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Autumn Semester 2022

Date / Time Speaker Title Location
23 September 2022
14:15-15:15 		Prof. Dr. Vladimir Dokchitser
University College London
Event Details

Number Theory Seminar

Title The parity conjecture
Speaker, Affiliation Prof. Dr. Vladimir Dokchitser, University College London
Date, Time 23 September 2022, 14:15-15:15
Location HG G 43
Abstract The Birch-Swinnerton-Dyer conjecture gives a formula for the rank of an elliptic curve or an abelian variety in terms of its L-function. The parity conjecture is the corresponding simpler prediction for the parity of the rank. As I will explain, it is much easier to use in practice and there is a wealth of explicit arithmetic phenomena that it predicts. I will end by discussing some of the recent results on the conjecture in the context of abelian surfaces and Jacobians of curves.
The parity conjectureread_more
HG G 43
28 October 2022
14:15-15:15 		Radu Toma
University of Bonn
Event Details

Number Theory Seminar

Title On the sup-norm of automorphic forms in higher rank
Speaker, Affiliation Radu Toma, University of Bonn
Date, Time 28 October 2022, 14:15-15:15
Location HG G 43
Abstract The theory of quantum chaos predicts strong bounds for the sup-norm of eigenfunctions of the Laplacian on certain negatively curved spaces, for example quotients of the upper half plane. On arithmetic quotients, the original breakthrough of Iwaniec and Sarnak (1995), and the work of many since then, has lead to much progress in producing bounds for Hecke-Maass forms, uniform in the eigenvalue, the volume of the space, or both (hybrid bounds). In higher rank, despite many improvements in the eigenvalue aspect, there is a notable scarcity of results in the volume or level aspect. In this talk, I aim to describe a soft technique for tackling the counting problem at the heart of the Iwaniec-Sarnak method. It relies on algebraic rigidity arguments and produces the first hybrid bounds on a family of compact arithmetic spaces of unbounded rank associated to division algebras.
On the sup-norm of automorphic forms in higher rankread_more
HG G 43
11 November 2022
14:15-15:15 		Prof. Dr. William Duke
UCLA
Event Details

Number Theory Seminar

Title On the analytic theory of isotropic ternary quadratic forms
Speaker, Affiliation Prof. Dr. William Duke, UCLA
Date, Time 11 November 2022, 14:15-15:15
Location HG G 43
Abstract I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is given explicitly in terms of local data determined by the orbit. Comparison with the well-known asymptotic for the corresponding count of all primitive zeros yields information on the distribution of the zeros in different orbits, the number of orbits r and the class number h of the genus of the form. For a certain special class of forms the distribution is shown to be uniform and a simple explicit formula is given for hr.
On the analytic theory of isotropic ternary quadratic formsread_more (CANCELLED)
HG G 43
18 November 2022
14:15-15:15 		Prof. Dr. William Duke
UCLA
Event Details

Number Theory Seminar

Title On the analytic theory of isotropic ternary quadratic forms
Speaker, Affiliation Prof. Dr. William Duke, UCLA
Date, Time 18 November 2022, 14:15-15:15
Location HG G 43
Abstract I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is given explicitly in terms of local data determined by the orbit. Comparison with the well-known asymptotic for the corresponding count of all primitive zeros yields information on the distribution of the zeros in different orbits, the number of orbits r and the class number h of the genus of the form. For a certain special class of forms the distribution is shown to be uniform and a simple explicit formula is given for hr.
On the analytic theory of isotropic ternary quadratic formsread_more
HG G 43
25 November 2022
14:15-15:15 		Prof. Dr. Yingkun Li
TU Darmstadt
Event Details

Number Theory Seminar

Title Harmonic Maass forms associated to CM newforms
Speaker, Affiliation Prof. Dr. Yingkun Li, TU Darmstadt
Date, Time 25 November 2022, 14:15-15:15
Location HG G 43
Abstract Fourier coefficients of harmonic Maass forms contain interesting arithmetic information. However, their rationality is only known in a few cases, some of which are related to deep questions such as the vanishing of derivative of certain L-functions. In this talk, I will discuss recent joint work with Stephan Ehlen and Markus Schwagenscheidt, where we used theta lifts to prove sharp rationality results about Fourier coefficients of harmonic Maass forms associated to CM newforms.
Harmonic Maass forms associated to CM newformsread_more (CANCELLED)
HG G 43
2 December 2022
14:15-15:15 		Prof. Dr. Anna von Pippich
Universität Konstanz
Event Details

Number Theory Seminar

Title The arithmetic volume of the moduli space of abelian surfaces
Speaker, Affiliation Prof. Dr. Anna von Pippich, Universität Konstanz
Date, Time 2 December 2022, 14:15-15:15
Location HG G 43
Abstract Let A_g denote the moduli stack of principally polarized abelian varieties of dimension g. The arithmetic volume of the compactification of A_g is defined to be the arithmetic degree of the metrized Hodge bundle. In 1999, Kühn proved a beautiful formula for the arithmetic volume of the compactification of A_1 in terms of special values of the Riemann zeta function. In this talk, we report on joint work with Barbara Jung generalizing Kühn's result to the case g=2.
The arithmetic volume of the moduli space of abelian surfacesread_more
HG G 43
9 December 2022
14:15-15:15 		Prof. Dr. Jan Hendrik Bruinier
TU Darmstadt
Event Details

Number Theory Seminar

Title Generating series of special divisors on orthogonal Shimura varieties
Speaker, Affiliation Prof. Dr. Jan Hendrik Bruinier, TU Darmstadt
Date, Time 9 December 2022, 14:15-15:15
Location HG G 43
Abstract A famous theorem of Gross-Kohnen-Zagier states that the generating series of Heegner divisors on a modular curve is a weight 3/2 modular form with values in the first Chow group. An analogous result for special divisors on orthogonal Shimura varieties was proved by Borcherds, and for higher codimension special cycles by Zhang, Raum and myself. After recalling some of these results, we report on possible extensions to special cycles on toroidal compactifications of orthogonal Shimura varieties.
Generating series of special divisors on orthogonal Shimura varietiesread_more
HG G 43

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