## Seminare und Vorträge im SS 2017

### am Dienstag, 18. April:

#### Oberseminar Zahlentheorie

Marc-Hubert Nicole (Institut de mathématique de Marseille (I2M), Université d'Aix-Marseilles), The Gross-Kohnen-Zagier theorem in p-adic families
Abstract: The Gross-Kohnen-Zagier theorem tells us that the relative positions of Heegner points on a modular curve X0(N) are Fourier coefficients of a Jacobi form of weight two. Hida families give us the means to p-adically interpolate Heegner points and classical modular forms with respect to the weight. In this talk, I will explain the p-adic variation of the Gross-Kohnen-Zagier theorem, viewed as an illustration of the p-adic Kudla program. For concreteness, I will emphasize statements of classical flavour that are both new and natural from our point of view.
Joint work in progress with M. Longo (Padova).

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 25. April:

#### Oberseminar Zahlentheorie

Igor Shparlinski (University of New South Wales), Billinear forms with Kloosterman sums
Abstract: We start with a general introduction to the area, which is nowadays known as Klostermania, where the main goal is showing nontrivial cancellations between Kloosterman sums in some families.
We outline some new bounds on bilinear sums with Kloosterman sums and also with some similar sums. In particular, these bounds improve some recent results of V. Blomer, E. Fouvry, E. Kowalski, Ph. Michel and G. Milicevic (2014-2016). As a result we improve the error term in the asymptotic formula for mixed moments of L-series associated with Hecke eigenforms.
We also discuss further extensions of this method (jointly with Tianping Zhang and Kui Liu), which improve some recent results of R. Nunes (2016).
Finally, we outline some possible arithmetic applications of these bounds.

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 2. Mai:

#### Oberseminar Zahlentheorie

Florian Luca (Univerity of Witwatersrand-MPIM-Ostrava), Typical size and cancellations among coefficients of modular forms
Abstract: We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato-Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)|≤n11/2(log n)-1/2+o(1)) for a set of n of asymptotic density 1, where Ï„(n) is the Ramanujan Ï„ func- tion. In comparison, the standard argument, based on the estimate for the number of prime divisors of a typical integer n, only leads to the bound |τ(n)|≤n11/2(log n)2+o(1) for almost all n.
We also discuss other arithmetic properties of coefficients of L-functions of elliptic curves.
This is joint work with Radziwiłł and Shparlinski.

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 9. Mai:

#### Oberseminar Zahlentheorie

Gerard van der Geer (Amsterdam), Modular forms for genus 2 and 3
Abstract: The best-known modular forms are elliptic modular forms; a natural generalization are vector-valued Siegel modular forms of genus g. For genus 2 and 3 these modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such vector-valued modular forms for genus 2 and 3. This is based on joint work with Jonas BergstrÃ¶m, Carel Faber and with Fabien ClÃ©ry.

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 30. Mai:

#### Oberseminar Zahlentheorie

TBD

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 13. Juni:

#### Oberseminar Zahlentheorie

Gerard Freixas (C.N.R.S. - Institut de MathÃ©matiques de Jussieu), On an analytic class number formula for fuchsian groups
Abstract: In this talk I will provide a geometric statement in the spirit of the Riemann-Roch theorem in Arakelov geometry, that can be understood as an analytic class number formula for the Selberg zeta function of an arbitrary fuchsian group of the first kind. In particular, I will describe the arithmetic and geometric content of this formula for PSL_2(Z).

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 27. Juni:

#### Oberseminar Zahlentheorie

Noriko Yui (Queen's University - MPIM), Supercongruences for rigid hypergeometric Calabi–Yau threefolds
Abstract: We give two proofs to the supercongruences for the fourteen rigid hypergeometric Calabi– Yau threefolds defined over Q. The existence of such supercongruences was conjectured (based on numerical evidence) by F. Rodriguez-Villegas in 2003. This is a joint work with Ling Long, Fang-Ting Tu and Wadim Zudilin.

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 4. Juli:

#### Oberseminar Zahlentheorie

TBD

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 11. Juli:

#### Oberseminar Zahlentheorie

Fabian VÃ¶lz (TU-Darmstadt), Non-holomorphic Eisenstein series as theta lifts
Abstract: Generalising the concept of classical non-holomorphic Eisenstein series associated to cusps, one can define elliptic Eisenstein series associated to points in the upper-half plane, and hyperbolic Eisenstein series associated to geodesics. These Eisenstein series can also be understood as weight 0 analogs of Zagier's famous cusp forms associated to classes of quadratic forms (later generalised by Bengoechea for negative discriminants). In my talk I will present realizations of these elliptic and hyperbolic Eisenstein series as theta lifts of non-holomorphic PoincarÃ© series, which generalizes a classical result on lifts of holomorphic PoincarÃ© series.

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 18. Juli:

#### Oberseminar Zahlentheorie

Wadim Zudilin (University of Newcastle), A magnetic double integral
Abstract: In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, Udo Ausserlechner (Infineon Technologies Austria AG) has come up with a remarkable double integral that can be viewed as a generalization of the classical elliptic "AGM" integral. In my talk I will discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections to modular forms. This is joint work with David Broadhurst (Open University, UK); it is dedicated to the memory of Jonathan Borwein (1951 - 2016).

14:00 Übungsraum 2 des Mathematischen Instituts

### am Dienstag, 25. Juli:

#### Oberseminar Zahlentheorie

Bernhard Heim (RWTH Aachen-German University of Technology in Oman-MPIM), Periodic Holomorpic Functions and Functional Equations
Abstract: Let $f$ be a holomorphic function on the complex upper half space. Let $f$ be periodic with $f(\tau +1) = f(\tau)$. In the talk we introduce a new type of functional equations and show how functions $f$ satisfying such functional equations are related to modular forms for $\Gamma_0(p)$, $p$ prime.

14:00 Übungsraum 2 des Mathematischen Instituts

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