Building Bridges: 1st EU-US conference on Automorphic Forms and related topics
Schedule
| Monday 6th | Tuesday 7th | Wednesday 8th | Thursday 9th | Friday 10th | |
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| 9:00 | Talks | Talks | Talks | Talks | |
| 9:30 | |||||
| 10:00 | Talks | ||||
| 10:30 | |||||
| 11:00 | Talks | Talks | Talks | ||
| Talks | |||||
| 11:30 | |||||
| 12:00 | |||||
| Lunch | |||||
| Lunch | |||||
| 12:30 | Lunch | Lunch | Lunch | ||
| 13:00 | |||||
| 13:30 | |||||
| 14:00 | Talks | Talks | Talks | Talks | Talks |
| 14:30 | |||||
| 15:00 | |||||
| 15:30 | |||||
| 16:00 | Talks | Talks | Talks | Talks | |
| 16:30 | |||||
| 17:00 | |||||
| 17:30 | |||||
| 18:00 | |||||
Talks and Abstracts
Download list of talks and list of abstracts as pdf-file.
List of talks on:
MondayTuesdayWednesdayThursdayFriday
Monday 6th
| 9:00 to 10:00 Jeff Hoffstein: TBA. | |
| This talk will begin with a survey of the Siegel zero and its interaction with the lives of the speaker and his former thesis adviser. Following this, we will look at shifted multiple Dirichlet series, and their applications to second moments of $GL(2)$ $L$-series twisted by characters. | |
| 10:10 to 10:30 Henrik Bachmann: Multiple Eisenstein series. | |
| Multiple zeta values are natural generalizations of the Riemann zeta values. They have arisen in various areas of mathematics such as number theory, algebraic geometry, knot theory and mathematical physics since the early 1990s. Riemann zeta values give the constant term in the Fourier expansion of the classical Eisenstein series. In the same way the recently introduced multiple Eisenstein series have the multiple zeta values as the constant term in their Fourier expansion. The case of double Eisenstein series was considered by Gangl, Kaneko and Zagier a few years ago. It was shown that the Double Eisenstein series fulfill the same relations as the associated double zeta values and that the Fourier expansion of these functions are of arithmetical interest. In the talk i want to give a brief introduction to multiple Eisenstein series and the computation of their Fourier expansions. | |
| 11:00 to 11:45 Jennifer Beineke: Oppenheim Summation and the Atkinson-Jutila Formula for the square of the Riemann Zeta Function. | |
| In a 1927 paper, Oppenheim generalized Vorono\"{\i}'s summation formula to obtain a representation for $D_a(x) = \sum_{n \le x} \sigma_a(n)$ in terms of Bessel functions. We will first describe a smooth version of Oppenheim summation. We will then discuss how its application can provide asymptotic results for moments of the Riemann zeta function. In particular, we will relate Oppenheim summation to a formula of Jutila, which is a modified version of Atkinson's 1949 formula for the error term in the asymptotic expansion of the second moment of $\zeta(s)$. This work is joint with Dan Bump. | |
| 12:00 to 12:30 Wenzhi Luo: Rankin-Selberg L-functions at the special points. | |
| The special L-value at the spectral point for the Rankin-Selberg convolutions of a holomorphic newform and an Maass form has interesting implication on the existence of cusp forms on generic hyperbolic surfaces. we derive a new asymptotic formula for its spectral second moment with a power saving in the remainder term. | |
| 14:00 to 14:45 Nils Skoruppa: On Computations of Half-Integral Weight Modular Forms. | |
| 15:00 to 15:30 Cristina Ballantine: Unitary groups over p-adic fields and biregular Ramanujan graphs. | |
| Expander graphs are well-connected yet sparse graphs. The expansion property of a regular graph is governed by the second largest eigenvalue of the adjacency matrix. One can consider quotients of the Bruhat-Tits building of GL(n), n=2,3, over a p-adic field and view them as graphs. In this context the relationship between regular expander graphs and the Ramanujan Conjecture is well understood and has led to the definition and construction of asymptotically optimal regular expanders called Ramanujan graphs. The notion of Ramanujan graph can be extended to bigraphs (i.e., biregular, bipartite graphs). In this talk I will use the representation theory of SU(3) over a p-adic field to investigate whether certain quotients of the associated Bruhat-Tits tree are Ramanujan bigraphs. I will show that a quotient of the Bruhat-Tits tree associated with a quasisplit form G of SU(3) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. (This is joint work with Dan Ciubotaru). | |
| 16:00 to 16:20 Howard Skogman: Some results on graph covers and zeta functions of graphs. | |
| We determine when a set of graph covers has a common cover and some results about the zeta functions of covering graphs. | |
| 16:30 to 17:00 Shaul Zemel: A Gross-Kohnen-Zagier Type Theorem for Higher-Codimensional Heegner Cycles. | |
| The multiplicative Borcherds singular theta lift is a well-known tool for obtaining automorphic forms with known zeros and poles on quotients of orthogonal spaces. This has been used by Borcherds in order to prove a generalization of the Gross-Kohnen-Zagier Theorem, stating that certain combinations of Heegner points behave, in an appropriate quotient of the Jacobian variety of the modular curve, like the coeffcients of a modular form of weight 3/2. The result of Borcherds proves that certain combinations of Heegner divisors on quotients of higher-dimensional orthognal spaces become coefficients of a modular form of weight $1+b_2$ , thus general-zing also the Hirzebruch-Zagier Theorem on Hirzebruch-Zagier divisors, and many other results. In many cases (such as those mentioned above) these quotients of orthognal spaces are Shimura varieties, hence admit variations of Hodge structures arising from certain Kuga-Sato type varieties (or universal families) lying over the Shimura variety. We show how that additive singular theta lift (which was also constructed by Borcherds) can be used in order to prove that the (fundamental cohomology classes of) higher codimensional Heegner cycles become, in certain quotient groups, coeffcients of modular forms as well. In particular, and the modular (or Shimura) curve case, these cycles are the the classical CM cycles inside (the product of) elliptic curves (or Abelian varieties with QM in the Shimura curve case). | |
Tuesday 7th
| 9:00 to 10:00 Charles Conley: Invariant Differential Operators. | |
| We will describe Helgason's method of computing algebras of invariant differential operators on homogeneous spaces, using various Jacobi group actions as examples. We will discuss in particular the factorization of invariant operators into first order covariant operators between different homogeneous vector bundles, and the computation of the central invariant operators by which the center of the universal enveloping algebra acts. This will be a survey talk, so we will do our best not to assume much prior knowledge of Lie theory. | |
| 10:10 to 10:30 Matthew Krauel: Jacobi Forms In The Theory Of Vertex Operator Algebras. | |
| 11:00 to 11:30 Amanda Folsom: Almost harmonic Maass forms and Kac-Wakimoto characters. | |
| We address a question of Kac, and explain the modular properties of certain characters due to Kac and Wakimoto pertaining to sl(m|n)^, where n is a positive integer. We prove that these characters are essentially holomorphic parts of generalizations of harmonic weak Maass forms. Using a different approach involving meromorphic Jacobi forms, this generalizes prior works which treat only the case n=1. This is joint work with Kathrin Bringmann (University of Cologne). | |
| 11:40 to 12:10 Ben Kane: Locally harmonic Maass foms and rational periods. | |
| In this talk, we will define certain functions which satisfy weight $2-2k$ modularity and are harmonic away from certain geodesics. These functions are connected to the kernel function of Kohnen and Zagier for the the Shimura and Shintani lifts (between weights $k+1/2$ and $2k$) through natural differential operators. Using these locally harmonic Maass forms, we give a new proof of the rationality of the periods of the weight $2k$ (hyperbolic Poincaré series) cusp forms appearing in Kohnen and Zagier's kernel function. This talk is based on joint work with Kathrin Bringmann and Winfried Kohnen. | |
| 14:00 to 14:45 Karl Mahlburg: Asymptotic Formulas for Stacks and Related Objects. | |
| A variety of combinatorial objects related to unimodal sequences have appeared in the literature, including integer partitions, concave and convex compositions, and stacks. The generating functions for these objects include theta functions, mock theta functions, and false theta functions, and a variety of techniques are necessary to understand them. I will present results on a recent unified study that describes the asymptotic behavior of all important examples. | |
| 15:00 to 15:30 Jeremy Lovejoy: The Bailey chain and mock theta functions. | |
| After reviewing the Bailey chain and classical constructions of families of modular $q$-hypergeometric multisums, we show how certain change-of-base formulas for Bailey pairs lead to families of \emph{mock modular} $q$-hypergeometric multisums. We also touch on relations between some of these multisums and Ramanujan's mock theta functions. | |
| 16:00 to 16:30 Larry Rolen: Integrality Properties of Symmetric Functions in Singular Moduli. | |
| In his work \emph{On Singular Moduli}, Zagier defined the class polynomial; a polynomial whose roots are traces of special values of modular functions (singular moduli). He then then showed that the trace of these values can be expressed in terms of coefficients of weight $3/2$ modular forms. This phenomena was explored further by Duke in Jenkins, who demonstrated conditions under which the traces of singular moduli for non-holomorphic modular functions must be integral. In this paper, we consider other symmetric functions in the singular moduli for non-holomorphic modular functions, and give conditions under which these symmetric functions are integral, or for which they are rational with an explicit bound on the denominator. | |
| 16:40 to 17:10 Yumiko Hironaka: Spherical functions on the space of p-adic unitary hermitian matrices. | |
| First I will explain about spherical functions on groups and homogeneous spaces, which are common eigenfunctions with respect to Hecke algebra acttion. Then, take $X$ be the space of unitary unramified hermitian matrices of size $2n$ over a $p$-adic field, and consider spherical functions on $X$. A typical one $\omega(x; z), \; x \in X, z \in \mathbb{C}^n$ is constructed by Poisson transform from relative invariants on $X$. The main term of $\omega(x; z)$ is expressed by a specialization of Hall-Littlewood symmetric polynomials of type $C_n$, and we can parametrize all spherical functions by using $\omega(x;z)$, which is $2^n$-dimensional for each $z \in \mathbb{C}^n$. | |
| 17:20 to 17:40 Yilmaz Simsek: Relations between theta-functions and Hardy-Berndt type sums. | |
| The main aim of this paper is to give relations between theta-functions and Hardy-Berndt type sums. Applying connection between Lambert series and Dedekind sums, the relation between theta-functions and Lambert series are studied. We also give some applications. | |
| 17:50 to 18:10 Guangshi Lv: On Fourier coefficients of automorphic forms. | |
| Fourier coefficients of automorphic forms are interesting and important objects in modern number theory. In this talk, the speaker will introduce some recent progress on the estimation of Fourier coefficients of automorphic forms. In particular, we shall talk about the integral order moments of Fourier coefficients of cusp forms and theta series. | |
Wednesday 8th
| 9:00 to 10:00 Jens Funke: Borcherds products and singular theta lifts for orthogonal and unitary groups. | |
| 10:10 to 10:30 Luke Stanbra: Theta lifting for SU(1,1). | |
| In this talk we consider a certain theta function for the special unitary group $SU(1,1)$ which gives rise to a theta lift from invariant functions for congruence subgroups of $SL_2(\mathbb{Z})$ to modular objects of weight 2. In particular, we study the lift of modular functions and relate this to the well known generating series of Faber. | |
| 11:00 to 11:20 Stephen Ehlen: CM values of Borcherds products and weight one harmonic weak Maass forms. | |
| We show that the values of Borcherds products on Shimura varieties of orthogonal type at certain CM points are given in terms of coefficients of the holomorphic part of weight one harmonic weak Maass forms. Furthermore, we investigate the arithmetic properties of these coefficients. As an example, we obtain an analog of the Gross-Zagier theorem on singular moduli. | |
| 11:30 to 12:00 Bernhard Heim: Identifying Borcherds lifts. | |
| In this talk we give a characterization of Borcherds lifts on O(2,n) by symmetries and state several applications. The image of Borcherds lifts can be characterized by the divisor. Since individual modular forms $F$ most of the time are given by Fourier expansion it is an interesting topic to read of such an expansion the lifting property. | |
| 12:10 to 12:30 Benjamin Linowitz: Coefficient growth for Hilbert modular forms. | |
| In 2010 Kohnen and Schmoll showed that if the Fourier coefficients a(n) of a modular form f of level N and even integral weight $k \ge 2$ satisfy the bound $a(n) \ll n^k/2$ then $f$ must be a cusp form. Kohnen and Martin have recently proven an analogous result for Siegel modular forms of weight $k$ and genus $2$ on the full Siegel modular group. In this talk we show that this result also holds for Hilbert modular forms, answering in the affirmative a question raised by Kohnen at the 26th Automorphic Forms Workshop. | |
| 14:00 to 14:45 Harold Stark: Poincare series and modular forms. | |
| 15:00 to 15:20 Soon-Yi Kang: TBA. | |
| 16:00 to 16:30 Anke Pohl: Period functions for Maass cusp forms via symbolic dynamics. | |
| Maass cusp forms are non-holomorphic eigenfunctions of the Laplace-Beltrami operator which are invariant under the action of a given lattice in PSL(2,R) and which decay rapidly towards any cusp. They were discovered in number theory in 1949 and soon seen to be essential for various fields, in particular, harmonic analysis, representation theory and physics. Despite intensive research, to date in general for a non-uniform lattice, no single Maass cusp form is explicitely known. For some lattices, Maass cusp forms have been shown to be linear isomorphic to so-called period functions. These are highly regular functions on (parts of) the real axis which satisfy a certain functional equation depending on the lattice. I will report on work in progress dedicated to deduce functional equations and definitions of period functions (for a wide class of non-uniform lattices) using symbolic dynamics for the geodesic flow. | |
| 16:40 to 17:10 Krzysztof Klosin: Congruences among automorphic forms on symplectic and unitary groups and the Bloch-Kato conjecture. | |
| Let $f$ and $g$ be two cusp forms of weights $k$ and $2$ respectively. We will present two theorems yielding "one direction" in the Bloch-Kato conjecture for the symmetric-square $L$-function of $f$ and the convolution $L$-function of $f$ and $g$. The method of the proof involves constructing "enough" congruences among automorphic forms on the groups ${\rm Sp}_4$ and ${\rm U}(2,2)$. The congruences involved are between forms with reducible Galois representation of particular type (Yoshida lifts and Maass lifts) and automorphic forms with irreducible Galois representations. We will also briefly discuss an extension of these results to the context of $p$-adic modular forms. Part of this is joint work with Mahesh Agarwal. | |
| 17:20 to 17:40 Paul Jenkins: Zeros of modular forms on Gamma_0(2). | |
| We study a canonical basis for spaces of weakly holomorphic modular forms of levels 2 and 3 and integer weight, and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain. This is joint work with Sharon Garthwaite. | |
Thursday 9th
| 10:00 to 10:20 Luis Lomeli: Functoriality for the classical groups over function fields and the Ramanujan conjecture. | |
| The Langlands-Shahidi method allows us to study automorphic L-functions of certain globally generic representations. We will give a presentation of the method for the split classical groups over the new case of a global function field, mentioning important differences that arise in contrast to the number field case. The converse theorem of Cogdell and Piatetski-Shapiro then allows us to obtain a functorial lift from globally generic cuspidal automorphic representations of a classical group to automorphic representations of an appropriate GL(N). Thanks to the work of Lafforgue on the Langlands conjecture for GL(N) over function fields, one obtains a proof of the Ramanujan conjecture by studying the image of functoriality. | |
| 10:30 to 11:00 Laura Peskin: Irreducible mod p representations of the metaplectic cover of SL_2(Q_p). | |
| 11:15 to 11:45 Jim Brown: Applications of the CAP ideal. | |
| Let f be a newform of weight 2k-2 and level N with N odd and square-free. In joint work with Mahesh Agarwal we show roughly half of the Bloch-Kato conjecture in this setting, namely, the size of the Shafarevich-Tate group of the Galois representation associated to f is bounded below by an appropriately normalized special value of the L-function associated to f. We accomplish this by studying congruences among automorphic forms on GSp(4). We present the theorem and discuss the necessary hypotheses. We will also present a conjecture about twisting special values to ensure they are p-adic units as well as numerical evidence for this conjecture. | |
| 12:00 to 12:30 Moshe Baruch: A Voronoi summation formula for Gaussian integers. | |
| We prove a Voronoi type summation formula for divisor functions associated with Gaussian integers. We use a method of Beineke and Bump to obtain the two sides of the summation formula as values of a certain Eisenstein series. This is joint work with D. Bump. Part of the work is joint with O. Bet Aharon. | |
| 14:00 to 15:00 David Yuen: Utility of Computations, and the Satake Compactification of the Paramodular Group. | |
| The majority of the talk will be on the utility of computations, including how the computer often asks us questions. The remainder of the talk will be on the cusp structure of the Paramodular Group in degree 2 for arbitrary levels. We give the 0-dimensional cusps and 1-dimensional cusps and how they cross. Applications to paramodular forms are given. | |
| 15:10 to 15:30 Soma Purkait: On Shimura Decomposition and Tunnell-like Formulae. | |
| Let k be an odd integer and N be a positive integer divisibe by 4. Let g be a newform of weight k - 1, level dividing N=2 and trivial character. We give an explicit algorithm for computing the space of cusp forms of weight k=2 that are `Shimura-equivalent' to g. Applying Waldspurger's theorem to this space allows us to express the critical values of the L-functions of twists of g in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms. | |
| 16:00 to 16:20 Barinder Banwait: Tetrahedral Elliptic Curves. | |
| I describe a strategy for finding "tetrahedral elliptic curves" over quadratic fields, that is, elliptic curves whose associated mod p representation, for some prime p, has projective image A4. | |
| 16:30 to 16:50 Kiminori Tsukazaki: Explicit isogenies of low degree. | |
| Let $E$ be an elliptic curve over a field $K$ and let $\ell$ be a prime. A $K$-rational $\ell$-isogeny of $E$ is a $K$-morphism $\phi$ of degree $\ell$ from $E$ to another elliptic curve $E'$ such that $\phi(O_E) = O_{E'}$. We are interested in finding $K$-rational $\ell$-isogenies of $E$. We describe a method that uses modular curves. The cases $\ell = 2,3,5,7$ and $13$ where the modular curve $X_0(\ell)$ has genus $0$ have been discussed by Cremona and Watkins. Following Elkies' method we will extend the method to the cases $l = 11,17,19,23,29,31,41,47,59$ and $71$ where $X_0^+(l)$, the quotient of modular curve $X_0(\ell)$ by Atkin-Lehner involution $w_{\ell}$, has genus $0$. | |
| 17:00 to 17:30 Fredrik Stromberg: Dimension formulas for vector-valued Hilbert modular modular forms. | |
| I will discuss some computational and theoretical aspects of dimension formulas for vector-valued Hilbert modular forms of integral and half-integral weight. | |
Friday 10th
| 9:00 to 9:30 Dubi Kelmer: Distribution of holonomy of Hilbert modular geodesics. | |
| I will describe the distribution of holonomy angles attached to closed geodesics on a Hilbert modular surface. To study their distribution I will use a trace formula on a space of hybrid Hilbert Maass-Modular forms. | |
| 9:40 to 10:10 Nuno Freitas: Fermat-type equations of signature (13,13,p) via Hilbert cuspforms. | |
| In this talk I am going to discuss how a modular approach via Hilbert cuspforms can be used to show that equations of the form $x^13 +y^13 = Cz^p$ have no non-trivial primitive solutions (a,b,c) such that $13 \not \mid c$ if $p > 4992539$. We will first relate a putative solution of the previous equation to the solution of another Diophantine equation with coefficients in $\mathbb{Q}(\sqrt{13})$ and then construct appropriate Frey-curves $E/\mathbb{Q}(\sqrt{13})$. Then by proving modularity of E we will be able to reach a contradiction by using a modular approach with Hilbert cuspforms over $\mathbb{Q}(\sqrt{13})$. | |
| 10:20 to 10:40 Armin Straub: An application of modular forms to short random walks. | |
| We consider random walks in the plane which consist of n steps of fixed length each taken into a uniformly random direction. Our interest lies in the probability distribution of the distance travelled by such a walk. While excellent asymptotic expressions are known for the density functions when n is moderately large, we focus on the arithmetic properties of short random walks. In the case of three and four steps, the density functions satisfy differential equations of modular origin. This intertwines with the combinatorics of the corresponding even moments and leads to hypergeometric evaluations of the density functions. Much less is known for the density in case of five random steps, but we use the modularity of the four-step case and the Chowla-Selberg formula to deduce its exact behaviour near zero. The talk will be based on joint work with Jonathan M. Borwein, James Wan, and Wadim Zudilin. | |
| 11:10 to 11:40 Anna v. Pippich: On elliptic analogues of the Takhtajan-Zograf metric on moduli spaces. | |
| The Takhtajan-Zograf form $\omega_{\mathrm{TZ}}$ on the moduli space of genus $g$ hyperbolic Riemann surfaces with $n$ marked points is constructed using non-holomorphic Eisenstein series associated to the marked points. In this talk, we report on a joint project with G. Freixas on an arithmetic Riemann-Roch isometry for singular metrics. The related local index theorem involves an analogue of the Takhtajan-Zograf metric employing so-called elliptic Eisenstein series. | |
| 11:50 to 12:20 Tim Huber: A theory of theta functions to the quintic base. | |
| In this lecture, properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta-function identity and counterparts of Jacobi's Principles of Duplication and Dimidiation. The resulting library of quintic transformation formulas is used to describe the action of Hecke operators of level five and more general quintic dissection operators. Central to the analysis is a new nonlinear coupled system of differential equations satisfied by the quintic theta functions. | |
| 14:00 to 14:30 Aleksandar Velizarov Petrov: A-expansions of Drinfeld Modular Forms of full level. | |
| In this talk, we introduce the notion of Drinfeld modular forms with A-expansions for ${\rm GL}_2(A)$. Our main result is the construction of an infinite family of cuspidal Drinfeld eigenforms with A-expansions, which generalizes results of Bartolome Lopez. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) examples of Drinfeld eigenforms that can be represented as `non-trivial' products of eigenforms; (iii) a restrictive multiplicity one result for Drinfeld modular forms with A-expansions. | |
| 14:40 to 15:00 Sebastián Herrero: On some cusp forms whose Fourier coefficients are special values of shifted Rankin-Selberg convolutions. | |
| Let $g$ be a fixed cusp form. The map sending any cusp form $f$ onto the product $fg$ is a linear operator. Its adjoint map (with respect to the usual Petersson inner product) was completely described by W. Kohnen in 1991. In this talk we will generalize such a result, as we consider the map sending any $f$ onto the n-th Rankin-Cohen bracket $[f, g]_n$. We will describe the adjoint map of this linear operator by computing the Fourier coefficients of its images. In our case, specia values of certain shifted Rankin-Selberg convolutions ``twisted" with some combinatorial expressions appear. | |
| 15:10 to 15:30 Nicole Raulf: QUE for Eisenstein series. | |
| In this talk we consider various aspects of quantum unique ergodicity for the Eisenstein series in the sense of Luo and Sarnak. This is joint work with Y. Petridis and M. Risager. | |
| 15:40 to 16:00 Soumya Bhattacharya: Finiteness of simple holomorphic eta products. | |
| An eta product is a finite product of the form $\prod_{d}\eta(d\tau)^{X_d}$ with $d\in\mathbb{N}$ and $X_d\in\mathbb{Z}$. An eta product $f(\tau)$ is $primitive$ if it is not of the form $h(\alpha\tau)$ for some eta product $h$ some integer $\alpha>1$. A holomorphic eta product is $irreducible$ if it is not the product of two holomorphic eta products. By a $simple$ holomorphic eta product we mean, a holomorphic eta product that is both primitive and irreducible. About 20 years back, Zagier gave a list of simple eta products of weight $\frac12$ and conjectured that the list is complete. He also conjectured that there are only finitely many simple holomorphic eta products of any given weight. Both of these conjectures were subsequently established by his student Gerd Mersmann. We shall talk about short proofs of this conjectures and we shall also see that, there are only finitely many simple holomorphic eta products of a given level (or equivalently, there are only finitely many irreducible holomorphic eta products of a given level). We shall give an explicit upper bound for the possible weights of an irreducible holomorphic eta product of a given level. | |
| 16:10 to 16:30 Onyekwere Chigozie: TBA. | |
| In this talk, I shall present a method for producing new upper bounds for the dimension of certain cohomology groups of arithmetic quotients of symmetric spaces. Suitably interpreted, these results give new upper bounds for spaces of automorphic forms of cohomological type. After explaining why non-trivial lower bounds are (in general) impossible to obtain, I will discuss how p-adically completed spaces of torsion classes can conjecturally be used as a substitute for classical automorphic forms in order to produce large families of Galois representations. | |