Titles and Abstracts

Speaker: Jan Bruinier

Title: Generating series of special divisors on arithmetic ball quotients

Abstract: A celebrated result of Hirzebruch and Zagier states that the generating series of Hirzebruch-Zagier divisors on a Hilbert modular surface is an elliptic modular form with values in the cohomology. We discuss some generalizations and applications of this result. In particular, we prove an analogue for special divisors on integral models of ball quotients. In this setting the generating series takes values in an arithmetic Chow group in the setting of Arakelov geometry. If time permits, we address some applications to arithmetic theta lifts and the Colmez conjecture. This is joint work with B. Howard, S. Kudla, M. Rapoport, and T. Yang.

Speaker: Scott Carnahan

Title: Generalized Monstrous Moonshine

Abstract: The subject known as Monstrous Moonshine began in the 1970s, when numerical computations suggested a relationship between representations of the monster simple group and modular functions obeying a genus zero property. The classical theories of modular functions on the complex upper half-plane and finite simple groups of symmetries were not previously thought to be closely related, and the initial observations were viewed with some skepticism. However, Borcherds’s 1992 proof of the Conway-Norton Monstrous Moonshine Conjecture, together with some preceding work, showed that ideas from theoretical physics, in particular conformal field theory, provide a bridge that links the two fields. Generalized Monstrous Moonshine is an enhancement proposed by Norton in 1987 that, from a physical standpoint, describes all possible twisted sectors of a conformal field theory with monster symmetry. This conjecture was recently proved following a program outlined by Hoehn.

Speaker: Dohoon Choi

Title: Congruences for weakly holomorphic modular forms

Abstract: In this talk, I will talk about special congruences for weakly holomorphic modular forms. Weakly holomorphic modular forms mean that they are modular forms but can have poles at some cusps. These modular forms have played important roles as generating functions for several objects such as the partition function, traces of singular moduli and so on. This talk will discuss on special congruences concerning with weakly holomorphic modular forms, which are motivated from congruences for the partition function studied by Ramanujan.

Speaker: SoYoung Choi

Title: Linear relations among half-integral weight Poincaré series

Abstract: We introduce an infinite family of half-integral weight Poincaré series coming from vector valued harmonic weak Maass forms, and investigate linear relations among the Poincaré series by applying a paring between weakly holomorphic modular forms and harmonic weak Maass forms together with properties of Maass Poincaré series. We also show that the Poincaré series are characterized by the Fourier coefficients of half-integral weight cusp forms. This is a joint work with Chang Heon Kim.

Speaker: YoungJu Choie

Title: Modular forms and cohomology

Abstract: Cohomology theory has been played a role in studying automorphic forms in number theory. Eichler-Shimura-Manin developed parabolic cohomology theory corresponding modular forms of integral weight. Bruggeman-Lewis-Zagier construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups. We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with real weight and parabolic cocycles. We impose no condition on the growth of the automorphic forms at the cusps. Our result concerns arbitrary cofinite discrete groups with cusps, and covers exponentially growing automorphic forms, like those studied by Borcherds, and like those in the theory of mock automorphic forms. A tool in establishing these results is the relation to cohomology groups with values in modules of "analytic boundary germs", which are represented by harmonic functions on subsets of the upper half-plane. This is a joint work with Bruggeman and Diamantis. This will be a survey talk.

Speaker: Stephan Ehlen

Title: The modular completion of certain (formal) generating series

Abstract: In this talk I will report on joint work with Kathrin Bringmann (Cologne) and Markus Schwagenscheidt (Darmstadt). In our work, we complete several (formal) generating series to non-holomorphic modular forms in two variables. For instance, we consider the formal generating series of CM traces (in τ) of the meromorphic function j’(z)/(j(z)-j(τ)). We show that this generating series can be completed to a non-holomorphic modular form of two variables τ, z∈ℍ which has weight 3/2 in τ for Γ_0(4) and weight 2 in z for SL_2(ℤ).

Speaker: Kazuhiro Hikami

Title: Quantum invariants of knots/3-manifolds and modular forms

Abstract: Recent studies show that quantum invariants of knots and 3-manifolds have rich structures related to geometry and number theory. We will discuss asymptotic behavior of quantum invariants and talk about their quantum modularity.

Speaker: Min-Joo Jang

Title: Quantum modular forms and singular combinatorial series

Abstract: Since Dyson defined the rank of a partition, a number of studies have been done on this statistic. For example, a celebrated result of Bringmann and Ono showed that the rank generating function is essentially a mock modular form. Andrews introduced k-marked Durfee symbols and more generally defined the ranks for them. In particular, when k = 1 one recovers Dyson’s rank. In this talk, we establish quantum modularity of this combinatorial series, the rank generating function for k-marked Durfee symbols. This is joint work with Amanda Folsom, Susie Kimport, and Holly Swisher.

Speaker: Seokho Jin

Title: The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product

Abstract: We determine asymptotic formulas for the Fourier coefficients of Jacobi forms expressed by infinite products with Jacobi theta functions and the Dedekind eta function. These are generalizations of results about the growth of the Fourier coefficients of Jacobi forms given by an inverse of Jacobi theta function to derive the asymptotic behavior of the Betti numbers of the Hilbert scheme of points on an algebraic surface by Bringmann-Manschot and about the asymptotic behavior of the χ_y-genera of Hilbert schemes of points on K_3 surfaces by Manschot-Rolon. And we get the algebraicity of the generating functions given by Göttsche for the Hilbert schemes associated to general algebraic surfaces.

Speaker: Ben Kane

Title: Regularized inner products and meromorphic modular forms

Abstract: In this talk, we consider a regularization of Petersson's inner product which is well-defined (and finite) between two meromorphic modular forms and agrees with Petersson's inner product whenever the latter exists. We take the inner product between a special family of meromorphic modular forms and show a connection with the automorphic Green's function and certain functions which are called polar harmonic Maass forms. We then discuss some applications of these polar harmonic Maass forms, including formulas and asymptotics for Fourier coefficients of meromorphic modular forms, construction of a basis of meromorphic modular forms of non-positive weight, and an algorithm to compute the divisor of a given meromorphic modular form given only its Fourier expansion. Most of the talk is joint work with Kathrin Bringmann and Anna von Pippich, while the first two applications are joint work with Kathrin Bringmann and the application to divisors of modular forms is joint work with Kathrin Bringmann, Steffen Loebrich, Ken Ono, and Larry Rolen.

Speaker: Masanobu Kaneko

Title: An explicit form of genus character L-functions of quadratic orders and its applications

Abstract: For general quadratic orders, the genus character L-functions are explicitly computed. As an application, we generalize a formula due to Hirzebruch-Zagier which expresses the class number of imaginary quadratic fields in terms of continued fraction expansion. This is a joint work with Yoshinori Mizuno.

Speaker: Byungchan Kim

Title: Congruences for a mock modular form on SL_2(ℤ) and the smallest parts function

Abstract: Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight 3/2 on SL_2(ℤ) modulo primes ℓ ≥ 5. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo ℓ. As another application, we show that congruences modulo ℓ for the smallest parts function are rare in a precise sense. This is a joint work with Scott Ahlgren.

Speaker: Chang Heon Kim

Title: Modularity of Galois traces of Weber's resolvents

Abstract: The values of the classical j-invariant at CM points are called singular moduli. Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier-Funke generalized his result to the sums of the values at Heegner points of modular functions on modular curves of arbitrary genus. In my previous work with Jeon and Kang, we dealt with certain modular functions (holomorphic cube root of j, Weber functions of level 48 and 72) whose values at CM points define class invariants. We related modified Galois traces of those invariants to modular traces of associated modular functions at Heegner points by using Shimura's reciprocity law, so that they are Fourier coefficients of weakly holomorphic modular forms of weight 3/2. In this talk, considering Weber's resolvents of level 5, we will construct new class invariants and show the modularity of their Galois traces. (This is a joint work with Ho Yun Jung, Soonhak Kwon and Yeong-Wook Kwon).

Speaker: Winfried Kohnen

Title: Shifted products of Fourier coefficients of cusp forms

Abstract: We will report on recent work regarding non-vanishing properties of shifted products of Fourier coefficients of cusp forms, both in the elliptic case (joint work with E. Hofmann) and in the Siegel case (joint work with J. Sengupta)

Speaker: Yingkun Li

Title: Modularity of generating series of winding numbers

Abstract: On a modular curve, there is an ample supply of closed geodesics. When the curve has genus zero, these geodesics are all null-homologous and it makes sense to consider the winding number of these geodesics with respect to two points on the modular curve. In this talk, we will consider modular properties of generating series formed out of these winding numbers. This is a joint work with J. Bruinier, J. Funke and O. Imamoglu.

Speaker: Subong Lim

Title: Hecke structures of weakly holomorphic modular forms and their algebraic properties

Abstract: Let S^!_k(Γ_1(N)) be the space of weakly holomorphic cusp forms of weight k on Γ_1(N) with an even integer k > 2 and M^!_k(Γ_1(N)) be the space of weakly holomorphic modular forms of weight k on Γ_1(N). Further, let z denote a complex variable and D:=(1/2πi)∂/∂z. In this talk, we construct a basis of the space S^!_k(Γ_1(N))/D^{k-1}(M^!_{2-k}(Γ_1(N))) consisting of Hecke eigenforms by using the Eichler-Shimura cohomology theory. Further, we show algebraicity of CM values of weakly holomorphic modular forms in the basis.

Speaker: Jeremy Lovejoy

Title: The colored Jones polynomial and Kontsevich-Zagier series for double twist knots

Abstract: In this talk I will explain how to use Bailey pairs to prove a formula for the colored Jones polynomial of the torus knots T_{(2,2t+1)}, where t is a positive integer. I will also discuss how a result of Takata leads to a formula for the colored Jones polynomial of the double twist knots K_{(-m,-p)} and K_{(-m,p)}, where m and p are positive integers. Among the applications are dualities at roots of unity between generalized Kontsevich-Zagier series and generalized U-functions. This is joint work with Kazuhiro Hikami in the torus knot case and Robert Osburn in the case of twist knots.

Speaker: Toshiki Matsusaka

Title: Polyharmonic weak Maass forms of higher depth for SL(2,ℤ)

Abstract: The space of polyharmonic Maass forms was defined by Lagarias-Rhoades, recently. They constructed its basis from the Taylor coefficients of the real analytic Eisenstein series. In this talk, we introduce polyharmonic weak Maass forms, that is, we relax the moderate growth condition at cusp, and we construct their generating set as a generalization of Lagarias-Rhoades’ works.

Speaker: Michael Mertens

Title: Modular forms of virtually real-arithmetic type

Abstract: The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated integrals. In this talk, we propose a unified framework for these notions as vector-valued modular forms with respect to a new class of arithmetic types which we call virtually real-arithmetic (vra) types. Some aspects of the theory of vra type modular forms such as rationality results for their Fourier and Taylor coefficients, Petersson pairings, and Hecke theory will be highlighted. This is joint work with Martin Raum.

Speaker: Robert Osburn

Title: Sequences, modular forms and period integrals

Abstract: It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. Recently, Brown introduced a program for understanding irrationality proofs of zeta values via period integrals. In this talk, we discuss how these arithmetic and modular properties persist for sequences associated to Brown’s period integrals. This is joint work with Dermot McCarthy and Armin Straub.

Speaker: Yoon Kyung Park

Title: Periods of modular forms of squarefree integer level

Abstract: Generalizing a result of Zagier for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms of level N, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level N. This is a joint work with Y. Choie and D. Zagier.

Speaker: Brandon Williams

Title: Poincaré square series for the Weil representation

Abstract: We calculate the Jacobi Eisenstein series of weight k ≥ 3 for a certain representation of the Jacobi group, and evaluate these at z = 0 to give coefficient formulas for a family of modular forms Q_{k,m,β} of weight k ≥ 5/2 for the (dual) Weil representation on an even lattice. The forms we construct have rational coefficients and contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for Q_{k,m,β} and the Eisenstein series of Bruinier and Kuss p-adically to get an efficient algorithm. The main application is in constructing automorphic products.

Speaker: Hwajong Yoo

Title: Non-optimal levels of a reducible mod l modular representation

Abstract: For a two-dimensional irreducible Galois representation arising from a cusp form, there is a notion of the optimal level introduced by Jean-Pierre Serre. His epsilon conjecture, proved by Kenneth Ribet, concerns about this. Later, Diamond and Taylor classified non-optimal levels of irreducible Galois representations. In this talk, we will discuss their counterpart, namely, non-optimal levels of a reducible mod l Galois representation arising from a weight 2 cusp form.

Speaker: Yichao Zhang

Title: Isomorphism of spaces of half-integral modular forms and related

Abstract: In this talk, we report an isomorphism between spaces of half-integral weight modular forms. Such isomorphism has been partially well-known for a long time and its integral-weight analogue was treated first by Bruinier and Bundschuh. We shall also see some related stuff such as Zagier duality on the Fourier coefficents and the calculation of Borcherds products.