10.05.2022 -- 16:00-17:00 (CEST, GMT+2)
Jaclyn Lang (Temple University) A modular construction of unramified p-extensions of $Q(N^{1/p})$
In his 1976 proof of the converse of Herbrand's theorem, Ribet used
Eisenstein-cuspidal congruences to produce unramified degree-p extensions of the
p-th cyclotomic field when p is an odd prime. After reviewing Ribet's strategy, we
will discuss recent work with Preston Wake in which we apply similar techniques to
produce unramified degree-p extensions of $Q(N^{1/p})$ when N is a prime that is
congruent to -1 mod p. This answers a question posed on Frank Calegari's blog.
03.05.2022 -- 16:00-17:00 (CEST, GMT+2)
Brandon Williams (RWTH Aachen) Free algebras of modular forms on ball quotients
We study algebras of modular forms on unitary groups of signature $(n, 1)$. We give a
sufficient criterion for the ring of unitary modular forms to be freely generated in
terms of the divisor of a modular Jacobian determinant. We use this to prove that a
number of rings of unitary modular forms associated to Hermitian lattices over the
rings of integers of ${Q}(\sqrt{ d})$ for $d = -1, -2, -3$ are polynomial algebras without relations. This is joint work with Haowu Wang.
26.04.2022 -- 16:00-17:00 (CEST, GMT+2)
Weibo Fu (Princeton) Growth of Bianchi modular forms
In this talk, I will establish a sharp bound on the growth of cuspidal
Bianchi modular forms. By the Eichler-Shimura isomorphism, we actually give a sharp
bound of the second cohomology of a hyperbolic three manifold (Bianchi manifold)
with local system arising from the representation $Sym^k \otimes \overline{Sym^k}$ of
$SL_2(C)$. I will explain how a p-adic algebraic method is used for deriving our
result.
Winter 2021-2022
01.02.2022 -- 15:00-16:00 (German time, GMT+1)
Vesselin Dimitrov (University of Toronto) The unbounded denominators conjecture
I will explain the ideas of the proof of the following recent
theorem, joint with Frank Calegari and Yunqing Tang:
A modular form for a finite index subgroup of $SL_2(Z)$ has a q-expansion
with bounded denominators if and only if it is a modular form for a
congruence subgroup.
I will also discuss some related open problems such as a hypothetical
$SL_2(F_q[t])$ analog of the theorem.
25.01.2022 -- 15:00-16:00 (German time, GMT+1)
Márton Erdélyi (Budapest University of Technology and Economics) Matrix Kloosterman sums
We study exponential sums arosing in the work of Lee and Marklof about
the horocyclic flow on the group $GL_n$. In many cases this sum can be
expressed with the help of classical Kloosterman sums. We give effective
bounds using the very basics of cohomological methods and get a nice
illustration of the general purity theorem of Fouvry and Katz. Joint
work with Árpád Tóth.
18.01.2022 -- 16:00-17:00 (German time, GMT+1)
Isabella Negrini (McGill) A Shimura-Shintani correspondence for rigid analytic cocycles
In their paper Singular moduli for real quadratic fields: a rigid analytic approach,
Darmon and Vonk introduced rigid meromorphic cocycles, i.e. elements of
$H^1(SL_2(Z[1/p]), M^\times)$ where $M^\times$ is the multiplicative group of rigid meromorphic
functions on the p-adic upper-half plane. Their values at RM points belong to narrow
ring class fields of real quadratic fiends and behave analogously to CM values of
modular functions on $SL_2(Z)\backslash\mathbf{H}$. In this talk I will present some progress towards
developing a Shimura-Shintani correspondence in this setting.
11.01.2022 -- 14:00-15:00 (German time, GMT+1)
Luis Garcia (UCL) Eisenstein cocycles and values of L-functions
There are several recent constructions by many authors of Eisenstein
cocycles of arithmetic groups. I will discuss a point of view on these constructions
using equivariant cohomology and equivariant differential forms. The resulting
objects behave like theta kernels relating the homology of arithmetic groups to
algebraic objects. As an application, I will explain the proof of some conjectures
of Sczech and Colmez on critical values of Hecke L-functions. The talk is based on
joint work with Nicolas Bergeron and Pierre Charollois.
14.12.2021 -- 15:00-16:00 (German time, GMT+1)
Patrick Bieker (TU Darmstadt) Modular units for orthogonal groups of
signature (2,2) and invariants for the Weil representation
We construct modular units for certain orthogonal groups in
signature (2, 2) using Borcherds products. As an input to the
construction we show that the space of invariants for the Weil
representation for discriminant groups which contain self-dual isotropic
subgroups is spanned by the characteristic functions of the self-dual
isotropic subgroups. This allows us to determine all modular units
arising as Borcherds products in examples.
07.12.2021 -- 15:00-16:00 (German time, GMT+1)
Manami Roy (Fordham) Dimensions for the spaces of Siegel cusp forms of level 4
Many mathematicians have studied dimension and codimension
formulas for the spaces of Siegel cusp forms of degree 2. The dimensions of
the spaces of Siegel cusp forms of non-squarefree levels are mostly now
available in the literature. This talk will present new dimension formulas
of Siegel cusp forms of degree 2, weight k, and level 4 for three
congruence subgroups. One of these dimension formulas is obtained using the
Satake compactification. However, our primary method relies on counting a
particular set of cuspidal automorphic representations of GSp(4) and
exploring its connection to dimensions of spaces of Siegel cusp forms of
degree 2. This work is joint with Ralf Schmidt and Shaoyun Yi.
30.11.2021 -- 15:00-16:00 (German time, GMT+1)
Riccardo Zuffetti (GU Frankfurt) Cones of codimension two special cycles
In the literature, there are several results on cones
generated by (effective, ample, nef...) divisors on (quasi-)projective
varieties. However, a little is known on cones generated by cycles of
codimension greater than one.
Let X be an orthogonal Shimura variety. In this talk, we consider the
cone $C_X$ generated by rational classes of codimension two special cycles
of X. We illustrate how to prove properties of $C_X$ by means of Fourier
coefficients of Siegel modular forms.
23.11.2021 -- 15:00-16:00 (German time, GMT+1)
William Chen (IAS) Connectivity of Hurwitz spaces and the conjecture of Bourgain,
Gamburd, and Sarnak
A Hurwitz space is a moduli space of coverings of algebraic
varieties. After fixing certain topological invariants, it is a classical
problem to classify the connected components of the resulting moduli space.
For example, the connectivity of the space of coverings of the projective
line with simple branching and fixed degree led to the first proof of the
irreducibility of M_g. In this talk I will explain a similar connectedness
result, this time in the context of SL(2,p)-covers of elliptic curves, only
branched above the origin. The connectedness result comes from combining
asymptotic results of Bourgain, Gamburd, and Sarnak with a new
combinatorial 'rigidity' coming from algebraic geometry. This rigidity
result can also be viewed as a divisibility theorem on the cardinalities of
Nielsen equivalence classes of generating pairs of finite groups. The
connectedness is a key piece of information that unlocks a number of
applications, including a conjecture of Bourgain, Gamburd and Sarnak on a
strong approximation property of the Markoff equation $x^2 + y^2 + z^2 - xyz$
= 0, a noncongruence analog of Rademacher's conjecture of the genus of
modular curves, tamely ramified 3-point covers in characteristic p, and
counting flat geodesics on a certain family of congruence modular curves.
16.11.2021 -- 15:00-16:00 (German time, GMT+1)
Paul Kiefer (TU Darmstadt) Orthogonal Eisenstein Series of Singular Weight
We will study (non-)holomorphic orthogonal Eisenstein series using
Borcherds' additive theta lift. It turns out that the lifts of
vector-valued non-holomorphic Eisenstein series with respect to the Weil
representation of an even lattice are linear combinations of
non-holomorphic orthogonal Eisenstein series. This yields their
meromorphic continuation and functional equation. Moreover we will
determine the image of this construction. Afterwards we evaluate the
non-holomorphic orthogonal Eisenstein series at certain special values
to obtain holomorphic orthogonal Eisenstein series and determine all
holomorphic orthogonal modular forms that can be obtained in this way.
09.11.2021 -- 15:00-16:00 (German time, GMT+1)
Lindsay Dever (Bryn Mawr) Distribution of Holonomy on Compact Hyperbolic 3-Manifolds
The study of hyperbolic 3-manifolds draws deep connections between number
theory, geometry, topology, and quantum mechanics. Specifically, the closed
geodesics on a manifold are intrinsically related to the eigenvalues of
Maass forms via the Selberg trace formula and are parametrized by their
length and holonomy, which describes the angle of rotation by parallel
transport along the geodesic. The trace formula for spherical Maass forms
can be used to prove the Prime Geodesic Theorem, which provides an
asymptotic count of geodesics up to a certain length. I will present an
asymptotic count of geodesics (obtained via the non-spherical trace
formula) by length and holonomy in prescribed intervals which are allowed
to shrink independently. This count implies effective equidistribution of
holonomy and substantially sharpens the result of Sarnak and Wakayama in
the context of compact hyperbolic 3-manifolds. I will then discuss new
results regarding biases in the finer distribution of holonomy.
02.11.2021 -- 15:00-16:00 (German time, GMT+1)
Nils Matthes (Copenhagen) Meromorphic modular forms and their iterated integrals
Meromorphic modular forms are generalizations of modular forms which are allowed
to have poles. Part of the motivation for their study comes from recent work of
Li-Neururer, Pasol-Zudilin, and others, which shows that integrals of certain
meromorphic modular forms have integer Fourier coefficients -- an arithmetic
phenomenon which does not seem to exist for holomorphic modular forms. In this
talk we will study iterated integrals of meromorphic modular forms and describe
some general algebraic independence results, generalizing results of
Pasol-Zudilin. If time permits we will also mention an algebraic geometric
interpretation of meromorphic modular forms which generalizes the classical fact
that modular forms are sections of certain line bundles, and describe the
occurrence of iterated integrals of meromorphic modular forms in computations of
Feynman integrals in quantum field theory.
26.10.2021 -- 15:00-16:00 (German time, GMT+2)
Jan H. Bruinier (TU Darmstadt) Arithmetic volumes of unitary Shimura varieties
The geometric volume of a unitary Shimura variety can be defined as the
self-intersection number of the Hodge line bundle on it. It represents an
important invariant, which can be explicitly computed in terms of special values
of Dirichlet L-functions. Analogously, the arithmetic volume is defined as the
arithmetic self-intersection number of the Hodge bundle, equipped with the
Petersson metric, on an integral model of the unitary Shimura variety. We show
that such arithmetic volumes can be expressed in terms of logarithmic derivatives
of Dirichlet L-functions. This is joint work with Ben Howard.
Summer 2021
06.07.2021 -- 15:00-16:00 (German time, GMT+2)
Lennart Gehrmann (Duisburg-Essen) Rigid meromorphic cocycles for orthogonal groups
I will talk about a generalization of Darmon and Vonk's notion of rigid meromorphic cocycles to the setting of orthogonal groups. After giving an overview over the general setting I will discuss the case of orthogonal groups attached to quadratic spaces of signature (3,1) in more detail. This is joint work with Henri Darmon and Mike Lipnowski.
29.06.2021 -- 15:00-16:00 (German time, GMT+2)
Claudia Alfes-Neumann (Bielefeld) Some theta liftings and applications
In this talk we give an introduction to the study of generating series of the traces
of CM values and geodesic cycle integrals of different modular functions.
First we define modular forms and harmonic Maass forms. Then we briefly discuss the
theory of theta lifts that gives a conceptual framework for such generating series.
We end with some applications of the theory.
22.06.2021 -- 10:00-11:00 (German time, GMT+2)
YoungJu Choie (Postech) A generating function of periods of automorphic forms
A closed formula for the sum of all Hecke eigenforms on $\Gamma_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level N is known. When N = 1 this was result given by Zagier in 1991. We discuss more general result regarding on this direction.
15.06.2021 -- 15:00-16:00 (German time, GMT+2)
Chao Li (Columbia University) Beilinson-Bloch conjecture and arithmetic inner product formula
For certain automorphic representations $\pi$ on unitary groups, we show
that if $L(s, \pi)$ vanishes to order one at the center $s=1/2$, then the
associated $\pi$-localized Chow group of a unitary Shimura variety is
nontrivial. This proves part of the Beilinson-Bloch conjecture for unitary
Shimura varieties, which generalizes the BSD conjecture. Assuming Kudla's
modularity conjecture, we further prove the arithmetic inner product
formula for $L'(1/2, \pi)$, which generalizes the Gross-Zagier formula. We
will motivate these conjectures and discuss some aspects of the proof. We
will also mention recent extensions applicable to symmetric power
L-functions of elliptic curves. This is joint work with Yifeng Liu.
08.06.2021 -- 15:00-16:00 (German time, GMT+2)
Claire Burrin (ETH) Rademacher symbols on Fuchsian groups
The Rademacher symbol is algebraically expressed as a conjugacy class
invariant quasimorphism PSL(2,Z) -> Z. It was first studied in connection to
Dedekind's eta-function, but soon enough appeared to be connected to class numbers
of real quadratic fields, the Hirzebruch signature theorem, or linking numbers of
knots. I will explain
(1) how, using continued fractions, Psi can be realized as the winding number for
closed curves on the modular surface around the cusp;
(2) how, using Eisenstein series, one can naturally construct a Rademacher symbol
for any cusp of a general noncocompact Fuchsian group;
(3) and discuss some new connections to arithmetic geometry.
01.06.2021 -- 15:00-16:00 (German time, GMT+2)
Yuya Murakami (Tohoku University) Extended-cycle integrals of the j-function for badly approximable numbers
Cycle integrals of the j-function are expected to play a role in the
real quadratic analog of singular moduli.
However, it is not clear how one can consider cycle integrals as a "continuous"
function on real quadratic numbers.
In this talk, we extend the definition of cycle integrals of the j-function from
real quadratic numbers to badly approximable numbers to seek an appropriate
continuity.
We also give some explicit representations for extended-cycle integrals in some
cases which can be considered as a partial result of continuity of cycle integrals.
25.05.2021 -- 15:00-16:00 (German time, GMT+2)
James Rickards (McGill University) Counting intersection numbers on Shimura curves
In this talk, we give a formula for the total intersection number of optimal
embeddings of a pair of real quadratic orders with respect to an indefinite
quaternion algebra over Q. We recall the classical Gross-Zagier formula for the
factorization of the difference of singular moduli, and note that our formula
resembles an indefinite version of this factorization. This lends support to the
work of Darmon-Vonk, who conjecturally construct a real quadratic analogue of the
difference of singular moduli.
18.05.2021 -- 15:00-16:00 (German time, GMT+2)
Sebastián Herrero (Pontifical Catholic University of Valparaiso) There are at most finitely many singular moduli that are S-units
In 2015 P. Habegger proved that there are at most finitely many
singular moduli that are algebraic units. In 2018 this result was made
explicit by Y. Bilu, P. Habegger and L. Kühne, by proving that there is
actually no singular modulus that is an algebraic unit. Later, this result
was extended by Y. Li to values of modular polynomials at pairs of singular
moduli.
In this talk I will report on joint work with R. Menares and J.
Rivera-Letelier, where we prove that for any finite set of prime numbers S,
there are at most finitely many singular moduli that are S-units. We use
Habegger's original strategy together with the new ingredient that for
every prime number p, singular moduli are p-adically disperse.
11.05.2021 -- 15:00-16:00 (German time, GMT+2)
Michael Mertens (University of Liverpool)
Weierstrass mock modular forms and vertex operator algebras
Using techniques from the theory of mock modular forms and harmonic Maass
forms, especially Weierstrass mock modular forms, we establish several dimension
formulas for certain holomorphic, strongly rational vertex operator algebras,
complementing previous work by van Ekeren, Möller, and Scheithauer. As an
application, we show that certain special values of the completed Weierstrass zeta
function are rational. This talk is based on joint work with Lea Beneish.
04.05.2021 -- 15:00-16:00 (German time, GMT+2)
Moni Kumari (Bar-Ilan University) Non-vanishing of Hilbert-Poincaré series
Modular forms play a prominent role in the classical as well as
in modern number theory. In the theory of modular forms, there is an
important class of functions called Poincaré series. These functions are
very mysterious and there are many unsolved problems about them. In
particular, the vanishing or non-vanishing of such functions is still
unknown in full generality. In a special case, the latter problem is
equivalent to the famous Lehmer's conjecture which is one of the classical
open problems in the theory. In this talk, I will speak about when these
functions are non-zero for Hilbert modular forms, a natural generalization
of modular forms for totally real number fields.
27.04.2021 -- 15:00-16:00 (German time, GMT+2)
Amanda Folsom (Amherst) Eisenstein series, cotangent-zeta sums, and quantum modular forms
Quantum modular forms, defined in the rationals, transform like
modular forms do on the upper half plane, up to suitably analytic error
functions. After introducing the subject, in this talk, we extend work of
Bettin and Conrey and define twisted Eisenstein series, study their period
functions, and establish quantum modularity of certain cotangent-zeta
sums. The Dedekind sum, discussed by Zagier in his original paper on quantum modular forms, is
a motivating example.
20.04.2021 -- 15:00-16:00 (German time, GMT+2)
Andreas Mono (University of Cologne) On a twisted version of Zagier's $f_{k, D}$ function
We present a twisting of Zagier's $f_{k,D}$ function by a sign function and a genus character. Assuming even and positive integral weight, we inspect its obstruction to modularity, and compute its Fourier expansion. This involves twisted hyperbolic Eisenstein series, locally harmonic Maass forms, and modular cycle integrals, which were studied by Duke, Imamoglu, Toth.
13.04.2021 -- 15:00-16:00 (German time, GMT+2)
Tiago Fonseca (University of Oxford) The algebraic geometry of Fourier coefficients of
Poincaré series
The main goal of this talk is to explain how to characterise Fourier
coefficients of Poincaré series, of positive and negative index, as certain
algebro-geometric invariants attached to the cohomology of modular curves, namely
their `single-valued periods'. This is achieved by a suitable geometric
reformulation of classic results in the theory of harmonic Maass forms. Some
applications to algebraicity questions will also be discussed.
Winter 2020-2021
27.01.2021 -- 16:00-17:00 (German time, GMT+1)
Tom Oliver (University of Nottingham) Twisting moduli, meromorphy and zeros
The zeros of automorphic L-functions are central to certain famous
conjectures in arithmetic. In this talk we will discuss the characterization of
Dirichlet coefficients, with a particular emphasis on applications to vanishing. The
primary focus will be GL(2), but we will also mention higher rank groups - namely,
GL(m) and GL(n) such that m-n=2.
20.01.2021 -- 16:00-17:00 (German time, GMT+1)
Jolanta Marzec (University of Kazimierz Wielki) Algebraicity of special L-values attached to Jacobi forms of higher index
The special values of motivic L-functions have obtained a lot of attention due to
their arithmetic consequences. In particular, they are expected to be algebraic up
to certain factors. The Jacobi forms may also be related to a geometric object
(mixed motive), but their L-functions are much less understood. During the talk we
associate to Jacobi forms (of higher degree, index and level) a standard L-function
and mention some of its analytic properties. We will focus on the ingredients that
come into a proof of algebraicity (up to certain factors) of its special values. The
talk is based on joint work with Thanasis Bouganis:
https://link.springer.com/article/10.1007/s00229-020-01243-w
13.01.2021 -- 16:00-17:00 (German time, GMT+1)
Johann Franke (University of Cologne) Rational functions, modular forms and cotangent sums
There are two elementary methods for constructing elliptic modular forms that
dominate in literature. One of them uses automorphic Poincare series and the other
one theta functions. We start a third elementary approach to modular forms using
rational functions that have certain properties regarding pole distribution and
growth. One can prove modularity with contour integration methods and Weil's
converse theorem, without using the classical formalism of Eisenstein series and
L-functions. This approach to modular forms has several applications, for example to
Eisenstein series, L-functions and Eichler integrals. In this talk we focus on some
applications to cotangent sums.
16.12.2020 -- 16:00-17:00 (German time, GMT+1)
Eugenia Rosu (University of Regensburg) Twists of elliptic curves with CM
We consider certain families of sextic twists of the elliptic curve
$y^2=x^3+1$ that are not defined over $\mathbb{Q}$, but over $\mathbb{Q}(\sqrt{-3})$. We compute a formula
that relates the value of the $L$-function $L(E_D, 1)$ to the square of a trace of a
modular function at a CM point. Assuming the Birch and Swinnerton-Dyer conjecture,
when the value above is non-zero, we should recover the order of the
Tate-Shafarevich group, and under certain conditions, we show that the value is
indeed a square.
09.12.2020 -- 16:00-17:00 (German time, GMT+1)
Ariel Pacetti (Universidad de Cordoba) $\mathbb{Q}$-curves, Hecke characters and some Diophantine equations
In this talk we will investigate integral solutions of the
equation $x^2+dy^2=z^p$, for positive values of $d$. To a
solution, one can attach a Frey curve, which happens to be a $ \mathbb{Q}$-curve. A result of Ribet implies that such a curve is related
to a weight $2$ modular form in $S_2(\Gamma_0(N),\epsilon)$.
Using Hecke characters we will give a precise formula for $N$
and $ \epsilon$ and prove non-existence of solutions in some cases. If time
allows, we will show how a similar idea applies to the equation $x^2+dy^6=z^p$.
02.12.2020 -- 16:00-17:00 (German time, GMT+1)
Gabriele Bogo (TU Darmstadt) Extended modularity arising from the deformation of Riemann surfaces
Modular forms appear in Poincare's work as solutions of certain
differential equations
related to the uniformization of Riemann surfaces.
In the talk I will consider certain perturbations of these differential
equations and
prove that their solutions are given by combinations of quasimodular forms
and Eichler integrals.
The relation between these ODEs and the deformation theory of Riemann
surfaces will be discussed.
By considering the monodromy representation of the perturbed ODEs one can
describe their solutions as components of vector-valued modular forms. This
leads to the general study of functions arising as components of
vector-valued modular forms attached to extensions of symmetric tensor
representations (extended modular forms). If time permits I will discuss
some examples, including certain functions arising in the study of
scattering amplitudes.
25.11.2020 -- 16:00-17:00 (German time, GMT+1)
Kathrin Maurischat (RWTH Aachen) Explicit construction of Ramanujan bigraphs
Ramanujan bigraphs are known to arise as quotients of Bruhat-Tits buildings for
non-split unitary groups U_3. However, these are only implicitly defined. We show
that one also obtains Ramanujan bigraphs in special split cases, and we give
explicit constructions. The proof is obtained by inspecting the automorphic
spectrum for temperedness, and for the construction we introduce the notion of
bi-Cayley graphs. This is joint work with C. Ballantine, S. Evra, B. Feigon, O.
Parzanchevski.
18.11.2020 -- 10:00-11:00 (German time, GMT+1)
Toshiki Matsusaka (Nagoya University) Two analogues of the Rademacher symbol
The Rademacher symbol is a classical object related to the transformation
formula of the Dedekind eta function. In 2007, Ghys showed that the Rademacher
symbol is equal to the linking number of a modular knot and the trefoil knot. In
this talk, we consider two analogues of Ghys' theorem. One is a hyperbolic analogue
of the Rademacher symbol introduced by Duke-Imamoglu-Toth. As they showed, the
hyperbolic Rademacher symbol gives the linking number of two modular knots. I will
give here some explicit formulas for this symbol. The other is the Rademacher symbol
on the triangle group. This symbol is defined from the transformation formula of the
logarithm of a cusp form on the triangle group, and gives the linking number of a
(triangle) modular knot and the (p,q)-torus knot. The latter part is a joint work
(in progress) with Jun Ueki (Tokyo Denki University).
11.11.2020 -- 16:00-17:00 (German time, GMT+1)
Christina Röhrig (Uni Köln) Siegel theta series for indefinite quadratic forms
Due to a result by Vigneras from 1977, there is a quite simple way to determine whether a certain theta series admits modular transformation properties. To be more specific, she showed that solving a differential equation of second order serves as a criterion for modularity. We generalize this result for Siegel theta series of arbitrary genus n. In order to do so, we construct Siegel theta series for indefinite quadratic forms by considering functions which solve an (n x n)-system of partial differential equations. These functions do not only give examples of Siegel theta series, but build a basis of the family of Schwartz functions that generate series which transform like modular forms.
04.11.2020 -- 16:00-17:00 (German time, GMT+1)
Michael Griffin (BYU) Class pairings and elliptic curves
28.10.2020 -- 16:00-17:00 (German time, GMT+1)
Alessandro Lägeler (ETH) Continued fractions and Hardy sums
As was shown by Hickerson in the 70's, the classical Dedekind sums s(d, c) can be represented as sums over the coefficients of the continued fraction expansion of the rational d / c. Hardy sums, the analogous integer-valued objects arising in the transformation of the logarithms of theta functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as coefficients of continued fractions has, however, been missing so far. In this talk, I will argue how one can fill this gap. As an application, I will present a new proof for the fact that the graph of the Hardy sums is dense in R x Z, which was previously proved by Meyer.
21.10.2020 -- 16:00-17:00 (German time, GMT+2)
Yunqing Tang (CNRS & Université Paris-Sud) Reductions of K3 surfaces via intersections on GSpin Shimura varieties
For a K3 surface X over a number field with potentially good reduction
everywhere, we prove that there are infinitely many primes modulo which the
reduction of X has larger geometric Picard rank than that of the generic fiber X. A
similar statement still holds true for ordinary K3 surfaces with potentially good
reduction everywhere over global function fields. In this talk, I will present the
proofs via the (arithmetic) intersection theory on good integral models (and its
special fibers) of GSpin Shimura varieties along with a potential application to a
certain case of the Hecke orbit conjecture of Chai and Oort. This talk is based on
joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik
and Ananth Shankar.
14.10.2020 -- 16:00-17:00 (German time, GMT+2)
Haowu Wang (MPIM Bonn) Root systems and free algebras of modular forms
In this talk we construct some new free algebras of modular forms. For
25 orthogonal groups of signature (2,n) related to irreducible root systems, we
prove that the graded algebras of modular forms on type IV symmetric domains are
freely generated. The proof is based on the theory of Weyl invariant Jacobi forms.
As an application, we show the modularity of formal Fourier-Jacobi expansions for
these groups. This is joint work with Brandon Williams.
Summer 2020
08.04.2020 - 15:00-16:00 (German time, GMT+2)
Danylo Radchenko (ETH Zurich) Universal optimality of the E8 and Leech lattices
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.
15.04.2020 -- 15:00-16:00 (German time, GMT+2)
Paloma Bengoechea (ETH Zurich) Periods of modular functions and Diophantine approximation
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.
22.04.2020 -- 15:00-16:00 (German time, GMT+2)
Jan Vonk (IAS Princeton) Singular moduli for real quadratic fields
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.
29.4.2020 -- 16:00-17:00 (German time, GMT+2)
Nick Andersen (Brigham Young University) Zeros of GL2 L-functions on the critical line
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.
06.5.2020 -- 10:00-11:00 (German time, GMT+2)
Soma Purkait (Tokyo Institute of Technology) Local Hecke algebras and new forms
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.
13.5.2020 -- 15:00-16:00 (German time, GMT+2)
Peter Humphries (University College London) Sparse equidistribution of hyperbolic orbifolds
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.
20.5.2020 -- 15:00-16:00 (German time, GMT+2)
Larry Rolen (Vanderbilt University) Periodicities for Taylor coefficients of half-integral weight modular forms
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.
27.5.2020 -- 15:00-16:00 (German time, GMT+2)
Olivia Beckwith (University of Illinois) Polyharmonic Maass forms and Hecke L-series
We study polyharmonic Maass forms and show that they are related to ray class extensions of real quadratic fields. In particular, we construct a basis for the space of polyharmonic Maass forms for Gamma(N) which is a generalization of a basis constructed by Lagarias and Rhoades for N=1. We show that twisted traces of cycle integrals of certain depth 2 polyharmonic Maass forms are central values of ray class Hecke L-series of real quadratic fields. This is ongoing joint work with Gene Kopp.
3.6.2020 -- 15:00-16:00 (German time, GMT+2)
Dan Fretwell (Bristol University) (Real Quadratic) Arthurian Tales
In recent years there has been a lot of interest in explicitly identifying the global Arthur parameters attached to certain automorphic forms. In particular Chenevier and Lannes were able to completely identify and prove the full lists of Arthur parameters in the case of level 1, trivial weight automorphic forms for definite orthogonal groups of ranks 8, 16 and 24 (not a simple task!).
One finds interesting modular forms hidden in these parameters (e.g. Delta and a handful of special Siegel modular forms of genus 2). Comparing Arthur parameters mod p proves/reproves various Eisenstein congruences for these special modular forms, e.g. the famous 691 congruence of Ramanujan and, more importantly, an example of a genus 2 Eisenstein congruence predicted by Harder (which, up to then, had not been proved for even a single modular form!).
In this talk I will discuss recent work with Neil Dummigan on extending the above to definite orthogonal groups over certain real quadratic fields and try to tell the analogous Arthurian tales (mysteries included).
10.6.2020 -- 15:00-16:00 (German time, GMT+2)
Martin Raum (Chalmers Technical University) Divisibilities of Hurwitz class numbers
Hurwitz class numbers, class numbers of imaginary quadratic fields, and
partition counts are among the most classic quantities in number theory,
and for each of them their factorizations, i.e. divisibilities, are
celebrated open questions. In the case of class numbers the
Cohen-Lehnstra Heuristics provides predictions of of statistical nature.
In the case of partition counts, Ramanujan's congruences have been known
since~1920 and extended by many including key results by Atkin in the
60ies and Ono around 2000.
We investigate the case of divisibility patterns as opposed to
divisibility statistics for Hurwitz class numbers. The Hurwitz class
number formula implies Ramanujan-type congruences, which in contrary to
known congruences for mock theta series can be supported on generating
series with non-holomorphic modular completion. We prove a result that
limits the possibilities of such "non-holomorphic" Ramanujan-type
congruences. When combined with experimental data, this provides
evidence that all of them arise from the class number formula.
24.6.2020 -- 15:00-16:00 (German time, GMT+2)
Sven Möller (Rutgers University) Eisenstein Series, Dimension Formulae and Generalised Deep Holes
of the Leech Lattice Vertex Operator Algebra
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.)
1.7.2020 -- 15:00-16:00 (German time, GMT+2)
Hao Zhang (Sorbonne Université) Elliptic cocycle for GL_N(Z) and Hecke operators
A classical result of Eichler, Shimura and Manin asserts that the map that assigns to a cusp form f its period polynomial r_f is a Hecke equivariant map. We propose a generalization of this result to a setting where r_f is replaced by a family of rational function of N variables equipped with the action of GLN(Z). For this purpose, we develop a theory of Hecke operators for the elliptic cocycle recently introduced by Charollois. In particular, when f is an eigenform, the corresponding rational function is also an eigenvector respect to Hecke operator for GLN. Finally, we give some examples for Eisenstein series and the Ramanujan Delta function.
8.7.2020 -- 15:00-16:00 (German time, GMT+2)
Shaul Zemel (Hebrew University of Jerusalem) Shintani Lifts of Nearly Holomorphic Modular Forms
The Shintani lift is a classical construction of modular
forms of half-integral weight from modular forms of even integral
weight. Soon after its definition it was shown to be related to
integration with respect to theta kernel. The development of the theory
of regularized integrals opens the question to what modular forms of
half-integral weight arise as regularized Shintani lifts of various
kinds of integral weight modular forms. We evaluate these lifts for the
case of nearly holomorphic modular forms, which in particular shows
that when the depth is smaller than the weight, the Shintani lift is
also nearly holomorphic. This evaluation requires the determination of
certain Fourier transforms, which are interesting on their own right.
This is joint work with Yingkun Li.
15.7.2020 -- 13:00-14:00 (German time, GMT+2)
Nikos Diamantis (University of Nottingham) Twisted L-functions and a conjecture by Mazur, Rubin and Stein
We will discuss analytic properties of L-functions twisted
by an additive character. As an implication, a full proof of a
conjecture of Mazur, Rubin and Stein will be outlined. This is a
report on joint work with J. Hoffstein, M. Kiral and M. Lee.
22.7.2020 -- 13:00-14:00 (German time, GMT+2)
Anna von Pippich (TU Darmstadt) An analytic class number type formula for the Selberg zeta function
In this talk, we report on an explicit formula for the special value at s=1 of the derivative of the Selberg zeta function for the modular group Γ. The formula is a consequence of a generalization of the arithmetic Riemann-Roch theorem of Deligne and Gillet-Soulé to the case of the trivial sheaf on the upper half plane modulo Γ, equipped with the hyperbolic metric. This is joint work with Gerard Freixas.
