**A newform theory for Hilbert Eisenstein series**

Ben Linowitz

Dartmouth

In his thesis, Weisinger developed a newform theory for Eisenstein series. This theory was later generalized to the Hilbert modular setting by Wiles. We extend the theory of newforms for Hilbert Eisenstein series. In particular we will show that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than 1/2. Additionally we will provide a number of applications of this newform theory. This is joint work with Tim Atwill.

**Universal Jacobi forms**

Nils Skoruppa

Universität Seigen

Every given Jacobi form φ whose index is a lattice L of rank > 1 gives rise to an infinite family of Jacobi forms of lower rank indices via a simple pullback procedure, However, certain Jacobi forms φ seem to be more special than others since they yield, for instance, as infinite family all Jacobi forms of weight one of given character on the full modular group with arbitrary scalar index. Similar observations can be made for higher weight, where this idea can then be extended to generate infinite families of elliptic cusp forms of weight 2 (which, in general, are not Hecke eigenforms though). In this talk, we present various examples for this phenomenon.

**Congruences of Apèry-like numbers via modular forms**

Brundaban Sahu

National Institute of Science Education and Research, India

We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we prove two recent conjectures of Chan, Cooper and Sica. Also we discuss a proof of supercongruences for a generalization of numbers which arise in Beukers' and Zagier's study of integral solutions of Apèry-like differential equations This is joint work with Robert Osburn (UCD, Dublin).

**Lifting of Hecke operators on line bundles over modular curves**

Abhishek Banerjee

Ohio State University

Given a principal congruence subgroup Γ = Γ(N) ⊆ **SL**_{2}(**Z**), Connes and Moscovici have introduced a modular Hecke algebra A(Γ) that incorporates both the pointwise multiplicative structure of modular forms and the action of the classical Hecke operators. It is well known that a Γ-modular form g of weight k may be described as a global section of the k-th tensor power of a certain line bundle p(Γ) : L(Γ) → Γ\H. The purpose of this talk is to develop a theory of modular Hecke algebras for Hecke correspondences between the line bundles L(Γ) that lift the classical Hecke correspondences between modular curves Γ\H.

**Explicit formulas for Drinfeld elliptic functions and supersingular j-invariants**

Matt Papanikolas

Texas A & M

Defined by Carlitz in the 1930's, the Carlitz module plays the role over function fields of the multiplicative group over number fields. Indeed the division values of its exponential function generate abelian extensions of the rational function field over a finite field. Likewise Drinfeld modules of higher ranks serve the role of elliptic curves, leading both to Drinfeld modular forms and higher dimensional Galois representations.

In this talk we will discuss new formulas for the exponential and logarithm functions of Drinfeld modules, which themselves are the starting point for analogues of elliptic functions and elliptic modular forms in this setting. These lead to explicit results on periods for Drinfeld modules and in turn new derivations of polynomials of supersingular j-invariants. Joint with Ahmad El-Guindy.

**Zeros of weakly holomorphic modular forms of level 2**

Paul Jenkins

Brigham Young University

We define a basis for weakly holomorphic modular forms of level 2 and integer weight, and show that most of the zeros of the basis elements lie on the lower boundary of a fundamental domain for Γ_{0}(2).

**The L-functions and modular forms database**

Fredrik Strömberg

Max Planck Institute for Mathematics

I will describe the background and some of the features the L-functions and modular forms database (LMFDB). The LMFDB is a large collaborative effort to make data and information related to L-functions and modular forms (of all kinds) available to a larger mathematical audience. The platform we have chosen is an interactive, database-driven website (www.lmfdb.org).

**Hypergeometric functions over finite fields and Siegel modular forms**

Dermot McCarthy

Texas A & M

Hypergeometric functions over finite fields exhibit many interesting properties. In particular, special values of these functions have been related to the Fourier coefficients of certain elliptic modular forms. Relationships with Siegel modular forms of higher degree are also expected.

We will outline recent work on proving an example of such a connection, whereby a special value of the hypergeometric function is related to an eigenvalue associated to a Siegel eigenform of degree 2.

This is joint work with Matt Papanikolas.

**Periods and functorial transfer**

Kimball Martin

University of Oklahoma

Period integrals of automorphic forms are related to special values of L-functions, Fourier coefficients of modular forms and Langlands functoriality. We will discuss some aspects of how periods behave under functorial transfer.

**Towards mixed Saito-Kurokawa lifts**

Dania Zantout

Clemson University

Let κ be an even integer and m an odd square-free positive integer. Starting with a newform f of weight 2κ-2 and level Γ(m) in S^{new, - } _{ 2κ-2 }(Γ(m)), one can associate to f two different Saito-
Kurokawa lifts F_{f} ∈ S_{κ}(Γ^{(2)}(m)) of congruence level Γ_{0}^{(2)}(m) and G_{f} ∈ S_{κ}(Γ(m)) of paramodular level Γ(m). Using representation theoretic methods, Schmidt had explained the existence of the two different Saito-Kurokawa lifts and proposed that if m is divisible by many primes, then f can have "mixed level" Saito-Kurokawa lifts. In this talk, we give a precise statement of the "mixed level" liftings. Towards the goal of realizing the mixed level Saito-Kurokawa lifting in an explicit linear version, we also give a lifting that generalize both the Maass lifting with level and Gritsenkos lifting.

**Eta-quotients and class numbers of imaginary quadratic fields**

Takeshi Ogasawara

Kyushu University

We consider a Hecke module generated by a specific eta-quotient of weight one, and give examples of numerical computation which make us conjecture that its dimension is equal to the class number of certain imaginary quadratic field.

**Numerical examples of Euler factors of Siegel modular forms of half-integral weight of degree three**

Shuichi Hayashida

Osaka University

A lifting from pairs of two elliptic modular forms to Siegel modular forms of half-integral weight of degree two was conjectured by T. Ibukiyama and myself. In 2010 this conjecture was solved under some conditions. In this talk I would like to give an evidence of a conjecture of lifting from three elliptic modular forms to Siegel modular forms of half-integral weight of degree three.

**Exact critical values of a symmetric fourth L function and Zagier's conjecture**

Tomoyoshi Ibukiyama

Osaka University

Around 1976, Zagier gave a conjecture on (five) critical vales of the symmetric 4-th L-function of the Ramanujan Delta function. There he wrote them by explicit rational numbers and cube of the Petersson inner product. We prove that the ratio of these 5 values are exactly as he conjectured. In order to prove this, we explicit calculate the critical values of the standard L-function of some vector valued Siegel cusp form and apply Kim-Ramakrishnan-Shahidi lifting. This is a joint work with Hidenori Katsurada.

**Geometry of numbers and universal quaternary quadratic forms**

Kate Thompson

University of Georgia

Applications of geometry of numbers (GoN) are prevalent throughout the last 200 years of number theory. It is well-known (perhaps first thanks to Ramanujan and Dickson) that there are nine diagonal positive definite quaternary integral quadratic forms of square discriminant which integrally represent all positive integers. The speaker, along with Pete L. Clark, Jacob Hicks and Nathan Walters, recently has used GoN to prove the universality of these forms. Generalizations of this technique to other known universal forms will be discussed. This work has its roots in the on-going UGA VIGRE group on GoN.

**On index-level change of Jacobi forms**

Hiroki Aoki

Tokyo University of Science

On the theory of Jacobi forms, it seems to be a close relation between their index and level, by the calculation of the trace of Jacobi new forms, given by Sakata. In this talk I would like to give an explicit operator changing these two parameters.

**Automorphic forms and sequence asymptotics**

Karl Mahlburg

LSU Baton Rouge

I will discuss asymptotic results for various combinatorial enumeration problems, including unimodal sequences, convex compositions, and stacks. The techniques for studying these objects include complex analysis, and also rely on the automorphic transformations of the generating functions, which involve theta functions, Jacobi theta functions, and quantum dilogarithms.

**Taylor coefficients of modular forms**

Cormac O'Sullivan

Bronx Community College and the CUNY Graduate Center

The Fourier coefficients of modular forms are well-known to contain useful arithmetic information. These Fourier coefficients may be thought of as Taylor coefficients at infinity. In joint work with Morten Risager, we begin to study the Taylor expansions of modular forms at points in the upper half plane, building on earlier work of Rodriguez Villegas and Zagier. At CM points we show these Taylor coefficients are non-zero and that they also have interesting arithmetic properties, many still unexplained.

**Holomorphic QUE for arbitrary levels**

Abishek Saha

ETH

Let f be a classical holomorphic newform of level q and even weight k. I will describe recent joint work with Paul Nelson and Ameya Pitale where we prove that the pushforward to the full level modular curve of the mass of f equidistributes as qk goes to infinity. This generalizes previous work by Holowinsky-Soundararajan (the case q = 1, k → ∞) and Nelson (the case qk → ∞ over squarefree integers q). Thus we settle the holomorphic quantum unique ergodicity conjecture in all aspects (for classical modular forms of trivial nebentypus). A potentially surprising aspect of our work is that we obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime.

**An extension of a proof of the Ramanujan congruences**

Holly Swisher

Oregon State University

Recently Lachterman, Schayer, and Younger published a new, elegant proof of the celebrated Ramanujan congruences for the partition function p(n). Their proof uses classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono. In this talk, we will begin with an introduction to partition theory, including its relationship to modular forms, and then give a method for generalizing Lachterman, Schayer, and Youngers proof to include Ramanujan congruences for multipartition functions pk(n) and Ramanujan congruences for p(n) for some prime powers.

**The bounded denominator conjecture for vector-valued modular forms**

Christopher Marks

PIMS at the University of Alberta

It has been understood since the fundamental work of Atkin and Swinnerton-Dyer in the 1970s that modular forms for noncongruence subgroups should have "unbounded denominators", and in fact the condition of having bounded denominators is conjecturaly equivalent to being a congruence modular form (with rational Fourier coefficients). I will explain how the theory of vector-valued modular forms provides an effective method of probing this conjecture, and present some recent evidence I have accumulated in the three-dimensional setting, which verifies a vector-valued generalization of the conjecture.

**Using mass formulas to enumerate definite quadratic forms of bounded class number**

Jonathan Hanke

University of Georgia

This talk will describe some recent results using exact mass formulas to determine all definite quadratic forms of small class number in n=3 variables, particularly those of class number one.

The mass of a quadratic form connects the class number (i.e. number of classes in the genus) of a quadratic form with the volume of its adelic stabilizer, and is explicitly computable in terms of special values of zeta functions. Comparing this with known results about the sizes of automorphism groups, one can make precise statements about the growth of the class number,
and in principle determine those quadratic forms of small class number.

We will describe some known results about masses and class numbers (over number fields), then present some new computational work over the rational numbers.

**Linear characters of Hilbert modular groups**

Hatice Boylan

Universität Siegen

We describe all linear characters of **SL**_{2} over arithmetic Dedekind domains. Our results are joint work with Nils Skoruppa and are part of a bigger project which aims to develop an arithmetic theory of Jacobi forms over number fields.

**On the average value of the divisor function**

Sheng-Chi Liu

Texas A & M

We will discuss the asymptotic evaluation of the average of the classical divisor function over values of a quadratic polynomial. We will explain how such a result can be used to study the first moment of certain Rankin-Selberg L-functions. This is joint work with Riad Masri.

**The Paramodular Conjecture**

Cris G. Poor

Fordham University

The Paramodular Conjecture of Brumer and Kramer is a testable modularity conjecture in degree 2. It associates abelian surfaces defined over **Q** of conductor N, whose endomorphisms defined over **Q** are trivial, to rational weight 2 paramodular newforms of level N that are not Gritsenko lifts. I will review the computational evidence for this conjecture.

**A uniform spectral gap for congruence covers of a hyperbolic manifold**

Lior Silberman

UBC

I will describe work with Dubi Kelmer on the first Laplace eigenvalue in towers of manifolds covered by real or complex hyperbolic space. All congruence quotients in a given dimension have a uniform spectral gap. We show how to deduce from this a uniform spectral gap for the family of congruence covers of a fixed arithmetic (non-congruence) manifold.

**Selberg trace formula**

Karen Taylor

Bronx Community College and the CUNY Graduate Center

A short expository talk on the derivation of the Selberg Trace Formula.

**A characterization of Siegel cusp forms of degree two by the growth of their Fourier coefficients**

Winfried Kohnen

University of Heidelberg

We prove a new charaterization of Siegel cusp forms of degree two in terms of the growth of their Fourier coeficcients. This is very recent joint work with Y. Martin.

**Arithmeticity of Rankin-Selberg kernels**

Dominic Lanphier

Western Kentucky University

Special values of the Rankin convolution L-functions of two different weight cusp forms well-known. The values are given in terms of well-known constants and a Petersson inner product of the higher-weight cusp form. This leads to a very nice closed form for the harmonic average of such values, where the average is taken over the space of cusp forms containing the higher-weight cusp form. We obtain a closed form for the harmonic average of the values over the lower-weight cusp form.