The following individuals have registered for the workshop on the website. Those with an asterisk plan to give a talk. Scroll down for abstracts.

Please notify the organizers regarding any oversights or omissions.

Scott Ahlgren* | UIUC |

Nick Andersen* | UIUC |

Hiroki Aoki* | Tokyo University of Science |

Olivia Beckwith* | Emory University |

Jeff Breeding-Allison* | Fordham University |

Jim Brown | Clemson University |

Luca Candelori* | LSU |

Ellen Eischen | University of Oregon |

Melissa Emory | University of Missouri |

Sharon Frechette* | College of the Holy Cross |

Dan Fretwell | University of Bristol |

Sharon Garthwaite* | Bucknell University |

Hugh Geller | Clemson University |

Jayce Getz* | Duke University |

Richard Gottesman | UC Santa Cruz |

Michael Griffin* | Princeton University |

Heekyoung Hahn* | Duke University |

Catherine Hsu* | University of Oregon |

Paul Jenkins* | Brigham Young University |

Andy Knightly* | University of Maine |

Abhash Kumar Jha* | NISER, Bhubaneswar India |

Kim Klinger-Logan* | University of Minnesota |

Krzysztof Klosin* | Queens College |

Aranya Lahiri | Indiana University |

Jaclyn Lang* | UCLA |

Robert Lemke Oliver* | Stanford University |

Huixi Li | Clemson University |

Jingbo Liu* | Wesleyan University |

Alicia Marino* | Wesleyan University |

Kimball Martin* | University of Oklahoma |

Dermot McCarthy* | Texas Tech University |

Anna Medvedovsky* | ICERM |

Michael Mertens* | Emory University |

Jaban Meher* | Indian Institute of Sciences |

Djordje Milicevic* | Bryn Mawr College |

Steven J. Miller | Williams College |

Tadashi Miyazaki* | Kitasato University |

Daniel Moore* | The Ohio State University |

Bharath Palvannan* | University of Washington |

Abhishek Parab | Purdue University |

Jason Polak* | McGill University |

Sudhir Kumar Pujahari* | Indian Institute of Science Education and Research |

Wissam Raji* | American University of Beirut |

Frank Romascavage, III | Bryn Mawr College |

Eugenia Rosu | UC Berkeley |

Jeremy Rouse | Wake Forest University |

Brundaban Sahu* | National Institute of Science Education and Research |

Howard Skogman* | The College at Brockport |

Karen Taylor* | Bronx Community College |

Kate Thompson | Davidson College |

Frank Thorne* | The University of South Carolina |

Jesse Thorner* | Emory University |

DJ Thornton* | Brigham Young University |

Tien Duy Trinh | Rutgers University |

Cindy Tsang | UC Santa Barbara |

Caroline Turnage-Butterbaugh* | Duke University |

Nahid Walji* | Universität Zürich |

Chen Wan* | University of Minnesota |

John Webb* | James Madison University |

**Scott Ahlgren**

Title: Kloosterman sums and Maass cusp forms of half integral weight for the modular group

Abstract: Kloosterman sums appear in many areas of number theory. We estimate sums of Kloosterman sums of half-integral weight on the modular group. Our estimates are uniform in all parameters in analogy with Sarnak and Tsimerman’s improvement of Kuznetsov's bound for the ordinary Kloosterman sums. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's famous formula for the partition function. This is joint work with Nick Andersen.

**Nick Andersen**

Title: The mock theta conjectures

Abstract: The mock theta 'conjectures' are two families of identities from Ramanujan's Lost Notebook involving the fifth order mock theta functions. These identities were first proven by Hickerson (Inventiones 1988) using Hecke-type $q$-series identities discovered by Andrews. We prove an equality between two vector-valued harmonic Maass forms of weight $1/2$ which encodes these identities, thus providing a simple, conceptual proof of the mock theta conjectures.

**Hiroki Aoki**

Title: A remark on the weights of Siegel modular forms

Abstract: I will talk about possible weights of modular forms, which is not only in integral numbers but in real numbers. First we review elliptic modular forms, then we investigate Jacobi forms, and finally we discuss about possible weights of modular forms of degree 2, by using previous results.

**Olivia Beckwith**

Title: The Number of Parts of Integer Partitions in Given Residue Classes

Abstract: In a previous work of the speaker and Michael Mertens, formulae were given for the number of parts of integer partitions that are $0\pmod{N}$, and for the difference between the number of parts in complementary residue classes. In a recent work, we improve on the first result to obtain asymptotic formulae for the number of parts in any given residue class.

**Jeff Breeding-Allison**

Title: Depth zero representations and dimensions of spaces of modular forms

Abstract: Computing finite-dimensional spaces of modular forms without knowing the dimension is an active area of research with many important applications. One such application is to prove modularity of certain abelian varieties. The Paramodular Conjecture, due to A. Brumer and K. Kramer, has drawn attention to spaces of paramodular cusp forms of weight two. We discuss an approach to computing dimensions of spaces of modular forms using depth zero representations of $G(k)$, where $G$ is a connected reductive group defined over a non-archimedean local field $k$ of characteristic $0$. We also discuss recent results on explicitly computing spaces of paramodular forms.

**Luca Candelori**

Title: The character of Riemann-Jacobi theta functions

Abstract: The classical transformation laws of theta functions involve a character from the theta group into the 8-th roots of unity, which can be computed explicitly using analytic techniques such as the Poisson summation formula. Using ideas of Deligne, we give a new group-theoretic construction of the square of this character, and discuss how this can be used to give new algebraic proofs of the transformation laws.

**Sharon Frechette**

Title: Appell Hypergeometric Functions over Finite Fields

Abstract: Using the dictionary between classical and finite field hypergeometric functions of one variable, recently developed by Fuselier, Long, Ramakrishna, Swisher and Tu, we define and develop a theory of multivariable finite-field hypergeometric functions. We establish transformations and symmetries of these multivariable functions, building analogues to classical results such as the cubic transformation for Appell hypergeometric functions shown by Koike and Shiga. We also explore the geometry of finite-field Appell hypergeometric functions through their relationship to the generalized family of Picard curves. This is joint work with Ling Long, Holly Swisher and Fang-Ting Tu.

**Dan Fretwell**

Title:Local origin congruences for elliptic modular forms.

Abstract: Congruences between modular forms have been studied extensively in the last century or so. The most famous of these is perhaps Ramanujan's congruence between the discriminant form and the weight $12$ Eisenstein series modulo $691$. One can show that the existence of this congruence is equivalent to the fact that $691$ is a regular prime (so that the class group of $\mathbb Q(\zeta_691)$ has an element of order $691$). Of course the simplest explanation of this is that $691$ divides the numerator of $\zeta(12)/\pi^{12}$, a quantity appearing in the constant term of $E_{12}$.

Various generalisations of Ramanujan's congruence have been made in recent years, giving lots of information on Galois representations associated to modular forms and strong evidence for the Bloch-Kato conjecture (a motivic generalization of the analytic class number formula).

In this talk I will introduce local origin congruences, a generalization of the above congruence to the case of level $N$ Eisenstein series and level $Np$ cusp forms. The primes in such congruences are shown not only to arise from zeta values but also Euler factors at $p$ (hence the term "local origin"). Time permitting we will see this in action for Hilbert modular forms.

**Sharon Garthwaite**

Title: Quantum Modular Forms from Eta-Theta Functions

Abstract: In his Ph.D thesis, Zwegers provides a recipe for building mock theta functions, providing context for Ramanujan's ``very interesting functions." This construction makes use of even and odd character theta series. More recently, Lemke Oliver classified all theta series that are also eta quotients. In this talk, we use Lemke Oliver's list and Zweger's recipe to systematically build families of mock theta functions associated to eta quotients. We find that these forms in fact fit into Zagier's theory of quantum modular forms.

**Jayce Getz**

Title: Limiting forms of trace formulae and triple product $L$-functions

Abstract: Langlands has proposed studying limits of trace formulae as an approach to proving Langlands functoriality in general. The proposal has only been carried out in very special cases corresponding more or less to the standard representation and symmetric square representation of $GL(2)$ and the tensor product of $GL(2)$ with itself. We explain how the analytic part of the proposal can be carried out in the case of the tensor product of three copies of $GL(2)$. Time permitting, we discuss the prospects of generalizing the approach to the tensor product of three copies of general linear groups of arbitrary rank.

**Michael Griffin**

Title: $p$-adic harmonic Maass forms.

Abstract: Harmonic Maass forms posses many intricate $p$-adic properties. Guerzhoy, Kent, and Ono, and others have extensively studied the $p$-adic properties of the $q$-series of related mock modular forms. Special values of Harmonic Maass forms also possess similarly interesting $p$-adic properties. In his doctoral thesis, Candelori investigated a theory of integer weight \emph{$p$-harmonic Maass forms} arising from the de Rham cohomology for $p$-adic modular forms. We consider a similar theory, although our approach and definitions differ somewhat from Candelori's. We construct $p$-adic analogues of classical harmonic Maass forms of weight $0$ and $1/2$ with square free level by means of the Hecke algebra. As in the classical case these forms are connected to positive weight modular forms by certain differential operators. Moreover, the coefficients of the half integral weight forms can be given as modular traces of weight $0$ forms over Heegner divisors. The complex harmonic Maass forms and their corresponding $p$-adic analogues may also be collected into an adelic theory. As an application, we consider elliptic curves $E/\mathbb{Q}$ with square free conductor. Building on work of Bruinier and Ono, we construct a function $H_E'$, whose vanishing at Heegner points determines the vanishing of central $L$ derivatives of quadratic twists of $E$.

**Heekyoung Hahn**

Title: Langlands' beyond endoscopy proposal and the Littlewood-Richardson semigroup

Abstract: Langlands' beyond endoscopy proposal for establishing functoriality motivates the study of irreducible subgroups of $\mathrm{GL}_n$ that stabilize a line in a given representation of $\mathrm{GL}_n$. Such subgroups are said to be detected by the representation. In this talk we study the important special case where the representation of $\mathrm{GL}_n$ is the triple tensor product representation $\otimes^3$. We prove a family of results describing when subgroups isomorphic to classical groups of type $B_{n}$, $C_n$, $D_{2n}$ are detected.

**Catherine Hsu**

Title: Two Classes of Number Fields with a Non-Principal Euclidean Ideal

Abstract: In 1979, Lenstra defined the Euclidean ideal, a generalization of the Euclidean algorithm. Just as the existence of a Euclidean algorithm for the ring of integers $\mathcal{O}_K$ in a number field $K$ implies a trivial class group, the existence of a Euclidean ideal $C$ in $\mathcal{O}_K$ implies a cyclic class group with generator $[C]$. By using certain growth results, Graves provided the first explicit example of a number field a non-principal Euclidean ideal. In this talk, we generalize Graves' techniques in order to introduce two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal.

**Paul Jenkins**

Title: Zeros of modular forms of half integral weight

Abstract: We study canonical bases for spaces of weakly holomorphic modular forms of level $4$ and half integer weight, and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for $\Gamma_0(4)$ lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincare' series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish. This is joint work with Amanda Folsom.

**Abash Jha**

Title:Rankin-Cohen brackets on Siegel modular forms of genus two and certain Dirichlet series.

Abstract: We compute the petersson scalar product of a Siegel cusp form $F$ with Rankin- Cohen bracket of a Siegel modular form $G$ and $T$-th Poinca{\'r}e series and show that this scalar product is the $T$-th Fourier coefficients of the image of the Siegel cusp form $F$ under adjoint of certain linear map constructed using Rankin- Cohen bracket and the value of this product is, up to a constant a special value of certain Dirichlet series of Rankin type associated with $F$ and $G.$ This is joint work with Brundaban Sahu.

**Kim Klinger-Logan**

Title: Self-Adjoint Operators and Zeros of L-functions

Abstract: Hilbert and Polya raised the possibility of proving RH by producing zeros of the zeta function among parameters $s$ for eigenvalues $\lambda_s=s(s-1)$ of suitable self-adjoint operators. A spark of hope towards finding an appropriate operator occurred when Haas (1977) numerically miscalculated eigenvalues for the invariant Laplacian and obtained zeros of zeta in his list of $s$-values. In 1981, Hejhal identified this numerical flaw and clarified what Haas had actually computed -- not genuine eigenvalues for the invariant Laplacian. Colin DeVerdiere speculated on a possible legitimization that languished for 30 years. Recent work of Bombieri and Garrett makes precise Colin DeVerdiere's speculation and proves that, while any discrete spectrum must have s-values among zeros of corresponding zeta functions, there is an operator-theoretic mechanism in play which has the regular behavior of zeta on the edge of the critical strip coerce discrete spectrum to be too regularly spaced to be compatible with Montgomery's pair-correlation conjecture. We show that similar mechanisms apply to more complicated periods of Eisenstein series. Exhibiting an $L$-function as a period of an Eisenstein series, I extend these results to a more hostile example in which non-compactness of the period presents further obstacles.

**Krzysztof Klosin**

Title: Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts

Abstract: Let $f$ be an automorphic Hecke eigenform on the unitary group $G=U(n,n)(\mathbf{A}_F)$, where $F$ is a totally real extension of $\mathbf{Q}$. In this talk we will discuss an $L$-value condition guaranteeing the existence of a congruence between $f$ and other Hecke eigenforms on $G$. We will apply this result to the case where $f$ is the Ikeda lift. Furthermore, we will discuss a non-vanishing result mod $p$ for Ikeda lifts and indicate potential applications to some new cases of the Bloch-Kato conjecture. This is joint work with Jim Brown.

**Jaclyn Lang**

Title: Images of Galois representations associated to Hida families

Abstract: We explain a sense in which Galois representations associated to non-CM Hida families have large images. This is analogous to results of Ribet and Momose for Galois representations associated to classical modular forms. In particular, we show how extra twists of the Hida family decreases the size of the image.

**Robert Lemke Oliver**

Title: Consecutive primes modulo q

Abstract: The primes are well-known to be equidistributed among the admissible residue classes modulo any integer q at least three. Here, we consider the distribution of consecutive primes modulo q, and we find considerable deviation from the natural prediction. We offer a conjectural explanation for this deviation, and we provide a conditional proof of this conjecture. This is joint work with Kannan Soundararajan.

**Jingo Liu**

Title: Representations of integral Hermitian forms by sums of norms

Abstract: In 1770, Lagrange proved the famous four square theorem, which says that each positive integer a can be represented as a sum of four squares. This theorem has been generalized in many directions since then. One interesting generalization is to consider the representations of integral quadratic forms of more variables by sums of squares. We define $g_{\mathbb Z}(n)$ to be the smallest number of squares whose sum represents all positive definite integral quadratic forms of $n$ variables over $\mathbb Z$ that are represented by some sums of squares. The existence of $g_{\mathbb Z}(n)$ and an explicit upper bound was given by Icaza in 1996. An improved upper bound was obtained later by Kim and Oh in 2005. Similarly, for Hermitian forms over the ring of integers $\mathcal O_E$ of imaginary quadratic field $E$, we define $g_{E}(n)$ to be the smallest number of norms whose sum represents all positive definite integral Hermitian forms of n variables over $\mathcal O_E$ that are represented by some sums of norms. In this talk, we will present a generalization of Kim and Oh's method and give an explicit upper bound for $g_{E}(n)$ for any imaginary quadratic field $E$ and positive integer $n$.

**Alicia Marino**

Title: Strictly k-Regular Integral Quadratic Forms

Abstract: An integral quadratic form is said to be strictly $k$-regular if it primitively represents all quadratic forms of $k$ variables that are primitively represented by its genus. We show that, for $k > 2$, there are finitely many inequivalent positive definite primitive integral quadratic forms of $k + 4$ variables that are strictly $k$-regular. Our result extends a recent finiteness result of Andrew Earnest et al. (2014) on strictly regular quadratic forms of $4$ variables.

**Kimball Martin**

Title: Eisenstein congruences and the Jacquet-Langlands correspondence

Abstract: We will explain how to use the Jacquet-Langlands correspondence to show, in a simple way, the existence of weight 2 elliptic or Hilbert cusp forms which are congruent to a weight 2 Eisenstein series. We will also discuss applications to L-values. This generalizes some results of Mazur and others.

**Dermot McCarthy**

Title: Multiplicative Relations for Fourier Coefficients of Degree 2 Siegel Eigenforms

Abstract: It is well known that the space of elliptic modular forms, of a given weight, has a basis of Hecke eigenforms which have multiplicative Fourier coefficients. While Hecke theory has been extended to Siegel modular forms, the results are not as straightforward as the elliptic case. In this talk, we will first outline the work of Andrianov on how the Hecke theory generalizes to spaces of degree 2 Siegel modular forms. Then we will discuss recent work on deriving simple multiplicative relations, which are analogous to the elliptic case, between certain Fourier coefficients of degree 2 Siegel eigenforms.

**Anna Medvedovsky**

Title: Lower bounds on dimensions of mod-p Hecke algebras

Abstract: Lower bounds on dimensions of mod-p Hecke algebras In 2012, Nicolas and Serre revived interest in the study of Hecke algebras on modular forms mod p when they proved that for $p = 2$ the completed Hecke algebra is a power series ring over F2 generated by T3 and T5. Their elementary arguments do not generalize directly to other primes, but their tools --- the Hecke recursion, the nilpotence filtration --- serve as the backbone of a new method-in-development, uniform and entirely in characteristic p, for understanding the structure of mod-p Hecke algebras. I will present this method (currently implemented for level one and $p = 2, 3, 5, 7, 13$) and compare it with the method of Bellaiche-Khare, which uses characteristic-zero theory to prove similar results for all $p\geq 5$.

**Jaban Meher**

Title: Products of eigenforms

Abstract: We will characterise all cases in which products of arbitrary numbers of nearly holomorphic eigenforms and products of arbitrary numbers of quasimodular eigenforms for the full modular group $SL_2(\mathbb Z)$ are again eigenforms.

**Michael Mertens**

Title: Partitions, singular moduli, and irreducible polynomials

Abstract: Recently, Bruinier and Ono found an algebraic formula for the partition function in terms of traces of singular moduli of a certain non-holomorphic modular function. In this talk, we show that the rational polynomial having these singuar moduli as zeros is (essentially) irreducible, settling a question of Bruinier and Ono. The proof uses careful analytic estimates together with some related work of Dewar and Murty, as well as extensive numerical calculations of Sutherland.

**Djordje Milicevic**

Title: $p$-adic analytic twists, modularity, and strong subconvexity

Abstract: One of the principal questions about automorphic L-functions are the so-called subconvex estimates on the size of their critical values, deeply arithmetic both in proofs and in the often spectacular consequences. In this talk, we will present our recent subconvexity bound for the central value of the L-function associated to a fixed cuspidal newform f twisted by a Dirichlet character chi of a high prime power conductor. From an adelic viewpoint, the analogy between this so-called "depth aspect" and the familiar t-aspect is particularly natural, as one is focusing on ramification at one (finite or infinite) place at a time.\\

We prove our results by exhibiting strong cancellation between the Hecke eigenvalues of f and the values of chi, which act as twists by exponentials with a p-adically analytic phase. Among the tools, we develop p-adic counterparts to Farey dissection and van der Corput estimates for exponential sums. This is joint work with Valentin Blomer.

**Steven J Miller**

Title: Biases in Moments of Elliptic Curve

Abstract: For non-CM families, Michel proved the second moment of the Fourier coefficents is $p^2 + O(p^{3/2})$. Cohomological arguments show that the lower order terms are of sizes $p^{3/2}, p, p^{1/2}$ and $1$. In every case we are able to analyze, the largest lower order term in the second moment expansion that does not average to zero is on average negative. We prove this bias conjecture for several large classes of families, including families with rank and unusual distributions of functional equation signs. We also identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point of elliptic curve L-functions. If time permits we'll discuss biases in other families of L-functions. This work is joint with Owen Barrett, Blake Mackall, Brian McDonald, Christina Rapti, Patrick Ryan, Caroline Turnage-Butterbaugh and Karl Winsor.

**Tadashi Miyazaki**

Title: Archimedean zeta integrals for $GL(3)\times GL(2)$

Abstract: We compute archimedean zeta integrals for Whittaker functions on $GL(3)$ and $GL(2)$ explicitly, and find Whittaker functions for which the archimedean zeta integrals coincide with the associated L-factors. This is a joint work with Professors Miki Hirano and Taku Ishii.

**Daniel Moore**

Title: A Tensor Product Theorem for Smooth Automorphic Representations

Abstract: Let $G = \mathbb{G}(\mathbb{A}_k)$ for $k$ a global field, $\mathbb{G}$ a reductive algebraic group defined over $k$, and $\mathbb{A}_k$ the adele ring of $k$. We will define the notion of smooth automorphic forms and automorphic representations on the former. In analogy with the lifting of $(\mathfrak{g},K)$-modules to their Casselman-Wallach globalizations in the theory of real reductive groups, we introduce a certain convolution algebra $\mathcal{S}(G)$ of ``Schwartz functions'' on $G$ and describe an equivalence of categories between smooth automorphic representations of $G$ and ``admissible'' $\mathcal{S}(G)$-modules. The algebra $\mathcal{S}(G)$ decomposes into a tensor product of analogous algebras $\mathcal{S}(G_v)$ over the places $v$ of $k$, and this permits us to discuss a tensor product theorem for admissible $\mathcal{S}(G)$-modules and, hence, for smooth automorphic representations, showing that every such representation $\pi$ factors into a tensor product $\otimes_v \pi_v$ of representations of the groups $\mathbb{G}(k_v)$. Our talk will be introductory in nature with a goal of introducing smooth automorphic representations to the audience for the first time and, time permitting, we will close by discussing some other aspects of the theory.

**Bharathwaj Palvannan**

Title: On the relationship between Selmer groups and factoring p-adic L-functions

Abstract: Dasgupta has proved a formula factoring a 3-variable Rankin-Selberg p-adic L-function. We prove the corresponding result involving Selmer groups. We will then explain a specialization result showing that Dasgupta's and our result are consistent with various main conjectures in Iwasawa theory.

**Jason Polak**

Title: Beginnings of Relative Endoscopy

Abstract: A key tool in establishing many cases of Langlands functoriality so far has been the Arthur-Selberg trace formula combined with the theory of endoscopy to express certain sums of orbital integrals on a group as sums of orbital integrals on smaller groups. Here, by group we mean the $F$-points of a reductive group where $F$ is a $p$-adic field. For relative trace formulae used to study distinguished automorphic representations, no such theory of endoscopy exists so far. In this talk I shall hint at the beginnings of such a theory by describing an orbital integral computation in the relative case. Then I'll describe the computation of dimension of a relative affine Springer fiber related to this orbital integral.

**Sudhir Pujahari**

Title: Distribution of gaps of eigenvalues/eigenangles of Hecke operators

Abstract: In this talk we will discuss the distribution of gaps of eigenvalues/eigenangles of Hecke operators acting on spaces of cusp forms of weight $k$ and level $N$. As an application we will see some evidence towards Maeda and Tsaknias conjectures. We will also see similar results for Hilbert modular forms and Maass forms.

**Wissam Raji**

Title: Period Functions of Half Integral Weight Modular Forms

Abstract: We study the Eichler cohomology associated with half- integral weight cusp forms using the Dedekind eta function $\eta(z)$ and the theta function $\theta(z)$. We prove that $\eta$-multiplication (resp. $\theta$-multiplication) gives an isomorphism between the space of cusp form of a half-integral weight and the cohomology group associated with the space $\eta$ (resp. $\theta$). We also show that there is an isomorphism between the direct sum of two spaces of cusp forms of half-integral weights and the cohomology group. (joint work with Dohoon Choi and Subong Lim.

**Eugenia Rosu**

Title: Integers that can be written as the sum of two cubes

Abstract: The Birch and Swinnerton-Dyer conjecture predicts that we have non-torsion rational points on an elliptic curve iff the L-function corresponding to the elliptic curve vanishes at 1. Thus BSD predicts that a positive integer N is the sum of two cubes if $L(E_N, 1)=0$, where $L(E_N, s)$ is the L-function corresponding to the elliptic curve $E_N: x^3+y^3=N$. Using methods from automorphic forms, we have computed several formulas that relate $L(E_N, 1)$ to the trace of a ratio of specific theta functions at a CM point . This offers a criterion for when the integer $N$ is the sum of two cubes. Furthermore, when $L(E_N, 1)$ is nonzero we get a formula for the number of elements in the Tate-Shafarevich group and show that it is a square in certain cases.

**Brundaban Sahu**

Title: Convolution sums of the divisor functions and the number of representations of certain quadratic forms

Abstract: We report about our recent results on the evaluation of some convolution sums of the divisor functions and their use in determining the number of representations of certain quadratic forms. By comparing with similar results, we also make some observations about the Fourier coefficients of certain cusp forms. This is a joint work with B. Ramakrishnan.

**Karen Taylor**

Title: Applications of hyperbolic fourier coefficients.

Abstract: In this talk we will give examples of identities obtained from specializing the exact formula for the nth hyperbolic fourier coefficient of a cusp form to the full modular group.

**Frank Thorne**

Title: Zeros of some L-functions outside the critical strip

Abstract: Zeta and L-functions are presumably not allowed to have any zeroes away from the critical line, let alone outside the critical strip. I will talk about some that do.

**Jesse Thorner**

Title: Bounded gaps between primes in Hecke equidistribution problems

Abstract: Using Duke's large sieve inequality for Hecke Grossencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution and explore several applications. For example, for any fixed $0<\epsilon<1/2$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<\epsilon\sqrt{p}$ and enumerate these gaps with lower bounds of the correct order of magnitude. We obtain similar results in the context of Hecke equidistribution for the Fourier coefficients of normalized Hecke eigenforms with complex multiplication.

**DJ Thornton**

Title: On Weakly Holomorphic Modular Forms in Certain Prime Power Levels of Genus Zero

Abstract: We examine canonical bases for certain spaces of weakly holomorphic modular forms of weight zero in levels 8, 9, and 16. We show that these genus zero bases exhibit many properties similar to those seen in prime genus zero bases.

**Caroline Turnage-Butterbaugh**

Title: Large gaps between zeros of Dedekind zeta-functions of a quadratic number field

Abstract: Let $K$ be a quadratic number field with discriminant $d$. The Dedekind zeta-funciton attached to $K$ can be expressed by $\zeta_{K}(s)=\zeta(s)L(s,\chi_d)$ for $s\ne1$, where $\zeta(s)$ is the Riemann zeta-function, the character $\chi_d$ is the Kronecker symbol associated to $d$, and $L(s,\chi_d)$ is the corresponding Dirichlet $L$-function. Using Hall’s Method with the twisted second moment of $\zeta_K(s)$, we show that there are infinitely many gaps between consecutive zeros of $\zeta_K(s)$ on the critical line which are greater than $2.866$ times the average spacing. This is joint work with Hung Bui and Winston Heap.

**Nahid Walji**Title: On the distribution of Hecke eigenvalues for GL2.

Abstract: Given a self-dual cuspidal automorphic representation for GL(2) over a number field, we establish the existence of an infinite number of Hecke eigenvalues that are greater than an explicit positive constant, and an infinite number of Hecke eigenvalues that are less than an explicit negative constant. This provides an answer to a question of Serre. We also consider analogous problems for cuspidal automorphic representations that are not self-dual.

**Chen Wan**Title: A multiplicity formula for the Ginzberg-Rallis model

Abstract: I will talk about a local multiplicity formula for the Ginzberg-Rallis model. Using that formula, one can prove the multiplicity one theorem for Vogan packet.

**John Webb**

Title: Fun facts about eta-quotients

Abstract: Let $\eta(z)$ be Dedekind's eta-function. For an integer $N\geq 1$, a function of the form $f(z)= \prod_{d\mid N} \eta(d z)^{r_d}$ with each $r_d \in \mathbb{Z}$ is called a level $N$ eta-quotient. We examine for which $N$ the entire graded algebra of holomorphic modular forms on $\Gamma_0(N)$ is generated by eta-quotients, as well as examine the feasibility of calculating bases of eta-quotients. We also show that any modular form in the space $M_k(\Gamma_0(N))$ with integral Fourier coefficients which is non-zero on the upper-half plane is an eta-quotient, generalizing a result of Kohnen. As an application, we calculate the cuspidal $\mathbb{Q}$-rational torsion subgroup of $J(X_0(2^n))$ for all $n\geq 1$.