## Upcoming

11.12.2019 - Lilian Matthiesen (KTH): Correlations of multiplicative functions and rational points in families

Abstract: In the first part of this talk I will discuss correlations of the form sum_{n,d < x} h_1(n) h_2(n+d) ... h_{r+1}(n+rd) (and generalisations thereof) for multiplicative functions h_1, ..., h_{r+1}. In particular, I will describe a set of conditions under which such correlations can be evaluated asymptotically as well as examples of functions satisfying these conditions. The second part of the talk will be about joint work with Dan Loughran on a question of Serre. By combining the analytic result on multiplicative functions mentioned above with ideas from algebraic geometry, we obtain (under suitable conditions) correct-order lower bounds for the number of varieties in a family over Q which have a rational point.

Abstract: In the first part of this talk I will discuss correlations of the form sum_{n,d < x} h_1(n) h_2(n+d) ... h_{r+1}(n+rd) (and generalisations thereof) for multiplicative functions h_1, ..., h_{r+1}. In particular, I will describe a set of conditions under which such correlations can be evaluated asymptotically as well as examples of functions satisfying these conditions. The second part of the talk will be about joint work with Dan Loughran on a question of Serre. By combining the analytic result on multiplicative functions mentioned above with ideas from algebraic geometry, we obtain (under suitable conditions) correct-order lower bounds for the number of varieties in a family over Q which have a rational point.

## Past seminars

27.11.2019 - Jan Gerken (Max Planck Institute for Gravitational Physics Potsdam): Differential equations for modular graph forms

Abstract: I will give a review of so-called modular graph forms, graph-labelled non-holomorphic modular forms that arise in the calculation of scattering amplitudes in string theory. Modular graph forms are defined as lattice sums and satisfy a host of algebraic and differential equations which were studied by Don Zagier, Francis Brown and others. In particular, these differential equations allow for a representation of modular graph forms in terms of iterated integrals over holomorphic Eisenstein series and this representation is assumed to be the correct

language to formulate an elliptic generalization of Brown’s single-valued map of multiple zeta values. Time permitting, I will introduce new generating functions for modular graph forms that satisfy Cauchy-Riemann and Laplace equations and hence have the potential to make the translation from lattice sums into iterated Integrals much easier. Based on arXiv:1911.03476 .Abstract: I will give a review of so-called modular graph forms, graph-labelled non-holomorphic modular forms that arise in the calculation of scattering amplitudes in string theory. Modular graph forms are defined as lattice sums and satisfy a host of algebraic and differential equations which were studied by Don Zagier, Francis Brown and others. In particular, these differential equations allow for a representation of modular graph forms in terms of iterated integrals over holomorphic Eisenstein series and this representation is assumed to be the correct

14.11.2019 - Marcus Berg (Karlstad): Massive deformations of Maass forms and Jacobi forms

Abstract: I review motivations and results from arXiv:1910.02745 in mathematical physics and number theory. Our main example is a one-parameter deformation of Kronecker-Eisenstein series.

30.10.2019 - Cecilia Salgado (Universidade Federal do Rio de Janeiro and MPIM Bonn): Mordell-Weil rank jumps and the Hilbert property

Abstract: Let X be an elliptic surface with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.

Abstract: Let X be an elliptic surface with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.

16.10.2019 - Anders Södergren (Chalmers/GU): Non-vanishing of cubic Dedekind zeta functions

Abstract: In this talk I will discuss the first steps towards understanding the amount of non-vanishing at the central point of cubic Dedekind zeta functions. In particular I will describe some of the challenges we face in trying to generalize well-known results for quadratic Dirichlet L-functions to the cubic case. This is work in progress with Arul Shankar and Nicolas Templier.

Abstract: In this talk I will discuss the first steps towards understanding the amount of non-vanishing at the central point of cubic Dedekind zeta functions. In particular I will describe some of the challenges we face in trying to generalize well-known results for quadratic Dirichlet L-functions to the cubic case. This is work in progress with Arul Shankar and Nicolas Templier.

02.10.2019 - Alice Pozzi (University College London): Rigid meromorphic cocycles at real multiplication points

Abstract: A rigid meromorphic cocycle is a class in the first cohomology of the group SL2(Z[1/p]) acting on the non-zero rigid meromorphic functions on the Drinfeld p-adic upper half plane by Möbius transformations. Rigid meromorphic cocycles can be evaluated at points of real multiplication, and their RM values conjecturally lie in the ring class field of real quadratic fields, suggesting a striking analogy with the classical theory of complex multiplication. In this talk, we discuss a special case of the conjecture, relating the RM value of the “Eisenstein” Dedekind-Rademacher cocycle to a Gross-Stark unit. We explain the connection with certain deformations of Hilbert Eisenstein series of weight one. This is work in progress with Henri Darmon and Jan Vonk.

Abstract: A rigid meromorphic cocycle is a class in the first cohomology of the group SL2(Z[1/p]) acting on the non-zero rigid meromorphic functions on the Drinfeld p-adic upper half plane by Möbius transformations. Rigid meromorphic cocycles can be evaluated at points of real multiplication, and their RM values conjecturally lie in the ring class field of real quadratic fields, suggesting a striking analogy with the classical theory of complex multiplication. In this talk, we discuss a special case of the conjecture, relating the RM value of the “Eisenstein” Dedekind-Rademacher cocycle to a Gross-Stark unit. We explain the connection with certain deformations of Hilbert Eisenstein series of weight one. This is work in progress with Henri Darmon and Jan Vonk.

26.09.2019 - Wee Teck Gan (National University of Singapore): Triality and Functoriality

Abstract: I will discuss some implications of the existence of the triality automorphism in the Langlands program, such as the demonstration of certain instances of functorial lifting. This is joint work with Gaetan Chenevier.

Abstract: I will discuss some implications of the existence of the triality automorphism in the Langlands program, such as the demonstration of certain instances of functorial lifting. This is joint work with Gaetan Chenevier.

25.09.2019 - Michael Björklund (Chalmers/GU): The quest for quasi-theta and the dreams of what lies beyond

Abstract: In the last year or so, Tobias Hartnick (Giessen/Karlsruhe) and I have tried to make sense of what a theta function associated to quasicrystal should be. We still do not know, but we have some ideas, and these will be discussed during the talk. No prior knowledge about quasicrystals (or theta functions) will be assumed.

Abstract: In the last year or so, Tobias Hartnick (Giessen/Karlsruhe) and I have tried to make sense of what a theta function associated to quasicrystal should be. We still do not know, but we have some ideas, and these will be discussed during the talk. No prior knowledge about quasicrystals (or theta functions) will be assumed.

11.09.2019 - Ezra Waxman (Technische Universität Dresden): Hecke Characters and the L-Function Ratios Conjecture

Abstract: A Gaussian prime is a prime element in the ring of Gaussian integers Z[i]. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Hecke proved that the Gaussian primes are equidistributed across sectors of the complex plane by making use of Hecke characters and their associated L-functions. In this talk I will present several applications obtained upon applying the L-functions Ratios Conjecture to this family of L-functions. In particular, I will present a refined conjecture for the variance of Gaussian primes across sectors, and a conjecture for the one level density across this family.

15.05.2019 - Olav Richter (University of North Texas): Ramanujan congruences for modular forms

Abstract: Congruences in the coefficients of automorphic forms have been the subject of much study. In this talk, I will discuss Ramanujan congruences for modular forms, which generalize Ramanujan’s original congruences for the partition function p(n).

Abstract: Congruences in the coefficients of automorphic forms have been the subject of much study. In this talk, I will discuss Ramanujan congruences for modular forms, which generalize Ramanujan’s original congruences for the partition function p(n).

08.05.2019 - Gergely Harcos (Rényi Institute of Mathematics and Central European University): On the global sup-norm of GL(n) cusp forms

Abstract: The sup-norm problem, a close relative of the subconvexity problem, has been studied widely for the group GL(2). There are fewer results for GL(n), and it was only recently that an explicit upper bound was derived for the global sup-norm of unramified Maass cusp forms there. I will discuss this result, which is joint work with Valentin Blomer and Peter Maga.

24.04.2019 - Matthew De Courcy-Ireland (École Polytechnique Fédérale de Lausanne): Kesten-McKay law for the Markoff surface mod p

Abstract: The Markoff surface is a cubic surface with the special feature that it is only quadratic in each variable separately. Its solutions modulo a prime number form a 3-regular graph where the edges correspond to exchanging the two roots of such a quadratic. We show that for increasingly large primes, the eigenvalues of these graphs asymptotically follow the same law as the eigenvalues of a random 3-regular graph. The proof is based on the method of moments and takes advantage of an action of GL(2,Z) on the Markoff surface. This is joint work with Michael Magee.

17.04.2019 - Patrick Meisner (KTH): Low-Lying Zeros and One Level Density of Families of L-Functions

Abstract: Understanding the zeros of L-functions is fundamental in solving many arithmetic problems. We define the one level density of an L-function to be the sum of all f(t) where the sum runs over all zeros 1/2+it of an L-function and f is an even Schwartz function. Katz and Sarnak predict that for nice families of L-functions, the one-level density should be controlled by random matrices. We will discuss the average of these one level densities for several families of L-functions defined over number fields and function fields.

10.04.2019 - Nancy Abdallah (Chalmers/GU): Cominbatorial invariance of KLV polynomials for fixed point free involutions

Abstract: Let S_2n be the symmetric group of permutations of {1,...,2n}, and let F_n be the subset of fixed point free involutions. To every interval [u,v] in the poset F_n ordered by the Bruhat order, we associate a KLV polynomial P_{u,v}. Using a combinatorial concept called Special Partial Matching or SPM, we prove that these polynomials are combinatorially invariant for upper intervals , i.e. for intervals [u,w_0] where w_0 is the maximal element of F_n. This gives a generalization of combinatorial invariance of the classical Kazhdan-Lusztig polynomials in the symmetric case. This is a joint work with Axel Hultman.

03.04.2019 - Caihua Luo (Chalmers/GU): Muller type irreducibility criterion for generalized principal series

Abstract: One of the aspects of the Langlands program concerns harmonic analysis, especially the analysis of the constituents of parabolic inductions. The first natural question is to see when an induced representation is irreducible. About 40 years ago, I. Muller gave a clean criterion for principal series. Building upon her work, we extend her criterion to generalized principal series. If time permits, we will also discuss an ambitious irreducibility conjecture for parabolic inductions.

27.03.2019 - Jörg Jahnel (University of Siegen): On integral points on open degree four del Pezzo surfaces

Abstract: I will report on investigations, together with Damaris Schindler (Utrecht), concerning algebraic and transcendental Brauer-Manin obstructions to integral points on complements of a hyperplane section in

degree four del Pezzo surfaces. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing zero over Q and over Z_p for all primes p, but not over Z.

degree four del Pezzo surfaces. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing zero over Q and over Z_p for all primes p, but not over Z.

06.03.2019 - Olga Balkanova (Chalmers/GU): Mixed moments of GL(2) and GL(3) L-functions

Abstract: We will discuss various techniques and ideas behind asymptotic analysis of the mixed moments of GL(2) and GL(3) L-functions, with special emphasis on the saddle point method and the Liouville-Green approximation. This is joint work with G. Bhowmik, D. Frolenkov and N. Raulf.

20.02.2019 - Beth Romano (University of Cambridge): Arithmetic Statistics via graded Lie algebras

Abstract: I will talk about recent work with Jack Thorne in which we find the average size of the Selmer group for a family of genus-2 curves by analysing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.

13.02.2019 - James Newton (King's College London): Potential automorphy over CM fields and applications

Abstract: I will discuss joint work with Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Scholze, Taylor and Thorne that establishes potential automorphy results for certain compatible systems of Galois representations over CM fields. This has applications to the Sato-Tate conjecture for elliptic curves over CM fields and the Ramanujan conjecture for weight zero cohomological automorphic representations of GL(2) over CM fields.

06.02.2019 - Morten Risager (University of Copenhagen): Counting lattices and geodesics

Abstract: The circle problem seeks to estimate the number of points of an orbit Gz inside a disc of growing radius R, where G acts discontinuously on a metric space X. Analogously the prime geodesic theorem seeks to estimate the number of prime geodesics on a Riemann surface with length less than R. We review various results connected to these problems in the arithmetic case emphasizing the connection to a conjecture on a spectral exponential sum. This is work in progress.

23.01.2019 - Jan Stevens (Chalmers/GU): The space of twisted cubics

Abstract: A twisted cubic is a smooth rational curve of degree three in projective 3-space. It is unique up to a change of coordinates, so there exists a 12 dimensional family of such curves in 3-space. A natural compactification is provided by the Hilbert scheme of space curves with Hilbert polynomial 3t+1. This Hilbert schemes has two smooth components, the smallest one of which contains the twisted cubics as an open subset. We describe a compactification with good geometric properties, that represents a functor. It is given by the space of Cohen-Macaulay curves, as introduced by Hønsen, parametrising Cohen-Macaulay curves together with a finite morphism to projective space which is assumed to be an isomorphism outside a finite number of points.