26th Automorphic Forms Workshop

April 26--29, 2012
University of British Columbia
Vancouver, BC

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Abstracts

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.

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.

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).

Given a principal congruence subgroup Γ = Γ(N) ⊆ SL₂(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.

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.

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 Γ₀(2).

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 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.

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.

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 Γ₀⁽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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

We describe all linear characters of SL₂ 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.

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 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.

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.

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

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.

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.