13.2.2024 -- 16:00-17:00 (CET, GMT+1)
Elisa Lorenzo Garcia (Université de Neuchâtel) On the conductor of Ciani plane quartics
In this talk we will discuss the determination of the conductor exponent of non-special Ciani quartics at primes of potentially good reduction in terms of their Ciani invariants.
As an intermediate step, we will provide a reconstruction algorithm to construct Ciani quartics with given invariants.
During the talk we will consider many particular examples and extensions of the presented results. (j.w.w. I. Bouw, N. Coppola and A. Somoza)
6.2.2024 -- 16:00-17:00 (CET, GMT+1)
Rachel Newton (King’s College London) Evaluating the wild Brauer group
The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points.
The Brauer--Manin pairing cuts out a subset of the adelic points, called the Brauer--Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer--Manin set is empty then we say there's a Brauer--Manin obstruction to the existence of rational points on X.
Computing the Brauer--Manin pairing involves evaluating elements of the Brauer group of X at local points.
If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p.
For p-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision.
Classifying Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Bloch and Kato.
Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer--Manin obstruction.
This is joint work with Martin Bright.
30.1.2024 -- 16:00-17:00 (CET, GMT+1)
Matthew de Courcy-Ireland (Stockholm University) Six-dimensional sphere packing and linear programming
This talk is based on joint work with Maria Dostert and Maryna Viazovska. We prove that the Cohn--Elkies linear programming bound is not sharp for sphere packing in dimension 6.
This is in contrast to Viazovska's sharp bound in dimension 8, even though it is believed that closely related lattices achieve the optimal densities in both dimensions.
The proof uses modular forms to construct feasible points in a dual program, generalizing a construction of Cohn and Triantafillou to the case of odd weight and non-trivial Dirichlet character.
Non-sharpness of linear programming is demonstrated by comparing this dual bound to a stronger upper bound obtained from semidefinite programming by Cohn, de Laat, and Salmon.
Our construction has vanishing Fourier coefficients along an arithmetic progression, which can be explained using skew self-adjointness of Hecke operators.
23.1.2024 -- 9:00-10:00 (CET, GMT+1)
Wee Teck Gan (National University of Singapore) The BZSV duality and the relative Langlands program
I will discuss a duality of Hamiltonian group varieties proposed in a recent preprint of Ben-Zvi, Sakellaridis and Venkatesh, which gives a new paradigm for the relative Langlands program.
I will then discuss a joint work with Bryan Wang on instances of this duality for certain Hamiltonian varieties which quantize to generalized Whittaker models.
16.1.2024 -- 16:00-17:00 (CET, GMT+1)
Pietro Mercuri (University of Rome - La Sapienza) Automorphism group of Cartan modular curves
We consider the modular curves associated to a Cartan subgroup of GL(2,Z/nZ) or to a particular class of subgroups of GL(2,Z/nZ) containing the Cartan subgroup as a normal subgroup.
We describe the automorphism group of these curves when the level is large enough.
If time permits, we give a sketch of the proof.
12.12.2023 -- 16:00-17:00 (CET, GMT+1)
Anke Pohl (University of Bremen) Resonances of Schottky surfaces
The investigation of L^2-Laplace eigenvalues and eigenfunctions for hyperbolic surfaces of finite area is a classical and exciting topic at the intersection of number theory, harmonic analysis and mathematical physics.
In stark contrast, for (geometrically finite) hyperbolic surfaces of infinite area, the discrete L^2-spectrum is finite.
A natural replacement are the resonances of the considered hyperbolic surface, which are the poles of the meromorphically continued resolvent of the Laplacian.
These spectral entities also play an important role in number theory and various other fields, and many fascinating results about them have already been found;
the generalization of Selberg's 3/16-theorem by Bourgain, Gamburd and Sarnak is a well-known example.
However, an enormous amount of the properties of such resonances, also some very elementary ones, is still undiscovered.
A few years ago, by means of numerical experiments, Borthwick noticed for some classes of Schottky surfaces (hyperbolic surfaces of infinite area without cusps and conical singularities) that their sets of resonances exhibit unexcepted and nice patterns, which are not yet fully understood.
After a brief survey of some parts of this field, we will discuss an alternative numerical method, combining tools from dynamics, zeta functions, transfer operators and thermodynamic formalism, functional analysis and approximation theory.
The emphasis of the presentation will be on motivation, heuristics and pictures. This is joint work with Oscar Bandtlow, Torben Schick and Alex Weisse.
5.12.2023 -- 16:00-17:00 (CET, GMT+1)
Min Lee (University of Bristol) Murmurations of holomorphic modular forms in the weight aspect
In April 2022, He, Lee, Oliver, and Pozdnyakov made an interesting discovery using machine learning – a surprising correlation between the root numbers of elliptic curves and the coefficients of their L-functions.
They coined this correlation 'murmurations of elliptic curves'. Naturally, one might wonder whether we can identify a common thread of 'murmurations' in other families of L-functions.
In this talk, I will introduce a joint work with Jonathan Bober, Andrew R. Booker and David Lowry-Duda, demonstrating murmurations in holomorphic modular forms.
28.11.2023 -- 16:00-17:00 (CET, GMT+1)
Peter Humphries (University of Virginia) Restricted Arithmetic Quantum Unique Ergodicity
The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds.
This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms.
I will discuss a variant of this problem in this arithmetic setting concerning the mass equidistribution of Hecke-Maass cusp forms on submanifolds of the modular surface.
21.11.2023 -- 16:00-17:00 (CET, GMT+1)
Steve Lester (King's College London) Around the Gauss circle problem
Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-R disc by its area is O(R1/2+o(1)).
One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square-root cancellation of sums of these random variables.
In this talk I will examine this heuristic and discuss how lattice points near the circle interact with one another.
This is joint work with Igor Wigman.
14.11.2023 -- 16:00-17:00 (CET, GMT+1)
Robin Zhang (Massachusetts Institute of Technology) Harris–Venkatesh plus Stark
The class number formula describes the behavior of the Dedekind zeta function at s=0 and s=1.
The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s=0 and s=1 in terms of units.
The Harris–Venkatesh conjecture describes the residue of Stark units modulo p, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader conjectures of Venkatesh, Prasanna, and Galatius.
In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.
7.11.2023 -- 16:00-17:00 (CET, GMT+1)
Xiaoyu Zhang (University Duisburg-Essen) Global theta correspondence mod p for unitary groups
Theta correspondence is a very important tool in Langlands program.
A fundamental problem in theta correspondence is the non-vanishing of the theta lifting of an automorphic representation.
In this talk, we would like to consider a mod p version of the non-vanishing problem for global theta correspondence for certain reductive dual pairs of unitary groups.
We approach this by looking at the Fourier coefficients of the theta lifting and reduce the problem to the equidistribution of unipotent orbits.
31.10.2023 -- 16:00-17:00 (CET, GMT+1)
Aaron Pollack (University of California San Diego) Arithmeticity of modular forms on G_2
Holomorphic modular forms on Hermitian tube domains have a good notion of Fourier expansion and Fourier coefficients.
These Fourier coefficients give the holomorphic modular forms an arithmetic structure: there is a basis of the space of holomorphic modular forms for which all Fourier coefficients of all elements of the basis are algebraic numbers.
The group G_2 does not have an associated Shimura variety, but nevertheless there is a class of automorphic functions on G_2 which possess a semi-classical Fourier expansion, called the quaternionic modular forms.
I will explain the proof that (in even weight at least 6) the cuspidal quaternionic modular forms possess an arithmetic structure, defined in terms of Fourier coefficients.
24.10.2023 -- 16:00-17:00 (CEST, GMT+2)
Aleksander Horawa (University of Oxford) Siegel modular forms and higher algebraic cycles
In recent work with Kartik Prasanna, we propose an explicit relationship between the cohomology of vector bundles on Siegel modular threefolds and higher Chow groups (aka motivic cohomology groups).
For Yoshida lifts of Hilbert modular forms, we use a result of Ramakrishnan to prove our conjecture.
For Yoshida lifts of Bianchi modular forms, we show that our conjecture implies the conjecture of Prasanna—Venkatesh.
Summer 2023
11.07.2023 -- 16:00-17:00 (CEST, GMT+2)
Mathilde Gerbelli-Gauthier (McGill University) Counting non-tempered automorphic forms using endoscopy
How many automorphic representations of level n have a specified
local factor at the infinite places? When this local factor is a discrete
series representation, this question is asymptotically well-undersertood as
n grows. Non-tempered local factors, on the other hand, violate the
Ramanujan conjecture and should be very rare. We use the endoscopic
classification for representations to quantify this rarity in the case of
cohomological representations of unitary groups, and discuss some
applications to the growth of cohomology of Shimura varieties.
04.07.2023 -- 16:00-17:00 (CEST, GMT+2)
Annika Burmester (Bielefeld University) A general view on multiple zeta values, modular forms and related q-series
Multiple zeta values and modular forms have a deep, partly
mysterious, connection. This can be seen in the Broadhurst-Kreimer
conjecture, which was made partly explicit by Gangl-Kaneko-Zagier in
2006. Further, multiple zeta values occur in the Fourier expansion of
multiple Eisenstein series as computed by Bachmann. We will study this
connection in more details on a formal level. This means, we review
formal multiple zeta values and then introduce the algebra G^f, which
should be seen as a formal version of multiple Eisenstein series, and
also multiple q-zeta values and polynomial functions on partitions
simultaneously. We will give a surjective algebra morphism from G^f into
the algebra of formal multiple zeta values.
27.06.2023 -- 16:00-17:00 (CEST, GMT+2)
Nina Zubrilina (Princeton University) Root Number Correlation Bias of Fourier Coefficients of Modular Forms
In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a
striking
oscillating pattern in the average value of the P-th Frobenius trace of elliptic
curves of
prescribed rank and conductor in an interval range. Sutherland discovered that this
bias
extends to Dirichlet coefficients of a much broader class of arithmetic L-functions
when
split by root number. In my talk, I will discuss this root number correlation bias when
the average is taken over all weight 2 modular newforms. I will point to a source of
this
phenomenon in this case and compute the correlation function exactly.
20.06.2023 -- 16:00-17:00 (CEST, GMT+2)
Salim Tayou (Harvard University) Mixed mock modularity of special divisors
Kudla-Millson and Borcherds have proved some time ago that
the generating series of special divisors in orthogonal Shimura
varieties are modular forms. In this talk, I will explain an extension
of these results to toroidal compactifications where we prove that the
generating series is a mixed mock modular form. More precisely, we
find an explicit completion using theta series associated to rays in
the cone decomposition. The proof relies on intersection theory at the
boundary of the Shimura variety. This recovers and refines recent results of Bruinier and Zemel.
The result of this talk are joint work with Philip Engel and Francois Greer.
13.06.2023 -- 09:00-10:00 (CEST, GMT+2)
Sander Zwegers (University of Cologne) Indefinite Theta Functions: something old, something new
In this talk we give an overview of the theory of indefinite
theta functions and discuss some recent results.
06.06.2023 -- 16:00-17:00 (CEST, GMT+2)
Ezra Waxman (University of Haifa) Artin's primitive root conjecture: classically and over Fq[T]
In 1927, E. Artin proposed a conjecture for the number of primes p ≤ x, for which g generates (ℤ/pℤ)x. By observing numerical
deviations from Artin's originally predicted asymptotic, Derrick and Emma Lehmer
(1957) identified the need for an additional correction factor; leading to a
modified conjecture that was eventually proved correct by Hooley (1967), under the
assumption of the Generalized Riemann Hypothesis (GRH). In this talk we discuss
several variants of Artin's conjecture: namely an "Artin Twin Primes Conjecture", as
well as an appropriate analogue of Artin's primitive root conjecture for algebraic
function fields.
30.05.2023 -- 10:30-11:30 (CEST, GMT+2)
Toshiki Matsusaka (Kyushu University) Discontinuity property of a certain Habiro series at roots of unity
The object of this talk is a family of q-series originating from Habiro's
work on the Witten-Reshetikhin-Turaev invariants. The q-series usually make sense
only when q is a root of unity, but for some instances, it also determines a
holomorphic function on the open unit disc. Such an example is Habiro's unified WRT
invariant H(q) for the Poincaré homology sphere. In 2007, Hikami observed its
discontinuity at roots of unity. More precisely, the value of H(ζ) at a root of
unity is 1/2 times the limit value of H(q) as q tends towards ζ radially within the
unit disc. In this talk, we give an explanation of the appearance of the 1/2 factor
and generalize Hikami's observations by using Bailey's lemma and the theory of
mock/false theta functions.
23.05.2023 No talk
16.05.2023 -- 16:00-17:00 (CEST, GMT+2)
Fabrizio Andreatta (University of Milan) Endoscopy for GSp(4) and rational points on elliptic curves
I report on a joint project with M. Bertolini , M.A. Seveso and R.
Venerucci, aimed at studying the equivariant BSD conjecture for rational
elliptic curves twisted by certain self-dual 4-dimensional Artin
representations in situations of odd analytic rank. We use the endoscopy
for GSp(4) to construct Selmer classes related to the relevant (complex
and p-adic) L-values via explicit reciprocity laws.
09.05.2023 -- 16:00-17:00 (CEST, GMT+2)
Sachi Hashimoto (MPI Leipzig) p-adic Gross-Zagier and rational points on modular curves
Faltings' theorem states that there are finitely many rational
points on a nice projective curve defined over the rationals of genus at
least 2. The quadratic Chabauty method makes explicit some cases of
Faltings' theorem. Quadratic Chabauty has recent notable success in
determining the rational points of some modular curves. In this talk, I
will explain how we can leverage information from p-adic Gross-Zagier
formulas to give a new quadratic Chabauty method for certain modular
curves. Gross-Zagier formulas relate analytic quantities (special values
of p-adic L-functions) to invariants of algebraic cycles (the p-adic
height and logarithm of Heegner points). By using p-adic Gross-Zagier
formulas, this new quadratic Chabauty method makes essential use of
modular forms to determine rational points.
02.05.2023 -- 16:00-17:00 (CEST, GMT+2)
Abhishek Saha (Queen Mary University of London) Mass equidistribution for Saito-Kurokawa lifts
The Quantum Unique Ergodicity (QUE) conjecture was proved in
the classical case for Maass forms of full level in the eigenvalue
aspect by Lindenstrauss and Soundararajan, and for holomorphic forms in
the weight aspect by Holowinsky and Soundararajan. In this talk, I will
discuss some joint work with Jesse Jaasaari and Steve Lester on the
analogue of the QUE conjecture in the weight aspect for holomorphic
Siegel cusp forms of degree 2 and full level. Assuming the Generalized
Riemann Hypothesis (GRH) we establish QUE for Saito-Kurokawa lifts as
the weight tends to infinity. As an application, we prove the
equidistribution of zero divisors of Saito-Kurokawa lifts.
25.04.2023 -- 16:00-17:00 (CEST, GMT+2)
Oguz Gezmis (Heidelberg University) Almost holomorphic Drinfeld modular forms
In his series of papers from 1970s, Shimura analyzed a
non-holomorphic operator, nowadays called the Maass-Shimura operator,
and later extensively studied almost holomorphic modular forms. He also
discovered their role on constructing class fields as well as the
connection with periods of CM elliptic curves. In this talk, our first
goal is to introduce their positive characteristic counterpart, almost
holomorphic Drinfeld modular forms. We further relate them to Drinfeld
quasi-modular forms which leads us to generalize the work of Bosser and
Pellarin to a certain extend. Moreover, we introduce the Maass-Shimura
operator $\delta_k$ in our setting for any nonnegative integer k
and investigate the relation between the periods of CM Drinfeld modules
and the values at CM points of arithmetic Drinfeld modular forms under
the image of $\delta_k$. If time permits, we also reveal how to
construct class fields by using such values. This is a joint work with
Yen-Tsung Chen.
18.04.2023 -- 16:00-17:00 (CEST, GMT+2)
Jan-Willem van Ittersum (MPIM Bonn) On quasimodular forms associated to projective representations of symmetric groups
We explain how one can naturally associate a quasimodular form to a
representation of a symmetric group. We determine its growth and explain
how this construction is applied to several problems in enumerative
geometry. Finally, we discuss the difference between linear and projective
representations. This is based on joint work with Adrien Sauvaget.
Winter 2022-2023
07.02.2023 -- 16:00-17:00 (CET, GMT+1)
Giulia Cesana (University of Cologne) Asymptotic equidistribution for partition statistics and topological invariants
Throughout mathematics, the equidistribution properties of certain objects
are a central theme studied by many authors. In my talk I am going to speak about a
joint project with William Craig and Joshua Males, where we provide a general
framework for proving asymptotic equidistribution, convexity, and log-concavity of
coefficients of generating functions on arithmetic progressions.
31.01.2023 -- 16:00-17:00 (CET, GMT+1)
Sandro Bettin (University of Genova) Continuity and value distribution of quantum modular forms
Quantum modular forms are functions f defined on the rationals whose period
functions, such as ψ(x):= f(x) - x-k f(-1/x) (for level 1), satisfy some
continuity properties. In the case of k=0, f can be interpreted as a Birkhoff sums
associated with the Gauss map. In particular, under mild hypotheses on G, one can
show convergence to a stable law. If k is non-zero, the situation is rather
different and we can show that mild conditions on ψ imply that f itself has to
exhibit some continuity property. Finally, we discuss the convergence in
distribution also in this case. This is a joint work with Sary Drappeau.
24.01.2023 -- 16:00-17:00 (CET, GMT+1)
Riccardo Salvati Manni (Sapienza University of Rome) Slope of Siegel modular forms: some geometric applications
We study the slope of modular forms on the Siegel space. We
will recover known divisors of minimal slope for g≤5 and we
discuss the Kodaira dimension of the moduli space of principally
polarized abelian varieties Ag (and eventually of the generalized
Kuga's varieties). Moreover we illustrate the cone of moving divisors
on Ag. Partly motivated by the generalized Rankin-Cohen bracket, we
construct a non-linear holomorphic differential operator that sends
Siegel modular forms to Siegel cusp forms, and we apply it to produce
new modular forms. Our construction recovers the known divisors of
minimal moving slope on Ag for g≤5.
17.01.2023 -- 09:00-10:00 (CET, GMT+1)
Soumya Das (Indian Institute of Science) Sup-norms of automorphic forms on average
Bounding the sup-norms of automorphic forms has been a very
active area in research in recent times.
Whereas lot of nice results are known for small rank groups, like
GL(2), almost nothing is known for, say, Siegel or Jacobi modular
forms of higher degrees. In this talk we aim to discuss some
conjectures and results in this area. We use either the theory of
Poincare series or averages of central values of L-functions to tackle
this problem. Our methods have the benefit of having a hands-on
approach and fits into many situations.
10.01.2023 -- 09:00-10:00 (CET, GMT+1)
Yota Maeda (Kyoto University) Deligne-Mostow theory and beyond
Ball quotients have been studied extensively in algebraic
geometry from the aspect of moduli spaces, and in number theory with
emphasis on the relation with modular forms. The Deligne-Mostow theory
gives them moduli interpretation through the isomorphism between the
Baily-Borel compactifications of them and certain GIT quotients.
In this talk, I will discuss whether the isomorphisms given by the
Deligne-Mostow theory are lifted to other compactifications from the
viewpoint of modular forms and pursue "better" compactifications.
Moreover, I will also clarify their connection with the recent
development in the minimal model program. This work is based on a joint work with Klaus Hulek (Leibniz University Hannover).
20.12.2022 -- 16:00-17:00 (CET, GMT+1)
Morten Risager (University of Copenhagen) Distributions of Manin's iterated integrals
We recall the definition of Manin's iterated integrals of a given length.
We then explain how these generalise modular symbols and certain aspects of the
theory of multiple zeta-values. In length one and two we determine the limiting
distribution of these iterated integrals. Maybe surprisingly, even if we can compute
all moments also in higher length we cannot in general determine a distribution for
length three or higher. This is joint work with Y. Petridis and with N. Matthes.
13.12.2022 -- 16:00-17:00 (CET, GMT+1)
Alex Dunn (Caltech) Bias in cubic Gauss sums: Patterson's conjecture
We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalised Riemann hypothesis)
concerning the bias of cubic Gauss sums.
This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846.
One important byproduct of our proof is that we show
Heath-Brown's cubic large sieve is sharp under GRH.
This disproves the popular belief that the cubic large sieve can be
improved.
An important ingredient in our proof is a dispersion estimate for cubic
Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term.
In this talk, I will introduce some aspects of the theory of arithmetic quantum chaos, focusing on the quantum unique ergodicity theorem for automorphic forms on the modular surface. Then I will give some results on effective decorrelation of Hecke eigenforms and the cubic moment of Hecke-Maass cusp forms. The proofs are based on the analytic theory of L-functions.
29.11.2022 -- 09:00-10:00 (CET, GMT+1)
Shouhei Ma (Tokyo Institute of Technology) Vector-valued orthogonal modular forms
I will talk about the theory of vector-valued modular forms on domains of type IV, with some emphasis on its algebro-geometric aspects.
22.11.2022 -- 16:00-17:00 (CET, GMT+1)
Eran Assaf (Dartmouth College) Orthogonal modular forms, Siegel modular forms and Eisenstein congruences
The theta correspondence between the orthogonal group and the symplectic group provides a cornerstone for studying Siegel modular forms via orthogonal modular forms. In this work, we make this correspondence completely explicit, with precise level structure for low to moderate even rank and nontrivial discriminant. Guided by computational discoveries, we prove congruences between eigenvalues of classical modular forms and eigenvalues of genuine Siegel modular forms, obtain formulas for the number of neighbors in terms of eigenvalues of classical modular forms, and formulate some conjectures that arise naturally from the data. This is joint work with Dan Fretwell, Colin Ingalls, Adam Logan, Spencer Secord, and John Voight
15.11.2022 -- 16:00-17:00 (CET, GMT+1)
Congling Qiu (Yale University) Modularity and automorphy of cycles on Shimura varieties
Algebraic cycles are central objects in algebraic/arithmetic geometry and problems around them are very difficult.
For Shimura varieties modularity of generating series with coefficients being algebraic cycles has been proved useful in the of study of algebraic cycles.
A closely related problem is the automorphy of representations spanned by algebraic cycles.
I will discuss the history of these problems some progress and applications.
01.11.2022 -- 16:00-17:00 (CET, GMT+1)
Manuel Müller (TU Darmstadt) The invariants of the Weil representation of SL2(ℤ)
The transformation behaviour of the vector valued theta function of a positive definite even lattice under the metaplectic group Mp2(ℤ) is described by the Weil representation.
This representation plays an important role in the theory of automorphic forms.
We show that its invariants are induced from 5 fundamental invariants.
Summer 2022
28.06.2022 -- 16:00-17:00 (CEST, GMT+2)
Yichao Zhang (Harbin Institute of Technology) Rationality of the Petersson Inner Product of Generally Twisted Cohen Kernels
Kohnen and Zagier showed that the Petersson inner product of Cohen
kernels at integers of opposite parity is rational in the critical strip. Later
Diamantis and O'Sullivan generalized such rationality to the Petersson inner
product with one of the two Cohen kernels acted by a Hecke operator. In this talk,
using Diamantis and O'Sullivan's twisted double Eisenstein series, we twist one of
the two Cohen kernels by a general rational number and prove a similar rationality
result. This is a joint work with Yuanyi You.
21.06.2022 -- 16:00-17:00 (CEST, GMT+2)
Jeff Manning (MPIM Bonn) The Wiles-Lenstra-Diamond numerical criterion over imaginary quadratic fields
Wiles' modularity lifting theorem was the central argument in
his proof of modularity of (semistable) elliptic curves over Q, and hence
of Fermat's Last Theorem. His proof relied on two key components: his
"patching" argument (developed in collaboration with Taylor) and his
numerical isomorphism criterion.
In the time since Wiles' proof, the patching argument has been generalized
extensively to prove a wide variety of modularity lifting results. In
particular Calegari and Geraghty have found a way to generalize it to prove
potential modularity of elliptic curves over imaginary quadratic fields
(contingent on some standard conjectures). The numerical criterion on the
other hand has proved far more difficult to generalize, although in
situations where it can be used it can prove stronger results than what can
be proven purely via patching.
In this talk I will present joint work with Srikanth Iyengar and
Chandrashekhar Khare which proves a generalization of the numerical
criterion to the context considered by Calegari and Geraghty (and
contingent on the same conjectures). This allows us to prove integral "R=T"
theorems at non-minimal levels over imaginary quadratic fields, which are
inaccessible by Calegari and Geraghty's method. The results provide new
evidence in favor of a torsion analog of the classical Langlands
correspondence.
14.06.2022 -- 16:00-17:00 (CEST, GMT+2)
Paul Nelson (IAS) The orbit method, microlocal analysis and applications to L-functions
L-functions are generalizations of the Riemann zeta function. Their
analytic properties control the asymptotic behavior of prime numbers in
various refined senses. Conjecturally, every L-function is a "standard
L-function" arising from an automorphic form. A problem of recurring
interest, with widespread applications, has been to establish nontrivial
bounds for L-functions. I will survey some recent results addressing this
problem. The proofs involve the analysis of integrals of automorphic
forms, approached through the lens of representation theory. I will
emphasize the role played by the orbit method, developed in a quantitative
form along the lines of microlocal analysis. The results/methods to be
surveyed are the subject of the following papers/preprints:
https://arxiv.org/abs/1805.07750
https://arxiv.org/abs/2012.02187
https://arxiv.org/abs/2109.15230
Determination of modular forms is one of the fundamental and
interesting problems in number theory. It is known that if the Hecke
eigenvalues of two newforms agree for all but finitely many primes, then
both the forms are the same. In other words, the set of Hecke eigenvalues
at primes determines the newform uniquely and this result is known as the
multiplicity one theorem. In the case of Siegel cuspidal eigenforms of
degree two, the multiplicity one theorem has been proved only recently in
2018 by Schmidt. In this talk, we refine the result of Schmidt by showing
that if the Hecke eigenvalues of two Siegel eigenforms of level 1 agree at
a set of primes of positive density, then the eigenforms are the same (up
to a constant). We also distinguish Siegel eigenforms from the signs of
their Hecke eigenvalues. The main ingredient to prove these results are
Galois representations attached to Siegel eigenforms, the Chebotarev
density theorem and some analytic properties of associated L-functions.
31.05.2022 -- 16:00-17:00 (CEST, GMT+2)
Nikos Diamantis (University of Nottingham) L-series associated with harmonic Maass forms and their values
We define a L-series for harmonic Maass forms and discuss their functional
equations. A converse theorem for these L-series is given. As an application, we
interpret as proper values of our L-functions certain important quantities that
arose in works by Bruinier-Funke-Imamoglu and Alfes-Schwagenscheidt, and which
they had philosophically viewed as "central L-values". This is joint work with M.
Lee, W. Raji and L. Rolen.
17.05.2022 -- 9:00-10:00 (CEST, GMT+2)
Sudhir Pujahari (NISER) Sato-Tate conjecture in arithmetic progressions for certain families of elliptic curves
In this talk we will study moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. In conclusion, we will obtain the Sato-Tate distribution for the trace of certain families of Elliptic curves. As a special case we will recover a result of Birch proving Sato-Tate distribution for certain family of elliptic curves. Moreover, we will see that these results follow from asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. This is a joint work with Kathrin Bringmann and Ben Kane.
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.
