic conductor. This has applications to various questions in number theory, for example, low-lying zeros in families of automorphic L-functions, but also yields an effective Weyl law for the underlying locally symmetric space. This is joint work with T. Finis.

Abstract: I will report on ongoing work, where I apply the Lefschetz fixed point theorem to local systems on the moduli space of abelian varieties of dimension at most 3, and use simple equalities in modular arithmetic, to study traces of Hecke operators on spaces of Siegel modular forms (of degree at most 3) modulo prime powers.

Abstract: Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi--Yau variety X, and some invariants extracted from a mirror family of Calabi--Yau varieties. After the physicists Bershadsky--Cecotti--Ooguri--Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems, and present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck--Riemann--Roch. I will explain a proof of the conjecture for Calabi--Yau hypersurfaces in projective space, based on the Riemann--Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang--Lu--Yoshikawa.