School and research conference
Modular Forms and Beyond
May 21 -26, 2018
Euler International Mathematical Institute, St. Petersburg, Russia
MONDAY
1) V. Golyshev (IITP RAS) Fibered motives and L-values (Prof. , 45 minutes)
We discuss techniques to fiber out pure and mixed periods over a torus (joint work with S. Bloch).
2) Nina Sachorova (NRU HSE, Laboratory of Mirror Symmetry and Automorphic forms)
Modular Cauchy kernel corresponding to the Hecke curve
I’ll talk about the construction of the the modular Cauchy kernel $\Xi_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=\gamma z_1, ~\gamma \in \Gamma_0(N) \right\}.$$ The function $\Xi_N(z_1, z_2)$ is used in two cases and for two different purposes. Using the Rankin-Selberg method, Don Zagier proved that the Hecke operator $T_k(m)$ on the space of cusp forms of weight $k>2$ can be defined by a kernel $\omega_m(z_1,\bar{z_2}, k)$. Firstly, we prove generalization of the Zagier theorem for the Hecke subgroups $\Gamma_0(N)$ of genus $g>0$. Namely, we obtain a kind of ``kernel function'' for the Hecke operator $T_N(m)$ on the space of the weight 2 cusp forms for $\Gamma_0(N)$, which is the analogue of the Zagier series $\omega_{m, N}(z_1,\bar{z_2}, 2)$. Secondly, we consider an elementary proof of the formula for the infinite Borcherds product of the difference of two normalized Hauptmoduls, ~$J_{\Gamma_0(N)}(z_1)-J_{\Gamma_0(N)}(z_2)$, for genus zero congruence subgroup $\Gamma_0(N)$. https://arxiv.org/pdf/1802.03299.pdf
3) Katya Stuken (NRU NSE)
Free algebras of the Hilbert modular forms
Let $d>0$ be square-free integer and $L_d$ be the Hilbert lattice, i.e. the even lattice of signature (2, 2) such that $L_d=\begin{pmatrix}0 & 1 \\1 & 0\\ \end{pmatrix} \oplus \begin{pmatrix} 2 & 1\\1 & \frac{1-d}{2}\\ \end{pmatrix}$ when $d=1 \pmod{4}$,
or $L_d = \begin{pmatrix}0 & 1 \\1 & 0 \\ \end{pmatrix} \oplus
\begin{pmatrix} 2 & 0 \\0 & -2d \\ \end{pmatrix}$ when $d=2,3\pmod{4}$.
Consider $\Gamma_d=O^+(L_d)$ and denote by $A(\Gamma_d)$
the algebra of $\Gamma_d$-automorphic forms. The main goal of the report is the following
Theorem: If the algebra $A(\Gamma_d)$ is free then $d \in \{2,3,5,6,13,21\}$.
4) D. Adler, The ring of weak Jacobi forms for D_8 root lattice (and D_n)
(NRU HSE, Laboratory of Mirror Symmetry and Automorphic forms)
For every quadratic lattice one can defines the notion of weak Jacobi forms associated with this lattice. In 1992 K. Withmuller proved that spaces of Jacobi forms associated with root lattices (except E_8) have the structure of a free algebra over the ring of modular forms. However, his proof is very complicated. In my talk I plan to introduce some constructions that help to solve this problem in case D_8 and obtain generators of the corresponding algebra in an explicit way. The talk will be based on joint results with Valery Gritsenko. These results are preparing to publication.
5) Haowu Wang (CEMPI, Lille)
Weyl invariant $E_8$ Jacobi forms
In 1992, Wirthmuller proved that for the lattice constructed from a classical root system (except $ E_8 $), the corresponding space of Jacobi forms which are invariant under the Weyl group is a polynomial algebra over the ring of modular forms. In the talk we will focus on discussing the remaining case $E_8$.
We will show that the space of Weyl invariant $E_8$ Jacobi forms is not a polynomial algebra and present a proper extension of Wirthmuller's theorem in the case. We will also give the generators of indices 2, 3, 4.
TUESDAY
6) Annalena Wernz (RWTH)
The isomorphism between the Hermitian modular group and O(2,4) .
The Hermitian modular group U(n, n; \mathcal{O}_K) of degree n over an imaginary quadratic field K = \mathbb{Q}(v?m) was introduced by Hel Braun in the 1940s as an analogue for the well known Siegel modular group. It is a subgroup of the special unitary group SU(n, n, \mathbb{C}) and for n = 2 it
is isomorphic to the orthogonal group O(2, 4). For m = 1, 2, 3, Kloecker showed in 2005 that the Hermitian modular group is isomorphic to a subgroup of O(2, 4). In my talk, I consider arbitrary m \neq 1, 3 and show that the Hermitian modular group is isomorphic to the discriminant kernel of
the orthogonal group O(2, 4). Furthermore, I compute the normalizer of the Hermitian modular group in the unitary group and show that it is isomorphic to the integral orthogonal group.
7) Haowu Wang (CEMPI Lille)
Non-existence of 2-reflective modular forms
An even lattice L of signature (2,n) is called 2-reflective if it admits a holomorphic modular form whose support of zero divisor is contained in the Heegner divisor determined by the (-2)-vectors in L. In this talk we give a formula expressing the weight of 2-reflective modular forms and prove that there is no 2-reflective lattice when $n\geq 15$ and $n\neq 19$
except the even unimodular lattices of signature (2,18) and (2,26).
8) Martin Woitalla (RWTH)
Coordinates for the graded ring of modular forms on the Cayley half-space of degree two
A result by Hashimoto and Ueda says that the graded ring of modular forms with respect to SO(2,10) is a polynomial ring in modular forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42. We show that one may choose Eisenstein series as generators. This is done by calculating sufficiently many Fourier coefficients of the restrictions to the Hermitian half-space. Moreover, we give several constructions of the skew-symmetric modular form of weight 252.
https://link.springer.com/article/10.1007%2Fs11139-017-9970-x
9) Aleksandr Kalmynin (NRU HSE, Laboratory of Mirror Symmetry and Automorphic forms)
Cohen-Kuznetsov series and intervals between numbers that are sums of two squares
The study of distribution of gaps between numbers that can be expressed as a sum of two perfect squares is a classical problem in analytic number theory, that dates back to Euler. In my talk I will present some new results that connect the distribution of gaps with Cohen-Kuznetsov construction for Jacobi-type forms and allow to improve the results on the moments of gaps between sums of two squares.
https://arxiv.org/pdf/1706.07380
10) Paul Kiefer (TU-Darmstadt)
Boundary Components of the Orthogonal Upper Half-Plane
In this talk we will introduce the boundary components of the orthogonal
upper half-plane and discuss results about their structure for the
orthogonal group and the corresponding discriminant kernel. We will see
that the $1$-dimensional boundary components are modular curves
intersecting only at the cusps and specify the corresponding congruence
subgroups. Afterwards we will count the number of $0$-dimensional and
$1$-dimensional boundary components.
WEDNESDAY
11) Dmitry Frolenkov (NRU HSE/MIAN, Moscow)
Convolution formula for the sums of generalized Dirichlet L-series
We will discuss the generalized Dirichlet series $L_n(s)$ that arise naturally
in various contexts, from the theory of modular forms to the Prime Geodesic Theorem.
On the one hand, the mean value of $L_n(1/2)$ determines the quality of the error term in asymptotic formulas for moments of symmetric square L-functions. On the other hand, investigation of the series $L_n(s)$ at the point 1 is ultimately related to the Prime Geodesic Theorem. Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet L-series. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square L-functions and an asymptotic
expansion for the average of central values of generalized Dirichlet L-series.
This is joint work with Olga Balkanova.
This work is published on arXiv https://arxiv.org/abs/1709.01365
THURSDAY
12) Yanbin Li (Tongji University, Shanghai)
Jacobi Forms of Squarefree Level and Eichler Orders in Definite Quaternion Algebra
We redefine the Eisenstein series of $J_{2,1} (N)$ (squarefree level) and bridge between them and Jacobi theta series associated to Eichler orders in definite quaternion algebra. Give simple calculation formulas for the traces of Brandt matrices and the type number of Eichler orders as two corollaries. These results generalize the formulas in "Counting zeros in quaternion algebras using jacobi forms"
DOI: https://doi.org/10.1090/tran/7575.
13) Stefan Ble (RWTH)
The Maass-Space and ultraspherical differential operators
The Maass-Space is a vector space of modular forms with some special relation to their fourier-coe cients. Andrianov proved the invariance of this space under hecke-operators by computing the fourier-coe cients, but nowadays there is some easier way to prove this. The image of the ultraspherical dierential operator is also able to characterizea Maass-form and the relevant Hecke-operaters to the image of the differential operator and the Maass-Space are commutative.
14) Felix Schaps (RWTH)
Eisenstein series for the orthogonal group O(2,n).
In my talk I consider Eisenstein series for the orthogonal group O(2,n) and their absolute convergence. H. Braun's proof of the absolute convergence of Siegel Eisenstein series in the 1930s can be transferred to the orthogonal setting. We have a look at the singular and non-singular part of the series. By this way, we also get a Fourier expansion.
15) Alexey Ustinov (Pacific National University, Institute of Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences)
An elementary approach to Somos-4 sequences
A sequence Somos-4 is defined by initial data $s_0$, $s_{1}$, $s_{2}$, $s_{3}$ and fourth-order recurrence
$$s_{n+2}s_{n-2}=\alpha s_{n+1}s_{n-1}+\beta s_n^2.$$
Usually properties of this sequence are studied by means of elliptic functions.
The talk will be devoted to the new elementary approach to Somos-4 sequences.
Hopefully it will be suitable for higher-rank Somos sequences corresponding to curves of higher genus.
16) Andreea Mocanu (University of Nottingham)
Level raising operators for Jacobi forms of lattice index
In this short talk, we introduce certain level raising operators defined on spaces of Jacobi forms of lattice index, which arise from isometries between lattices. We discuss some of their properties and the connection with a theory of newforms for Jacobi forms.
17) Xiong Ran, (Tongji university, Shanghai, China)
Fourier coefficients of Jacobi Eisenstein series of lattice index.
18) Galina Voskresenskaya (Samara University)
Eta-function in modern investigations
Investigations on eta-functions meet with mathematical studies in different topics --- number theory, group theory, complex analysis, combinatorial analysis and others. These studies has been done during 150 years --- from classical works of Kronecker and Dedekind to modern articles on large sporadic groups representations and the theory of superstrings. In this talk we describe properties of Dedekind`s eta--function, constructions arising from it and their applications to various topics of the number theory and algebra. We describe their connections with the theory of group representations and their role in investigations of the structure of spaces of modular forms.
Especially we consider the special class of modular forms --- eta--products with multiplicative coefficients. The main topic is the theory of Frame-shape correspondence. We shall speak about new results and open problems of the theory.
[1] G.V. Voskresenskaya, Finite simple groups and multiplicative eta—products // Zapiski POMI, 375, p. 71-91, 2010.
[2] G.V. Voskresenskaya, Dedekind eta--functions in modern investigations// VINITI, 136, p. 103--137, 2017
[3] G.V. Voskresenskaya, Decomposition of spaces of modular forms // Mat.Zametki, 99, N 6, p. 851-860, 2016.
FRIDAY
19) Adrian Hauffe-Waschbusch (RWTH)
Growth of Fourier coefficients of Hermitian modular forms.
Bocherer and Das published a paper on the growth of the Fourier
coefficients of Siegel modular forms in 2016. In this paper they gave a
condition on the growth of the Fourier coefficient which implies that
the modular forms are cusp forms. I transfered this result to Hermitian
modular forms on imaginary-quadratic number fields.