## Seminare und Vorträge im WS 2016/2017

### am Montag, 17. Oktober:

#### Oberseminar Zahlentheorie

Fabian Clery (Max Planck Institute, Siegen), Construction of vector-valued Siegel modular forms of degree 2
Abstract: We will explain how to construct vector-valued Siegel modular forms of degree 2 by using the invariant theory of binary sextics. All along the talk, we will stress on concrete examples to show the efficiency of this method. This is a joint work with Carel Faber and Gerard van der Geer.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 24. Oktober:

#### Oberseminar Zahlentheorie

Christian Kassel (Universite de Strasbourg), The Hilbert scheme of n points on a torus and modular forms
Abstract: In recent joint work with Christophe Reutenauer (UQAM), we explicitly computed the zeta function of the Hilbert scheme of n points on a two-dimensional torus. The computation involves a family of polynomials with nice properties: they are palindromic, their coefficients are non-negative integers and their values at 1 and at roots of unity of order 2, 3, 4 and 6 can be expressed in terms of well-known modular forms.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 31. Oktober:

#### Oberseminar Zahlentheorie

Giacomo Cherubini (MPIM), Mean square bounds in the prime geodesic theorem
Abstract: I will start with explaining what is the prime geodesic theorem emphasizing analogies and differences with the prime number theorem. I will also mention what is the currently best known pointwise bound on the remainder, due to Soundararajan and Young. Then I will explain an ongoing work to obtain upper bounds on the mean square of the remainder.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 7. November:

#### Oberseminar Zahlentheorie

Michael Griffin (Köln), P-adic harmonic Maass functions and lifts to integer weight forms
Abstract: In his paper “On traces of singular moduli”, Zagier showed that twisted traces of modular functions over CM elliptic curves give rise to coefficients of modular forms of half-integer weight. This works was extended to harmonic Maas forms of higher weights and level by Miller–Pixton by studying formulas of Maass-Poincaré series, and by Bruinier–Funke and others who realized these lifts as theta lifts. Duke–Jenkins studied integrality results of such lifts, and their work and that of Zagier shows the existence of striking p-adic properties. In general Harmonic Maass forms have presumably transcendental coefficients. However the transcendence can be controlled, and the forms demonstrate properties connected to p-adic modular forms, as studied for instance by Guerzhoy–Kent–Ono, Candelori, and others. Some similar properties of half-integral weight forms have also recently been studied by Bringmann–Guerzhoy–Kane. These properties appear to respect the lifts, suggesting the possibility of a p-adic theory of theta lifts. However, CM traces do not make sense over general p-adic modular forms which may not be defined for supersingular elliptic curves. We address this issue by constructing families of p-adic functions analogous to harmonic Maass functions which converge on the full supersingular locus. These functions are constructed by means of the Hecke algebra. The lifts described above extend to these p-adic harmonic Maass forms in a natural way, and the results interpolate the coefficients of half-integer weight forms.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 21. November:

#### Oberseminar Zahlentheorie

Valentin Blomer (Universität Göttingen), Small gaps in the spectrum of the rectangular billiard
Abstract: We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio alpha, in comparison to the corresponding quantity for a Poissonian sequence. The proofs use various tools from elementary number theory, diophantine approximation and analytic number theory. This is joint work with Bourgain, Radziwill and Rudnick.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 21. November:

#### Oberseminar Zahlentheorie

Siddarth Sankaran, Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet-Langlands correspondence.
Abstract: This is joint work with Gerard Freixas, in which we compute and compare arithmetic intersection numbers on Shimura varieties attached to inner forms of GL(2) over a real quadratic field. We'll see how to compute such an intersection on a twisted version of a Hilbert modular surface, and how the arithmetic Riemann-Roch formula intervenes to generate formulas for arithmetic volumes of Shimura curves.

14:00 Stefan Cohn-Vossen Raum des Mathematischen Instituts

### am Montag, 28. November:

#### Oberseminar Zahlentheorie

Kathrin Maurischat (Uni-Heidelberg), On Sturm's operator and holomorphic projection
Abstract: Sturm's operator realizes the holomorphic projection of L2-functions or of (non-holomorphic) modular forms to there holomorphic spectral shares in terms of Fourier coefficients. We recall well-known results for large weight for Siegel modular forms and give new results for smaller weights. We show that there are cases in which Sturm's operator fails. We also give parallel results for vector valued weights.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 5. December:

#### Oberseminar Zahlentheorie

Árpád Toth (Eötvös Loránd University), Extensions of the Katok-Sarnak formula
Abstract: The Katok-Sarnak formula expresses the cycle integral of the Shimura lift of a weight 1/2 Maass form in terms of the Fourier coefficients of the form. I will talk about an extension of this formula. This is joint work with Duke and Imamoglu.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 12. December:

#### Oberseminar Zahlentheorie

Lola Thompson (Oberlin, MPIM), Prime Gaps
Abstract: We will give an overview of the exciting 2013 papers on bounded gaps between primes and the results that grew out of them. We will examine “bounded gaps” theorems in various settings and discuss applications to other problems in number theory. Parts of this talk are based on joint work with subsets of the following co-authors: Abel Castillo, Chris Hall, Robert J. Lemke Oliver, and Paul Pollack.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 19. December:

#### Oberseminar Zahlentheorie

Jan Bruinier (TU-Darmstadt), Generating series of special divisors on arithmetic ball quotients
Abstract: We report on joint work with B. Howard, S. Kudla, M. Rapoport, and T. Yang. A celebrated result of Hirzebruch and Zagier states that the generating series of Hirzebruch-Zagier divisors on a Hilbert modular surface is an elliptic modular form with values in the cohomology. The goal of this talk is to prove an analogue for special divisors on integral models of ball quotients. In this setting the generating series takes values in an arithmetic Chow group. If time permits, we discuss some applications to arithmetic theta lifts and the Colmez conjecture.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Mittwoch, 4. Januar:

#### Oberseminar Zahlentheorie

Caner Nazaroglu (Univerity of Chicago), Indefinite Theta Series
Abstract: Theta functions for definite signature lattices constitute a rich source of modular forms and arise in physics and mathematics in various contexts. Theta series for indefinite signature lattices and their modular completions are first worked out in the signature (n-1,1) case by Zwegers in a seminal work. More recently, this construction is generalized to generic signatures and the main topic of this talk will be such generalizations. I am going to first review ordinary theta series both to set the stage and to draw attention to properties relevant to their different signature generalizations. I will then describe their generic signature generalizations and point out how they give rise to higher degree mock modular forms. I will lastly give examples from mathematics and physics where such functions appear.

14:00 Seminarraum 3 des Mathematischen Instituts

### am Montag, 9. Januar:

#### Oberseminar Zahlentheorie

Fredrik Strömberg (Nottingham), Dimension formulas for vector-valued Hilbert modular forms
Abstract: I will present results from joint work with Nils Skoruppa on explicit dimension formulas for vector-valued Hilbert modular forms. In particular I will discuss some of the most interesting computational and theoretical challenges in the evaluation of the different terms.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 16. Januar:

#### Oberseminar Zahlentheorie

Jitendra Bajpal (Göttingen), Vector Valued Modular Forms
Abstract: All of the famous modular forms (e.g. Dedekind eta function) involve a multiplier, this multiplier is a 1-dimensional representation of the underlying group. This suggests that a natural generalization will be matrix valued multipliers, and their corresponding modular forms are called vector valued modular forms. These are much richer mathematically and more general than the (scalar) modular forms. In this talk, classification of vector valued modular forms for any genus zero Fuchsian group of the first kind will be discussed. The connection between vector valued modular forms and Fuchsian differential equations will be explained.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 23. Januar:

#### Oberseminar Zahlentheorie

Joseph Oesterle (Institut Mathématiques de Jussieu-Paris Rive Gauche), Multiple zeta values and Apéry-like sums
Abstract: Multiple zeta values are the numbers $\sum _{{n}_{1}>\cdots >{n}_{r}>0}{n}_{1}^{-{a}_{1}}\cdots {n}_{r}^{-{a}_{r}}$ and multiple Apéry-like sums the numbers $\sum _{{n}_{1}>\cdots >{n}_{r}>0}{\left(\begin{array}{c}2{n}_{1}\\ {n}_{1}\end{array}\right)}^{-1}{n}_{1}^{-{a}_{1}}\cdots {n}_{r}^{-{a}_{r}}$ where a1 … ar are positive integers, with a1 > 1. I shall give an account of the following theorem, recently proved under my supervision by Akhilesh Parol: any multiple zeta values can be expressed as a Z-linear combination of multiple Apéry-like sums. Moreover, with an additional requirement about tails of the series, this expression is unique (and explicit). The simplest example is Euler’s famous formula $\sum _{n>0}{n}^{-2}=3\sum _{n>0}{\left(\begin{array}{c}2n\\ n\end{array}\right)}^{-1}{n}^{-2}.$

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 30. Januar:

#### Oberseminar Zahlentheorie

No Seminar.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Montag, 6. Februar:

#### Oberseminar Zahlentheorie

Jens Funke (University of Durham), Indefinite theta series
Abstract: In this talk, we discuss recent developments in the construction of indefinite theta series from a geometric point of view. This generalises work of Zwegers to higher weight and arbitrary signature.

11:30 Übungsraum 2 des Mathematischen Instituts

### am Mittwoch, 15. Februar:

#### ABKLS Seminar

14:00-18:00 ENC-D 224

### am Dienstag, 21. Februar:

#### Oberseminar Zahlentheorie

Jonathan Schürr, Some twisted denominator identities of the fake monster algebra
Abstract: We describe lattices and discriminant forms (especially the Leech lattice and the fixed point sublattices), vector valued modular forms for the Weil representation of SL(2,Z), eta products and theta series. These are used in the construction of twisted denominator identities as automorphic products of singular weight for the fake monster algebra. This is done with a method of direct construction from Scheithauer. It starts with classical modular forms associated to the specific group element. Those are then lifted to vector valued modular forms on a discriminant form associated to the fixed point lattice. Borcherds singular theta correspondence then gives rise to the automorphic product we looked for.

11:00 Stefan Cohn-Vossen Raum des Mathematischen Instituts (Raum 313)

### am Donnerstag, 2. März:

#### Oberseminar Zahlentheorie

Peng Yu (University of Wisconsin), CM values of special functions on Shimura varieties of orthogonal type
Abstract: Gross-Zagier formula has inspired a lot of interesting work. One of them is the study of CM values of special functions on Shimura varieties. I will first define special CM cycles on Shimura variety of orthogonal type and show how to compute CM values of automorphic Green functions associated to these cycles. The theorem can be further extended to a proof of a generalized Gross-Zagier formula that relates Faltings heights of CM cycles with the central derivative of a Rankin-Selberg convolution L-function. Its application to Siegel 3-fold case gives CM values of even Siegel theta functions and Rosenhain λ-invariants, which has practical applications to genus two curve cryptography.

14:00-15:00 Seminarraum 3 des Mathematischen Instituts (Raum 314)

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