# Seminar der Arbeitsgruppe Algebra: Wintersemester 2018/19

## 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.