# Seminar of the Algebra Group: Winter term 2018/19

## Relations between Fourier coefficients of Siegel modular forms

Tuesday, October 16, 2018, 15:15 – 16:15, Room S2|15 401

**Speaker:** Jolanta Marzec (TU Darmstadt)

**Abstract:** Fourier coefficients of Siegel modular forms have played a prominent role in number theory. For example, in case of classical modular forms (degree $n=1$), the coefficients are equal to Hecke eigenvalues and are linked to a number of rational points on suitable elliptic curves. In the case of Siegel modular forms of degree $2$ they carry information about certain central critical values of quadratic twists of associated spinor $L$-functions (B\"ocherer's conjecture). Of special importance are \emph{fundamental Fourier coefficients}: their non-vanishing is related to existence of `nice' models for the associated global automorphic representations, and consequently enabled to prove analytic and algebraic properties for $L$-functions for $\mathrm{GSp}_4\times\mathrm{GL}_2$.

Even though a lot of work has been devoted to Siegel modular forms of degree $2$, their Fourier coefficients – especially in case of non-trivial level – lack good understanding. Therefore, motivated by the properties described above and the problem of determination of Siegel modular forms by an interesting subset of Fourier coefficients, we derive relations between `simple' and `more complicated' Fourier coefficients, and indicate their applications.

## CM values, cycle integrals, and polyharmonic Maass forms

Thursday, October 25, 2018, 15:20 – 17:00, Room S2|15 244

**Speaker: **Toshiki Matsusaka (Kyushu University / Universität zu Köln)

**Abstract**: In 2011 Duke-Imamoglu-Toth showed the generating functions of traces of cycle integrals and CM values of modular functions are (poly)harmonic Maass forms. I give a generalization of their work to polyharmonic Maass forms.

## Applications of sieve methods

Tuesday, October 30, 2018, 15:15 – 16:15, Room S2|15 401

**Speaker:** Christian Elsholtz (TU Graz)

**Abstract:** Apart from the ancient sieve of Eratosthenes, modern sieve theory started in 1915, when Viggo Brun showed that there are not too many twin primes, so that the sum of reciprocals 1/5+1/7+1/11+1/13 +1/17+1/19+… is finite, or convergent. In fact, it can be proved that for any fixed (a_1, …, a_k) the number of k-tuples of primes (n+a_1,n+a_2,…, n+a_k) with n <x is at most C_{k,a_1,…,a_k} x/(log x)^k. Using the large sieve, one can analyze what happens, when k is not a constant. If for example k=log x, one can give the following upper bound for the number of such tuples: O(x/exp( log x log log log x/log log x)). When k is (log x)2, onbe gets the dramatically better bound O(x^(2/3)), and when k increases this approaches x^(1/2). From this one can conclude that the set of primes cannot be written as a sumset P=A+B+C, even if one allows finitely many mistakes. In this talk we give a survey of such upper bounds for the prime k-tuple problem in its full range, from very small k to very large k.

## Newform theory from Jacobi forms of lattice index

Tuesday, November 27, 2018, 14:00 – 15:00, Room S2|15 401

**Speaker:** Andreea Mocanu (University of Nottingham)

**Abstract:** I will give a brief introduction to Jacobi forms, including some examples and their relation to other types of modular forms. After that, I will discuss some of the ingredients that go into developing a theory of newforms for Jacobi forms of lattice index, namely Hecke operators, level raising operators and orthogonal groups of discriminant modules.

## Non-existence of reflective modular forms

Tuesday, December 18, 2018, 15:15 – 16:15, Room S2|15 401

**Speaker**: Haowu Wang (Université Lille)

**Abstract:** Reflective modular forms are holomorphic functions whose divisors are determined by rational quadratic divisors associated to reflective vectors. This type of modular forms has applications in arithmetic, geometry and Lie algebra. The classification of reflective modular forms is an old open problem and has been investigated by several mathematicians. In this talk, I introduce an approach based on the theory of Jacobi forms to classify reflective modular forms on lattices of general level. I give an explicit formula to express the weight of reflective modular forms. I also prove two non-existence results. The first one is that there is no lattice of signature (2, n) having a 2-reflective modular form when n > 14 with three exceptions. The second is that there is no lattice of signature (2, n) admitting a reflective modular form when n >22 except the unimodular lattice of signature (2,26).

## Automorphic representations of symplectic groups

Tuesday, January 8, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker:** Jolanta Marzec (TU Darmstadt)

**Abstract:** How to associate an automorphic representation of GSp(2n) to a Siegel modular form of degree n. Tomorrow we will focus on spherical representations.

## p-adic dynamics of Hecke operators

Friday, January 11, 2019, 14:15 -15:15, Room S2|15 401

**Speaker:** Eyal Goren (McGill University)

**Abstract:** Let p and q be distinct primes. In a joint work with Payman Kassaei (Kingâ€™s College) we are studying questions concerning the dynamics of the Hecke operator T_q in its action on the modular curves X_1(N), where (N, pq) = 1, in the p-adic topology. Using Serre-Tate coordinates and isogeny volcanoes for points of X_1(N) with ordinary reduction, and using the graphs of supersingular elliptic curves and the Gross-Hopkins period map for points with supersingular reduction, we are able to get rather precise understanding of orbits of points. If time allows, I will also discuss the work of Herrero, Menares and Rivera-Letelier that is complementary to ours.

## Automorphic representations of symplectic groups, II

Monday, March 11, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker:** Jolanta Marzec (TU Darmstadt)

**Abstract:** Part III: We continue studying an automorphic representation associated to a Siegel modular form of degree n. This time we will construct a component at the archimedean place.

## Generalized Kontsevich-Zagier series via knots

Tuesday, April 23, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker:** Robert Osburn (University College Dublin)

**Abstract: **Over the past two decades, there has been substantial interest in the overlap between quantum knot invariants, q-series and modular forms. In this talk, we discuss one such instance, namely an explicit q-hypergeometric expression for the colored Jones polynomial for double twist knots. As an application, we generalize a duality at roots of unity between the Kontsevich-Zagier series and the generating function for strongly unimodal sequences. This is joint work with Jeremy Lovejoy (Paris 7 and Berkeley).

## Non-Archimedean integrals on the Hitchin Fibration

Tuesday, May 14, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker:** Dimitri Wyss (Université Paris 6)

**Abstract:**

Based on mirror symmetry considerations, Hausel and Thaddeus conjectured an equality between 'stringy' Hodge numbers for moduli spaces of SL_n/PGL_n Higgs bundles. We prove this conjecture using non-archimedean integration in the sense of Denef-Loeser and Batyrev and the duality of generic Hitchin fibers.

In a more arithmetic context, similar ideas lead to a new proof of the geometric stabilization theorem for anisotropic Hitchin fibers, a key ingredient in the proof of the fundamental lemma by Ngô. This is joint work with Michael Groechenig and Paul Ziegler.

## tba

Tuesday, May 21, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker** Mark Feldmann (Universität Münster)

**Abstract:** tba

## Fourier coefficients of automorphic forms: p-adic analysis and arithmetic at the cusps

Tuesday, June 25, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker** Andrew Corbett (University Exeter, UK)

**Abstract:** The information buried in the Fourier coefficients of automorphic forms is not often readily accessible; that is part of their endear. We tackle a selection of problems in number theory whose solution is related a careful understanding of such Fourier coefficients at different cusps. We bring forth a p-adic framework which may be utilised in the various problems.

## Purity for the Brauer group of singular schemes

## A Geometric Theta Correspondence for Picard Modular Surfaces

Tuesday, July 9, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker** Robert Little (Oxford University)

**Abstract:** For a noncompact Picard Modular Surface X, the machinery of Kudla & Millson give us a degree 2 cohomology class $\theta_L(\tau)$ on X, which is a holomorphic modular form in the variable $\tau$ . Following work of Cogdell, Hirzebruch-Zagier and Funke-Millson, we aim to extend this cohomology class to the boundary components of a compactification of X, which will allow us to create a holomorphic, modular, non-trivial and compactly supported degree 2 cohomology class on X. This work uses techniques from representation theory, arithmetic geometry and real analysis; we will also be able to give concrete examples of holomorphic modular forms given by pairing non-compact homology on X.

## CM values of modular functions and factorization

Tuesday, October 29, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker:**Yingkun Li (TU Darmstadt)

**Abstract: **The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.

## Estimates of holomorphic cusp forms associated to co-compact arithmetic subgroups

Tuesday, November 26, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker:** Anilatmaja Aryasomayajula (IISER Tirupati)

**Abstract:** In 1995, Iwaniec and Sarnak computed estimates of Hecke eigen Mass forms associated to co-compact arithmetic subgroups of SL_{2}(R). Adapting the arguments of Iwaniec and Sarnak to the holomorphic setting, Das and Sengupta derived sub-convexity bounds of holomorphic cusp forms associated the co-compact arithmetic subgroups considered by Iwaniec and Sarnak. In a recent work of ours, we have combined the approach of Das and Sengupta with a heat kernel approach to improve the sub-convexity bounds of Dad and Sengupta.

## Formalising perfectoid spaces

Tuesday, December 17, 2019, 15:15 – 16:15, Room S2|15 401

**Speaker:** Johan M. Commelin (Albert-Ludwigs-Universität Freiburg)

**Abstract:** Computers have revolutionised many parts of modern society. For example: CAD (computer-aided design) software has pervasively changed the way that architects and designers perform their jobs. Mathematical research has not been left alone: ranging from the rapid dissemination via arXiv and the unlimited communication via MathOverflow to computation involving computer algebra systems and even controversial computer-generated proofs of the four colour theorem and the boolean Pythagorean triples problem. However, mathematical practice is still mostly the romantic pencil-and-paper practice that it was before the invention of the transistor. Together with Kevin Buzzard and Patrick Massot, I have formalised the definition of a perfectoid space: we have explained perfectoid spaces to a computer system. In this talk I will explain what “formalising” means, including a live demonstration of the system that we used. I will also make some remarks on how far we are removed from a future of meaningful computer-aided mathematical theorem proving.

## Wilton estimates and the fine structure of Fourier coefficients

Tuesday, January 14, 2020, 15:15 – 16:15, Room S2|15 401

**Speaker**: Edgar Assing, University Bonn

**Abstract:** We will discuss the correlation between Fourier coefficients of modular forms and additive characters. The study of such correlations has a long history starting from an estimate due to Wilton.

However, if the level of the modular form is large new features arise. We will explain these phenomena by carefully studying the local Whittaker functions associated to our modular form. Further we will draw a connection to the sup-norm problem, which has seen much attention in recent years.

# A Gross-Kohnen-Zagier theorem for non-split Cartan curves

Tuesday, January 21, 2020, 15:15 – 16:15, Room S2|15 401

**Speaker**: Dr. Daniel Kohen, Universität Duisburg – Essen

**Abstract**: Let E/Q be an elliptic curve of conductor p^2 and odd analytic rank. The famous Gross-Kohnen-Zagier theorem states that, as we vary throughout the imaginary quadratic fields in which p is split, the collection of Heegner points generate an at most 1 dimensional space and that their relative position on the Mordell-Weil group of E is encoded in the Fourier coefficients of a certain Jacobi form of weight 2 and index p^2.

For imaginary quadratic fields in which p is inert we can construct special points on E via a modular parametrization from a non-split Cartan modular curve, generalizing the classical construction of Heegner points. In this talk we prove that the position on E of the special points arising from non-split Cartan curves is reflected in the Fourier coefficients of a Jacobi form of weight 6 and a certain lattice index of rank 9. In order to prove this, we construct an explicit even lattice L of signature (2, 1) obtained by studying the properties of the special points on non-split Cartan curves and we apply Borcherds’ striking generalization of the G-K-Z theorem to show that our special points are the coefficients of a vector valued modular form of weight 3/2. Using Nikulin’s theory of discriminant forms and lattices we show the connection with the desired Jacobi form. finally, we provide an explicit example of this construction. This is joint work with Nicolas Sirolli (Universidad de Buenos Aires).

## Zeta-Functions of varieties and Galois representations, and their meromorphic continuation

Tuesday, January 29, 2020, 15:15 – 16:15, Room S2|15 401

**Speaker:** Prof. Dr. Eugen Hellmann (Universität Münster)

**Abstract**: In arithmetic geometry the numbers of zeros of a set of polynomials (with integral coefficients) modulo each prime are usually put together into a so-called zeta-function. This is a function of one complex variable converging on some right complex half space. The functions arising this way are expected to have a meromorphic continuation to the complex plane and to satisfy a functional equation.

I will explain the meaning of this in easy examples and explain the parallel conjecture for Galois representations.

If time permits I will mention results concerning the approach to this problems via modular forms (and their higher dimensional generalizations).

## Cubic sum problem and singular moduli

Tuesday, February 11, 2020, 15:15 – 16:15, Room S2|15 401

**Speaker:** Dr. Hongbo Yin (MPIM Bonn)

**Abstract**: One interesting question in number theory is to determine whether an integer can be written as the sum of two rational cubes. In this talk, I will introduce this problem's history and progress. I will also talk about my recent work on this problem where the theory of singular moduli comes in surprisingly.

## tba

Tuesday, May 5, 2020, 15:15 – 16:15, Room S2|15 401

**Speaker**: tba