The following individuals have registered for the workshop on the website. Those with an asterisk plan to give a talk. See the list of abstracts at the bottom of the page.

Please notify the organizers regarding any oversights or omissions.

 Mahesh Agarwal University of Michigan-Dearborn Scott Ahlgren* University of Illinois Nickolas Andersen* University of Illinois Cristina Ballantine* College of the Holy Cross Owen Barrett* Yale University Mike Belfanti* Ohio State University Jim Brown Clemson University Luca Candelori Louisiana State University Abel Castillo* University of Illinois at Chicago Kevin Childers* Brigham Young University Britain Cox* Purdue University Rachel Davis* Purdue University Robert Dorward Oberlin College Ellen Eischen* University of North Carolina at Chapel Hill Jorge Florez CUNY Graduate Center Cameron Franc* University of Michigan Sharon Frechette* College of the Holy Cross Holley Friedlander* Dickinson College Michael Griffin* Emory University Anna Haensch* Duquesne University Alia Hamieh* Queen's University Paul Jenkins* Brigham Young University Rodney Keaton* University of Oklahoma Tomoya Kiyuna* Kyushu University Krzysztof Klosin* Queens College and the Graduate Center (City University of New York) Andrew Knightly* University of Maine Gene Kopp University of Michigan Jeff Lagarias University of Michigan Dominic Lanphier* Western Kentucky University Benjamin Linowitz University of Michigan Michael Lipnowski Duke University Blake Mackall* Williams College Michael Mertens* Emory University Djordje Milicevic* Bryn Mawr College Steven J. Miller* Williams College Trung Hieu Ngo* Texas A&M University at Qatar Kevin Nowland Ohio State University Kyle Pratt* University of Illinois at Urbana-Champaign Wendell Ressler* Franklin & Marshall College James Ricci* Daemen College Olav Richter* University of North Texas Jeremy Rouse* Wake Forest University Malcolm Rupert* University of Idaho Brundaban Sahu* School of Mathematical Sciences, National Institute of Science Education and Research Karam deo Shankhadhar* Universidad de Chile Tom Shemanske Dartmouth College Howard Skogman The College at Brockport (SUNY Brockport) Naomi Tanabe* Queen's University Karen Taylor* Bronx Community College Kate Thompson* Davidson College Lola Thompson Oberlin College Jesse Thorner* Emory University Stephanie Treneer* Western Washington University Amanda Tucker* SUNY Geneseo Nahid Walji* University of California, Berkeley Karl Winsor* University of Michigan Tian An Wong* CUNY Graduate Center Fan Zhou* Ohio State University
Scott Ahlgren: Mock modular forms and the smallest parts function
The first part of the talk will introduce mock modular forms and some of the roles which they play in number theory. The second part will focus on a particular mock modular form of weight 3/2 which is related to the Dedekind eta function and which is of combinatorial interest (as the generating function for the “smallest parts” partition function introduced by George Andrews). I will discuss recent work about the coefficients of this mock modular form (much of which is joint with Nick Andersen).

Tian An: The Hitchin fibration and tensor products lifts
The Hitchin fibration is a geometric construction which played a key role in the proofs of the fundamental lemmas of Langlands-Shelstad and Jacquet-Rallis in positive characteristic. Specifically, the Lefschetz trace formula relates its cohomology to orbital integrals. In this talk I will give an outline of this method, and sketch the geometric analogues for a tensor product lifting of automorphic forms on GL(n) x GL(m).

Owen Barrett, Steven J. Miller and Karl Winsor: Large gaps between zeros of GL(2) $L$-functions
Let $L(s, f)$ be an $L$-function associated to a primitive form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates from Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moment of derivatives of $L(1/2 + it, f)$ and use it to show that there are infinitely many gaps between consecutive zeros of $L(s, f)$ along the critical line that are at least $\sqrt{3}$ times the average spacing. Using general pair correlation results of Murty and Perelli for primitive GL(2) $L$-functions and a technique of Montgomery, we also prove that there are infinitely many gaps between consecutive zeros of $L(s, f)$ along the critical line that are smaller than 0.83 times the average spacing. This is joint work with Brian McDonald, Patrick Ryan and Caroline Turnage-Butterbaugh.

Owen Barrett and Steven J. Miller: Lowest Zeros of GL(2) $L$-functions
The Katz-Sarnak philosophy dictates that the spacing statistics of the zeros of Hecke cusp form $L$-functions averaged within families coincide with those of eigenvalues of particular random matrices in the limit as the level tends to infinity. While these predictions have been supported by multiple density and correlation results across various families, S. J. Miller discovered a significant disagreement for finite conductors in the numerical data for elliptic curve $L$-functions. We use the $L$-functions Ratios Conjectures to calculate the 1-level density for the family of even quadratic twists of a holomorphic cuspidal newform for large but finite level and recover leading and lower-order terms. Then, using the arithmetic of these lower-order terms and the discretization of the $L$-functions at the central point, we extend a random matrix model proposed by Due\~nez, Huynh, Keating, Miller, and Snaith for elliptic curves to Hecke cusp forms of weight $k>2$. This is joint work with Patrick Ryan.

Nickolas Andersen: Mock modular forms of weight 5/2 and partitions
We study the coefficients of a natural basis for the space of mock modular forms of weight $5/2$ on the full modular group. The “shadow” of the first element of this infinite basis encodes the values of the partition function $p(n)$. We show that the coefficients of these forms are given by traces of singular invariants. These are values of modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoglu, and Toth on cycle integrals of the $j$-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

Cristina Ballantine: Ramanujan bigraphs - explicit constructions
We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of a particular inner form of $SU_3(\mathbb{Q}_p)$. To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size. The Ramanujan property of the graph is governed by the representations of the inner form.

Mike Belfanti: An introduction to Arthur's invariant trace formula

Luca Candelori: On the transformation laws of algebraic theta functions
As analytic functions, theta functions possess a functional equation thanks to the Poisson summation formula. In this talk, we give an algebraic meaning to this functional equation, by viewing theta functions as sections of appropriate vector bundles over the moduli stack of elliptic curves. This algebraic interpretation can also be used as a starting point for an algebro-geometric theory of modular forms taking values in the Weil representation, and in particular of modular forms of half-integral weight.

Abel Castillo: An effective version of Hilbert's Irreducibility Theorem
Let $f(X, t_1, \ldots ,t_k)$ be a polynomial in $X$ with coefficients in $\mathbb{Z}[t_1,\ldots, t_k]$. Hilbert's Irreducibility Theorem tells us that for "almost all" integer specializations of $(t_1, \ldots ,t_k)$, the resulting polynomial in $X$ has the "largest possible" Galois group over $\mathbb{Q}$. Effective versions of Hilbert's Irreducibility Theorem typically give upper bounds for the number of integer specializations of bounded height that fail to have the largest possible Galois group over $\mathbb{Q}$. In this talk we will discuss an upper bound for specializations whose Galois group is a fixed subgroup $H$, where the bound becomes stricter for smaller choices of $H$. This is joint work with Rainer Dietmann.

Kevin Childers: S_4-extensions with a common cubic subfield
Given a fixed non-normal cubic extension of the rational numbers with specified ramification, we explore the relationships between S_4-extensions lying above the cubic extension and the corresponding octahedral Galois representations.

Britain Cox: Local Components of Automorphic Forms
We discuss work on how explicit local representations may be combined into an automorphic form.

Rachel Davis: Affine Galois representations attached to elliptic curves
Affine general linear Galois representations attached to elliptic curves are generalizations of the $\ell$-adic Tate representations. We study surjectivity, modularity, and deformation theory in the context of these representations. This is joint work with Edray Goins.

Ellen Eischen: An application of Serre-Tate theory to the construction of p-adic families of automorphic forms
One approach to constructing p-adic L-functions relies on constructing certain p-adic families of Eisenstein series. In turn, certain explicit constructions of p-adic families of Eisenstein series (such as the constructions of J.-P. Serre for modular forms, N. Katz for Hilbert modular forms, and the speaker for automorphic forms on unitary groups of signature (n,n)) rely on p-adically interpolating Fourier coefficients of Eisenstein series. How can one proceed in situations in which there are not Fourier expansions? We will discuss a modified approach, namely the use of "Serre-Tate expansions", which will be introduced in the talk. Using Serre-Tate theory, the condition on the signature in the speaker's construction of families of Eisenstein series mentioned above can be removed. Part of this work is joint with A. Caraiani, J. Fintzen, E. Mantovan, and I. Varma.

Cameron Franc: Three-dimensional imprimitive representations of the modular group and their associated modular forms
Let G be the modular group. Then G has finitely many one-dimensional representations, and the kernel of every two-dimensional representation of G with finite image has a congruence subgroup as its kernel. In three-dimensions one finds representations of G with finite image, but whose kernels are noncongruence subgroups. In this talk we will present joint work with Geoff Mason that gives a clean classification of the (infinite collection) of three-dimensional imprimitive representations of G, and we will describe the finite subset of these that have congruence kernel. Using earlier joint work this allows us to give concrete expressions for an infinite collection of noncongruence modular forms in terms of generalized hypergeometric series, and to prove a strong form of the unbounded denominators conjecture in these cases.

Sharon Frechette: Orbital Integrals and Shalika Germs for sl_n and sp_{2n}
Shalika germs were introduced as a tool for studying orbital integrals, objects that play a large role in harmonic analysis on p-adic groups. The Shalika germ expansion expresses regular semisimple orbital integrals in terms of nilpotent ones, in a neighborhood of the origin. Exact values of Shalika germs elude computation, except for those of a few Lie algebras of small rank. We prove that Shalika germs on sl_n and sp_{2n} belong to a class of motivic functions defined by Cluckers and Loeser by means of a first-order language of logic. The proof involves Nevins’ combinatorial matching between two parametrizations of nilpotent orbits, one involving partitions, and DeBacker’s parametrization arising from the Bruhat-Tits building. As a result, we establish bounds on the Shalika germs that are uniform in $p$. This is joint work with Julia Gordon and Lance Robson.

Holley Friedlander: Weyl group multiple Dirichlet series over the rational function field
Weyl group multiple Dirichlet series (WMDS) are Dirichlet series in several complex variables with analytic continuation and a group of functional equations isomorphic to the Weyl group. These series conjecturally arise as Whittaker coefficients of Eisenstein series on metaplectic groups and have applications in number theory including moments of $L$-functions. Given a global field, one may use combinatorial data from the associated root system to construct WMDS. One such approach is the averaging method of Chinta and Gunnells. In this talk, we discuss the structure of twisted Weyl group multiple Dirichlet series defined over the rational function field. In particular, we show that these series may be written as a sum of simpler local series and thus are completely determined by a relatively small number of coefficients.

Anna Haensch: Connected Graphs for Operators on Quaternary Codes
Codes can be viewed as a lattices via a classical construction, and consequently, many of the concepts of lattice theory can be adapted to the setting of codes. One particularly interesting association exists between weight-enumerators for codes and theta-series for lattices, which brings with it some of the tools of modular forms. There is a well defined analogue of Heck-operators for theta series in the setting of codes over finite fields, namely, the Kneser-Hecke-operator. In this talk we will discuss a similar construction for codes over finite chain rings, in particular, exploring the graph associated to the Kneser-Hecke-operator.

Alia Hamieh: Determining Hilbert modular forms by the central values of Rankin-Selberg convolutions
Let $g$ be a Hilbert modular form. We show that the central values $\{ L(f\otimes g, 1/2): f\in\mathcal{F}\}$ uniquely determine $g$. Here $\mathcal{F}$ is a carefully chosen infinite family of Hilbert modular forms that vary in the weight aspect. This is a joint work with Naomi Tanabe.

Paul Jenkins: Zeros of weakly holomorphic modular forms of half integral weight
We show that elements of a canonical basis for the space of weakly holomorphic modular forms of half integral weight on Gamma_0(4) in Kohnen's plus space have many zeros on the lower boundary of the fundamental domain. Thus, at many places on this arc, Zagier's Eisenstein series and certain Poincare' series have equal values. This is joint work with Amanda Folsom.

Rodney Keaton: Level Stripping of Degree 2 Siegel Modular Forms
In this talk we present a method for stripping prime powers from the level of a degree 2 Siegel modular form while preserving a certain congruence. As an application, we consider four dimensional Galois representations.

Tomoya Kiyuna: Kaneko-Zagier type equation for Jacobi forms
Kaneko and Zagier introduced a second-order ordinary differential equation for elliptic modular forms. Modular and quasimodular solutions of the equation was studied by Kaneko and Koike. In this talk, we carry out similar studies for Jacobi forms. First, we introduce a fourth-order partial differential equation for Jacobi forms. Next, we construct Jacobi form solutions of the equation.

Krzysztof Klosin: The p-adic Maass lift
The Maass lift is an automorphic form on the unitary group U(2,2) which is a CAP form arising from a modular form on GL(2). In this talk we will report on a recent joint work with Tobias Berger aimed at constructing a p-adic version of the lift, i.e., one which allows us to lift a Hida family of elliptic modular forms to a p-adic analytic family of automorphic forms on U(2,2).

Andrew Knightly: An asymptotic Petersson formula for GSp(2n)
We give an asymptotic Petersson formula for cuspidal representations of GSp(2n) of level N tending to infinity. As an application we obtain a weighted equidistribution result for the Satake parameters as a fixed place p.

Dominic Lanphier: Values of Rankin convolution L-functions and modularity
Several authors have shown how central values of certain L-functions attached to a cuspform on GL(2), and twisted by a GL(1) or a GL(2) form, can uniquely determine the cuspform. We show how certain critical values of convolution L-functions can uniquely determine a cuspform and in fact how such values can be used to establish modularity of certain functions. We also give a relatively simple test for modularity.

Michael Lipnowski: Twisted limit formula for cyclic base change

Blake Mackall, Steven J. Miller and Karl Winsor: Universal Lower-Order Biases in Elliptic Curve Fourier Coefficients
Let $\mathcal{E}: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic curves over $\mathbb{Q}(T)$, with $A(T),B(T)\in \mathbb{Z}(T)$, and consider the $k$th moments $A_{k,\mathcal{E}}(p) := \sum_{t \bmod p} a_{\mathcal{E}_t}(p)^k$ of the Fourier coefficients $a_{\mathcal{E}_t}(p) := p + 1 - \#\mathcal{E}_t(\mathbb{F}_p)$. Rosen and Silverman proved a conjecture of Nagao relating the first moment $A_{1,\mathcal{E}}(p)$ to the rank of the family over $\mathbb{Q}(T)$, and Michel proved the second moment is $A_{2,\mathcal{E}}(p) = p^2 + O(p^{3/2})$. Cohomological arguments show the lower order terms are always of sizes $p^{3/2}, p, p^{1/2}$ and 1. The bound of $p^{3/2}$ cannot be improved upon as there exist families with second moment lower order terms of size $p^{3/2}$. In every family we are able to analyze, the largest lower order term in the second moment formula that does not average to zero is negative. We prove this bias conjecture'' for several large classes of families, including families with rank, complex multiplication, and unusual distribution of signs. We identify all lower order terms in large classes of families, shedding light on the objects controlling these terms. The negative bias in these terms has implications towards the excess rank conjecture and the behavior of zeros near the central point of elliptic curve $L$-functions.

Michael Mertens: Special values of shifted convolution Dirichlet series
In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution L-functions by defining shifted convolution L-functions. In this talk, we investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and mixed mock modular'' forms. For a special case, we also show this mock modular form to be a linear combination of weakly holomorphic p-adic modular forms.

Djordje Milicevic: On moments of twisted L-functions
In addition to providing statistical and intrinsic information about the underlying family of automorphic forms, asymptotic formulas with a power saving for moments of central values of L-functions are an essential ingredient in analytic approaches to questions of arithmetic importance such as upper bounds, nonvanishing, or extreme values. In this talk, I will present recent asymptotic formulas for moments in several families of twisted L-functions with all primitive characters modulo q, with a power saving in q. We use a variety of methods, using the full power of spectral theory of GL(2) automorphic forms to treat a possibly highly unbalanced shifted convolution problem, as well as arithmetic techniques (including q-Weyl differencing, Burgess—Karatsuba method, Riemann Hypothesis for curves over finite fields, and independence of Kloosterman sheaves) to prove large-sieve-type and estimates on bilinear forms in Kloosterman sums in critical ranges. This is joint work with Blomer, Fouvry, Kowalski, and Michel.

Trung Hieu Ngo: Quantum modular forms and partition functions
A quantum modular form is a function on the rational numbers whose modular obstruction gives rise to an interesting analytic function. We introduce the notion of a renormalized duo of q-hypergeometric series, which is related to the theory of quantum Maass forms of Bruggeman, Lewis, and Zagier and which gives a method to construct quantum modular forms. We discuss how to use special quantum modular forms and the Hardy-Ramanujan-Wright circle method to give new asymptotic results for partition functions. This is joint work with Yingkun Li and Rob Rhoades.

Kyle Pratt: Coefficient bounds for level 2 cusp forms
We give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+\epsilon})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the coefficients of certain level 2 modular functions. This is joint work with Paul Jenkins.

Wendell Ressler: Conjugacy classes for the Hecke groups and related binary quadratic forms
We give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. We conclude that there are finitely many conjugacy classes of a given trace in any Hecke group, and that there are finitely many equivalence classes of related hyperbolic $$\mathbb{Z}[\lambda]$$-binary quadratic forms.

James Ricci: Regular Ternary Quadratic Polynomials of Fixed Conductor
In 1924, Helmut Hasse established a local-to-global principle for representations of rational quadratic forms. Unfortunately, an analogous local-to-global principle does not hold for representations over the integers. An integral quadratic polynomial is called regular if such a principal exists; that is if it represents all the integers which are represented locally by the polynomial itself over Z_p for all primes p as well as over the reals. In this talk we will give necessary conditions that provide a finiteness result for positive ternary regular quadratic polynomials. This work generalizes the analogous finiteness results for positive definite regular ternary quadratic forms by G.L. Watson and for ternary triangular forms by W.K. Chan and B.-K. Oh.

Olav Richter: Siegel modular forms mod p
I will report on recent joint work with Martin Westerholt-Raum on congruences of Siegel modular forms of arbitrary degree. Specifically, I will discuss the ring structure of Siegel modular forms modulo a prime p, I will present a characterization of U(p)-congruences of Siegel modular forms, and I will give Sturm bounds of Siegel modular forms.

Jeremy Rouse: Trinomials defining quintic number fields
Given a quintic number field $K/\mathbb{Q}$, we consider the problem of determining all trinomials $x^{5} + ax + b$ with $a, b \in \mathbb{Q}$ that have a root in $K$. These polynomials are parametrized by the rational points of a genus $4$ curve $C_{K}$. Moreover, there is a map from $C_{K}$ to a genus one curve $E$ defined over a (in general) degree $10$ extension of $\mathbb{Q}$. In some cases, we are able to determine the Mordell-Weil group of $E$ and use elliptic curve Chabauty to determine all such trinomials. In particular, if $K= \mathbb{Q}[z]$, where $z^{5} - 5z + 12 = 0$, then $f(x) = x^{5} - 5x + 12$ is the only trinomial (up to equivalence) with a root in $K$.

Malcolm Rupert: Towards an Explicit Theta Lift from Hilbert Modular Forms to Siegel Paramodular Forms

Brundaban Sahu: Jacobi Cusp Forms and the adjoint of linear maps constructed with the Rankin-Cohen brackets
Given a fixed Jacobi cusp form, we consider a family of linear maps between the spaces Jacobi cusp forms by using the Rankin-Cohen brackets and then we compute the the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Jacobi forms constructed using this method involve special values of certain Dirichlet series associated to Jacobi cusp forms. This is a generalization of the work due to W. Kohnen, S. D Herrero in case of elliptic modular forms to the case of Jacobi cusp forms earlier also done by H. Sakata for a special case. This is a joint work with Abhash Kumar Jha.

Karam deo Shankhadhar: Converse theorem for Jacobi forms

Naomi Tanabe: A Note on Special Values of Rankin-Selberg Convolutions for Hilbert Modular Forms
In this talk, we study the non-vanishing of the central values of Rankin-Selberg convolutions for Hilbert modular forms $f$ and $g$, where $g$ is fixed and $f$ varies over certain infinite families. This is a joint work with Alia Hamieh.

Karen Taylor: Modular Forms: View from a Hyperbolic Element
In (1941), Petersson gave a uniform treatment of parabolic, hyperbolic and elliptic Poincare series. Peterson was the first to give, the now classsical, expression for the fourier coefficients of Poincare series as a special value of a Dirichlet series involving Kloosterman sums and Bessel functions. In the classical case both the fourier series and the Poincare series are relative to a parabolic element. In this talk, we extend the treatment to give all four cases that occur when both the Fourier series and the Poincare series can be hyperbolic or parabolic.

Jesse Thorner: Applications of a Log-Free Zero Density Estimate for Automorphic L-functions
The first proofs of nontrivial bounds on the least prime in an arithmetic progression or the prime number theorem for short intervals required deep information about the density of zeros of L-functions close to the line Re(s) = 1. For automorphic L-functions satisfying the Ramanujan-Petersson conjecture, we obtain zero density results of comparable strength that allow us to prove various automorphic analogues of bounding the least prime in an arithmetic progression or counting primes in short intervals. We discuss applications to the Sato-Tate conjecture for a non-CM elliptic curve over a totally real field. This is joint work with Robert Lemke Oliver.

Stephanie Treneer: Weierstrass points on $X_0^+(p)$ and supersingular $j$-invariants
A Weierstrass point on a compact Riemann surface of genus $g$ is a point at which some holomorphic differential vanishes to order at least $g$. Weierstrass points on the modular curve $X_0(p)$, when reduced modulo the prime $p$, have been shown to correspond in a precise way to the complete set of $j$-invariants of supersingular elliptic curves in characteristic $p$. Here we investigate Weierstrass points on the quotient space $X_0^+(p)$ and give a relationship between their reductions modulo $p$ and the quadratic irrational supersingular $j$-invariants in characteristic $p$.

Amanda Tucker: Roots of modular units
We prove that if a modular unit has a pth root that is again a modular unit, then the level of that root is at most p times the level of the original unit.

Nahid Walji: Matching densities for automorphic representations
Given a pair of automorphic representations for GL(n)/Q, we define their matching density to be the density, if it exists, of the set of places at which the Hecke eigenvalues of the two automorphic representations are equal. We show, under the strong Artin conjecture, that the set of matching densities of such pairs of cuspidal automorphic representations (for all n), is dense in the interval [0, 1].

Fan Zhou: New Types of Voronoi Formulae on GL(n)
The Voronoi formula is an analog of Poisson summation formula for automorphic forms. Previously, one Voronoi formula was known for Maass forms on GL(n). It has great application in analytic number theory of automorphic forms and their L-functions. We discover new types of Voronoi formulae for automorphic forms on GL(n) for n>=4. There are [n/2] different Voronoi formulae on GL(n), which are summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums of dimension k=1,2,...,[n/2].