# Seminar der Arbeitsgruppe Algebra

## Relations between Fourier coefficients of Siegel modular forms

Dienstag, der 16. Oktober 2018, 15:15 – 16:15 Uhr, Raum S2|15 401

**Referentin:** 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

Donnerstag, den 25. Oktober 2018, 15:20 – 17:00 Uhr, Raum S2|15 244

**Referent: **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

Dienstag, der 30. Oktober 2018, 15:15 – 16:15 Uhr, Raum S2|15 401

**Referent:** 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

Dienstag, den 27. November 2018, 14:00 – 15:00 Uhr, Raum S2|15 401

**Referentin:** 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, Raum 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:** We study Weil group representations over the coefficient feld Qp and establish certain equivalences of categories in the flavor of Fontaine's classification of p-adic representations of the absolute Galois group. If we restrict to crystalline (or de-Rham) Weil group representations, we can describe the category of these Weil group representations in terms of generators. More precisely it is generated as an abelian tensor category by the full subcategory of Galois group representations and finite unramified inductions of the character Qp(| · |) given by Artin's reciprocity law.

## 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

Dienstag, 14. Januar 2020, 15:15 – 16:15 Uhr, Raum S2|15 401

**Speaker**: Edgar Assing, Universität 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).

## On the derived category of the Iwahori-Hecke Algebra

Tuesday, January 28, 2020, 15:15 – 16:15, Room S2|15 401

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

**Abstract: **As a special case of the local Langlands correspondence (for GL_n) there is a bijection between the simple modules over the Iwahori-Hecke algebra and so called L-parameters. I will describe how this bijection should extend to an exact functor from the derived category of the Iwahori-Hecke algebra to the derived category of coherent sheaves on a stack of L-parameters. The main idea is to relate the construction of a family interpolating the (modified) local Langlands correspondence (following Emerton and Helm) to parabolic induction. If time permits I will mention some p-adic aspects of this construction.

## 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.

## Higher pullbacks of orthogonal modular forms

Dienstag, den 18. Februar 2020, 15:15 – 16:15 Uhr, Raum S2|15 401

**Referent:** Brandon Williams (TU Darmstadt)

**Abstract:** We apply differential operators to modular forms on orthogonal groups $\mathrm{O}(2,\ell)$ to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form $\phi$ are theta lifts of partial development coefficients of $\phi$. For certain lattices of signature $(2,2)$ and $(2,3)$, for which there are interpretations as Hilbert-Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

## Universal optimality of the E8 and Leech lattices

Wednesday, April 08,.2020, 15:00-16:00, ZOOM**Speaker: **Danylo Radchenko (ETH Zurich)

**Abstract:** I will talk about the recent proof of universal optimality of the E8 and Leech lattices (and explain what universal optimality means). While the statement itself does not involve any automorphic forms, the key ingredient in the proof is a new kind of interpolation formula for radial Fourier eigenfunctions which turns out to be intimately related to certain vector-valued modular forms for SL(2,Z). The talk is based on a joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Maryna Viazovska.

## Periods of modular functions and Diophantine approximation

Wednesday, April 15 2020, 15:15 – 16:15, ZOOM**Speaker:** Paloma Bengoechea (ETH Zurich)

**Abstract: **The „value“ of Klein's modular invariant j at a real quadratic irrationality w has been recently defined using the period of j along the geodesic associated to w in the hyperbolic plane. Works of Duke, Imamoglu, Toth, and Masri establish analogies between these values and singular moduli when they are both gathered in traces. We will talk about the distribution of the values j(w) individually, according to the diophantine approximation of w. Some of our results were conjectured by Kaneko. This is joint work with O. Imamoglu.

## Singular moduli for real quadratic fields

Wednesday; April 22 20202, 15:15 -16:00, ZOOM**Speaker:** Jan Vonk (IAS Princeton)

**Abstract:** In the early 20th century, Hecke studied the diagonal restrictions of Eisenstein series over real quadratic fields. An infamous sign error caused him to miss an important feature, which later lead to highly influential developments in the theory of complex multiplication (CM) initiated by Gross and Zagier in their famous work on Heegner points on elliptic curves. In this talk, we will explore what happens when we replace the imaginary quadratic fields in CM theory with real quadratic fields, and propose a framework for a tentative 'RM theory', based on the notion of rigid meromorphic cocycles, introduced in joint work with Henri Darmon. I will discuss several of their arithmetic properties, and their apparent relevance in the study of explicit class field theory of real quadratic fields. This concerns various joint works, with Henri Darmon, Alice Pozzi, and Yingkun Li.

# tba

Dienstag, den 28. April 2020, 15:15 – 16:15, Raum S2|15 401

**Speaker**: Soumya Das (Indian Institute of Science)

## Zeros of GL2 L-functions on the critical line

Wednesday, April 29 2020, 16:00 – 17:00, ZOOM**Speaker:** Nick Andersen (Brigham Young University)

**Abstract: **We use Levinson’s method and the work of Blomer and Harcos on the GL2 shifted convolution problem to prove that at least 6.96% of the zeros of the L-function of any holomorphic or Maass cusp form lie on the critical line. This is joint work with Jesse Thorner.

## Abgesagt – tba

Dienstag, den 05. Mai 2020, 15:15 – 16:15, Raum S2|15 401

**Speaker**: Ashay Burungale (Caltech, USA)

## Local Hecke algebras and new forms

Wednesday, May 6 2020, 10:00 – 11:00, ZOOM**Speaker: **Soma Purkait (Tokyo Institute of Technology)

**Abstract: **We describe local Hecke algebras of GL_2 and double cover of SL_2 with certain level structures and use it to give a newform theory. In the integral weight setting, our method allows us to give a characterization of the newspace of any level as a common eigenspace of certain finitely many pair of conjugate operators that we obtain from local Hecke algebras. In specific cases, we can completely describe local Whittaker functions associated to a new form. In the half-integral weight setting, we give an analogous characterization of the newspace for the full space of half-integral weight forms of level 8M, M odd and square-free and observe that the forms in the newspace space satisfy a Fourier coefficient condition that gives the complement of the plus space. This is a joint work with E.M. Baruch.

## Sparse equidistribution of hyperbolic orbifolds

Wednesday, May 13 2020, 15:00-16:00, ZOOM**Speaker:** Peter Humphries (University College London)**Abstract**: Duke, Imamoḡlu, and Tóth have recently constructed a new geometric invariant, a hyperbolic orbifold, associated to each narrow ideal class of a real quadratic field. Furthermore, they have shown that the projection of these hyperbolic orbifolds onto the modular surface equidistributes on average over a genus of the narrow class group as the fundamental discriminan of the real quadratic field tends to infinity. We discuss a refinement of this result, sparse equidistribution, where one averages over smaller subgroups of the narrow class group: we connect this to cycle integrals of automorphic forms and subconvexity for Rankin-Selberg L-functions. This is joint work with Asbjørn Nordentoft.

A handout for the talk is available on the website of the seminar: https://www2.mathematik.tu-darmstadt.de/~international_seminar/.

## Periodicities for Taylor coefficients of half-integral weight modular forms

Wednesday, May 20 2020, 15:00-16:00, ZOOM **Speaker:** Larry Rolen (Vanderbilt University)

**Abstract:** Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $\tau=i$ of the classical Jacobi theta function. Here, in joint work with Michael Mertens and Pavel Guerzhoy, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular forms of half-integral weight.

## Eisenstein Series, Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra

Wednesday, June 24 2020, 15:00 – 16:00 Zoo,

**Speaker:** Sven Möller (Rutgers University)

**Abstract: **Conway, Parker and Sloane (and Borcherds) showed that there is a natural bijection between the Niemeier lattices (the 24 positive-definite, even, unimodular lattices of rank 24) and the deep holes of the Leech lattice, the unique Niemeier lattice without roots. We generalise this statement to vertex operator algebras (VOAs), i.e. we show that all 71 holomorphic VOAs (or meromorphic 2-dimensional conformal field theories) of central charge 24 correspond to generalised deep holes of the Leech lattice VOA. The notion of generalised deep hole occurs naturally as an upper bound in a dimension formula we obtain by pairing the character of the VOA with a certain vector-valued Eisenstein series of weight 2. (This is joint work with Nils Scheithauer.)