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.
| Daniel Adams* | Brigham Young University |
| Scott Ahlgren* | University of Illinois |
| Ali Al-Saedi* | Oregon State University |
| Nick Andersen | University of California at Los Angeles |
| Allison Arnold-Roksandich* | Oregon State University |
| Matt Boylan | University of South Carolina |
| Jeffery Breeding-Allison* | Boston College |
| Jim Brown | Clemson University |
| Jorge Flórez* | City University of New York-Graduate Center |
| Dan Fretwell* | University of Bristol |
| Kohta Gejima* | Osaka University |
| Hugh Geller | Clemson University |
| Richard Gottesman* | University of California at Santa Cruz |
| Anna Haensch* | Duquesne University |
| Catherine Hsu* | University of Oregon |
| Vicki Iba* | Brigham Young University |
| Kevin James | Clemson University |
| Marie Jameson | University of Tennessee |
| Paul Jenkins | Brigham Young University |
| Cihan Karabulut* | William Paterson University |
| Rodney Keaton | East Tennessee State University |
| Ryan Keck* | Brigham Young University |
| Kim Klinger-Logan* | University of Minnesota |
| Andrew Knightly* | University of Maine |
| Hankun Ko | Brigham Young University |
| Arvind Kumar* | Harish-Chandra Research Institute |
| Dominic Lanphier | Western Kentucky University |
| Huixi Li* | Clemson University |
| Tianyi Mao* | City University of New York-Graduate Center |
| Kimball Martin | University of Oklahoma |
| Jolanta Marzec* | Durham University |
| Toshiki Matsusaka* | Kyushu University |
| Jaban Meher | National Institute of Science Education and Research |
| Djordje Milićević* | Bryn Mawr College |
| Steven Miller* | Williams College |
| Andreea Mocanu* | University of Nottingham |
| Grant Molnar* | Brigham Young University |
| Thomas Morrill* | Oregon State University |
| Eric Moss* | Brigham Young University |
| Michalis Neururer* | University of Nottingham |
| Abhishek Parab* | Purdue University |
| Krzysztof Pawelec* | Pennsylvania State University |
| Ameya Pitale* | University of Oklahoma |
| Neha Prabhu* | Indian Institute of Science, Education, and Research,Pune |
| Sudhir Pujahari* | Harish-Chandra Research Institute |
| Wissam Raji* | American University of Beirut |
| Daniel Reiss* | University of Idaho |
| Angelo Rendina* | University of Sheffield |
| Olav Richter | University of North Texas |
| Jeremy Rouse | Wake Forest University |
| Manami Roy* | University of Oklahoma |
| Malcolm Rupert* |
University of Idaho |
| Gunja Sachdeva* | Indian Institute of Science, Education, and Research,Pune |
| Adrienne Sands* | University of Minnesota |
| Ralf Schmidt | University of Oklahoma |
| Thomas Shemanske | Dartmouth College |
| Alok Shukla* | University of Oklahoma |
| Saurabh Singh* | Indian Statistical Institute |
| Howard Skogman | The College at Brockport |
| Fredrik Stromberg* | University of Nottingham |
| Long Tran | University of Oklahoma |
| Kit Vander Wilt* | Brigham Young University |
| Siddhesh Wagh* | University of Oklahoma |
| Merrill Warnick* | Brigham Young University |
| Jordan Wiebe* | University of Oklahoma |
| Ka Lun Wong* | University of Hawaii at Manoa |
| Hui Xue | Clemson University |
| Shaoyun Yi | University of Oklahoma |
Title: Canonical bases for spaces of weakly holomorphic modular forms with fractional Fourier coefficients or Weierstrass points at infinity
Abstract: Let $M_k^!(N)$ be the space of weakly holomorphic modular forms of level $N$. Let $M_k^\#(N) \subset M_k^!(N)$ be the subspace of forms which can have poles only at infinity, and $S_k^\#(N) \subset M_k^\#(N)$ be the subspace of forms which vanish at all cusps other than infinity. We examine and construct canonical bases for $M_k^\#(N)$ and $S_k^\#(N)$ in levels $N = 52$ and $N = 64$. In particular, it will be shown that the coefficients of these basis elements satisfy Zagier duality.
Scott Ahlgren
Title: A polyharmonic Maass form of depth 3/2 for SL_2(Z)
Abstract: Duke, Imamoglu, and Tóth constructed a polyharmonic Maass form whose Fourier coefficients
encode real quadratic class numbers. A more general construction of such forms was given by Bruinier, Funke, and Imamoglu. Here we give a direct construction of such a form for the full modular group and study the properties of its coefficients. This is joint work with Nickolas Andersen and Detchat Samart.
Ali Al-Saedi
Title: An infinite family of restricted plane overpartition congruences modulo 4
Abstract: In 2009, Corteel, Savelief and Vuletic generalized the concept of overpartitions to a new object called plane overpartitions.
In recent work, the author defines a restricted form of plane overpartitions called $k$-rowed plane overpartions and proves a method to obtain congruences for this and many other types of combinatorial generating functions. In this paper, we employ elementary generating function manipulations to prove an infinite family of k-rowed plane overpartition congruences modulo 4 for all $k>1$.
Allison Arnold-Roksandich
Title: Counting eta-quotients of prime level
Abstract: It is known that all modular forms on $\SL_2(Z)$ can be expressed as a rational function in $\eta(z)$, $\eta(2z)$ and $\eta(4z)$. By using known theorems, and calculating the order of vanishing, it is possible to compute the eta-quotients for a given level. Using this count, it is possible to figure out how many eta-quotients are linearly independent and using the dimension formula, we can figure out a subspace spanned by the eta-quotients. In this talk, we discuss the case where $N=p$ a prime.
Jeffery Breeding-Allison
Title: Saito-Kurokawa and Borcherds liftings
Abstract: In some cases, a Saito-Kurokawa or arithmetic lifting from certain spaces of Jacobi forms to spaces of Siegel modular forms of degree two can be used to construct a Borcherds or exponential lifting. In this talk, we describe some examples of this construction and a special case where this construction fails.
Jorge Flórez
Title: Explicit reciprocity laws for higher local fields
Abstract: In this talk we present a generalization of Kolyvagin's explicit reciprocity laws to higher local fields. This involves explicit formulas for the generalized Kummer pairing associated to an arbitrary formal group, in terms of multidimensional p-adic derivations, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. In the specific case of a Lubin-Tate formal group, these formulas give a higher-dimensional version of the explicit reciprocity laws of Artin-Hasse, Iwasawa and Wiles.
Dan Fretwell
Title: Paramodular Eisenstein congruences for GSp(4)
Abstract: There are many congruences known to exist between elliptic modular forms. The most well known of these is Ramanujan's congruence between the discriminant function and the weight 12 Eisenstein series modulo 691. Such congruences give us a wealth of information about residual Galois representations and class numbers of cyclotomic fields.
One can define the notion of "Eisenstein congruence" for automorphic forms over any reductive group. Certain Eisenstein congruences for GSp4, predicted by Harder, have been difficult to find and have resisted proof for decades. In this talk we will use the theory of algebraic modular forms to find new evidence for a paramodular version of this conjecture. We will also see theoretical justification for expecting paramodular forms to satisfy the congruence.
Kohta Gejima
Title: An explicit formula of the unramified Shintani functions for (GSp(4),GL(2)\times_{\GL(1)}GL(2))
Abstract: Let $F$ be a non-archimedean local field of characteristic zero. We give an explicit formula of Shintani functions on $\GSp_4(F)$. This explicit formula is a natural generalization of the explicit formula of Shintani functions on the split orthogonal group $\SO_5(F) \simeq \PGSp_4(F)$ due to Kato--Murase--Sugano. As an application, we evaluate a local zeta integral of Murase--Sugano type, which turns out to be the spin $L$-factor of $\GSp_4$.
Richard Gottesman
Title: The arithmetic of vector-valued modular forms
Abstract: Vector valued modular forms are generalizations of modular forms with a character. They form a graded module over the ring of modular forms. I will explain how understanding the structure of the module of vector valued modular forms allows one to show that the component functions of vector valued modular forms are solutions to certain ordinary differential equations. In some cases, these differential equations have explicit solutions and one then obtains the q-series expansions of the vector valued modular forms. No previous knowledge of vector valued modular forms will be assumed.
Catherine Hsu
Title: Higher congruences between newforms and Eisenstein series of squarefree level
Abstract: Let $p\geq 3$ be prime. For squarefree level N>6, we use a commutative algebra result of Berger, Klosin, and Kramer to bound the depth of Eisenstein congruences modulo $p$ (from below) by the $p$-adic valuation of the numerator of $\varphi(N)/24$. We then show that if $N$ has at least three prime factors and some prime $p\geq 5$ divides $\varphi(N),$ the Eisenstein ideal is not locally principal. Time-permitting, we will illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert-Samuel multiplicities.
Vicki Iba and Merrill Warnick
Title: Congruences for coefficients of modular functions in levels 6 and 12
Abstract: We construct canonical bases for spaces of weakly holomorphic modular forms for levels 6 and 12 with poles only at infinity. We show that Zagier-type duality holds in these levels. We show that certain coefficients are highly divisible by primes not dividing the level, and observe congruences modulo powers of 2 and 3.
Cihan Karabulut
Title: Eisenstein cocycles for GL(n) and values of L-functions in imaginary quadratic extensions
Abstract: We generalize Sczech's Eisenstein cocycle for $\GL(n)$ over totally real extensions of $\mathbb{Q}$ to finite extensions of imaginary quadratic fields. By evaluating the cocyle on certain cycles, we parametrize complex values of Hecke $L$-functions previously considered by Colmez, giving a cohomological interpretation of his algebraicity result on special values of the $L$-functions.
Ryan Keck and Eric Moss
Title: Congruences of modular forms in level 2 with poles at 0
Abstract: Let $M_k^\flat(2)$ be the space of weakly holomorphic modular forms that are holomorphic away from 0. We give congruences modulo powers of 2 of the Fourier coefficients for weight 0 forms in this space, answering a question posed by Andersen and Jenkins.
Kim Klinger-Logan
Title: A spectral interpretation of zeros of certain functions
Abstract: We prove that all the zeros of a certain family of meromorphic functions are on the critical line Re($s$)=1/2, by relating the zeros to the discrete spectrum of unbounded self-adjoint operators. For example, for $h(s)$ a meromorphic function with no zeros in Re($s$)$>1/2$ so that $h(s)$ is real-valued on $\mathbb{R}$ and $a_s:=h(1-s)/h(s) << |s|^{1-\epsilon}$ in Re($s$)$>1/2$, the only zeros of $1 + a_s$ or of $1 - a_s$ are on the critical line. We will use spectral theory and extend results of Lax-Phillips and ColinDeVerdiere to establish this result. This simplifies ideas of W. Muller, J. Lagarias, M. Suzuki, H. Ki, O. Velasquez Castanon, D. Hejhal, L. de Branges and P. Taylor.
Andrew Knightly
Title: Weighted distribution of low-lying zeros of GL(2) L-functions
Abstract: We consider several infinite families of cusp forms of increasing weight and/or level. In each case it was shown by Iwaniec, Luo and Sarnak that the low-lying zeros of the associated $L$-functions are orthogonally distributed. We show that if the zeros are weighted by central $L$-values, the distribution becomes symplectic (for a restricted class of test functions). This is an application of certain weighted equidistribution results for Hecke eigenvalues, arising from the relative trace formula. This is joint work with Caroline Reno.
Arvind Kumar
Title: The Adjoint map of the Serre derivative and its applications
Abstract: We compute the adjoint of the Serre derivative map with respect to the Petersson scalar product by using existing tools of nearly holomorphic modular forms. The Fourier coefficients of a cusp form of integer weight $k$, constructed using this method, involve special values of certain shifted Dirichlet series associated with a given cusp form $f$ of weight $k+2$. As application, we get an asymptotic bound for the special values of these shifted Dirichlet series and also relate these special values with the Fourier coefficients of $f$. We also give a formula for the Ramanujan tau function in terms of the special values of the shifted Dirichlet series associated to the Ramanujan delta function.
Huixi Li
Title: Mixed Level Saito-Kurokawa Liftings
Abstract: The Saito-Kurokawa lifting is a Hecke equivariant map from $S_{2k-2}(\textrm{SL}_2(\mathbb{Z}))$ to $S_{k}((\textrm{Sp}_4(\mathbb{Z})$. Classically, there are two generalizations of the Saito-Kurokawa lifting, one with respect to congruence groups, and the other with respect to paramodular groups. The construction of these lifts was given in a representation theoretic context by Schmidt in 2007. Incidentally, this representation theoretic construction easily leads to the construction of mixed level Saito-Kurokawa liftings. For arithmetic applications one would like to have a classical construction of these mixed level liftings. Part of this was done by Zantout in 2013, namely, she gave a lifting from $J_{k,m}^{\text{cusp}}(\Gamma_0(N))$ to $S_{k}(\Gamma_{N}[m])$. Recently we constructed the other part of the mixed level lifting arithmetically, i.e., the lift from the elliptic forms to the Jacobi forms, and we proved using representation theory that the mixed level Saito-Kurokawa liftings produced via representation theory and classically are the same up to a constant multiple. This is joint work with Jim Brown.
Tianyi Mao
Title: The Distribution of Integers in a Totally Real Cubic Field
Abstract: Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit group. The main term can be expressed in terms of Gr\"{o}ssencharacter L-functions.
Title: Height zeta function and multiple Dirichlet series (5 minute talk)
Abstract: I will talk about potential use of period integrals of Eisenstein series in study of the height distribution of integer tuples in certain number fields. This is an unfinished work.
Jolanta Marzec
Title: On L-functions attached to Jacobi forms of higher index
Abstract: Jacobi forms have been studied by several people and it has been known that they enjoy many similar properties to those possessed by Siegel modular forms. Therefore it is natural to ask whether the same holds for the associated L-functions. During the talk we will briefly introduce Jacobi forms and explain how one can use a doubling method to associate to them a (standard) L-function and deduce/prove its analytic and algebraic properties. This is joint work with Thanasis Bouganis.
Toshiki Matsusaka
Title: Traces of CM values of McKay-Thompson series
Abstract: In 1992, R. Borcherds proved the Monstrous Moonshine conjecture. According to Conway-Norton’s observation, the Fourier cofficients of the j-function can be expressed in terms of the degrees of the irreducible representations of the monster group. Moreover, in some cases, it is observed that the coefficients of McKay-Thompson series are expressed in terms of the degrees for other sporadic simple groups. For example, the coefficients of level 2 McKay-Thompson series T_2A are simple linear combinations of the degrees of irreducible representations of the baby monster group.
On the other hand, in 1996, M. Kaneko showed that the Fourier coefficients of the j-function can be expressed in terms of the traces of CM-values of the j-function. In this talk, we generalize the Kaneko’s result to the square-free level N McKay-Thompson series for the genus zero groups Gamma0(N) and Gamma*0(N).
Jaban Meher
Title: Zeros of L-functions attached to half-integral weight cusp forms
Abstract: In this talk we will prove that $L$-functions attached to certain cusp forms of half-integral weight has infinitely many zeros on the critical line. This is a joint work with S. Pujahari and K. Srinivas.
Djordje Milićević
Title: Counting cusp forms
Abstract: Central to the modern analytic theory of automorphic forms (such as the classical holomorphic modular forms) is the notion of a family. Several definitions of a family have been proposed, all of which involve a finite set of cusp forms on a reductive linear group (such as $\GL(2)$), described by a natural condition and expanding in size. The cardinality of the expanding set acts as an essential characteristic of a family. In this talk, which will emphasize the underlying intuition, I will present new asymptotic results on counting automorphic forms in the universal families and Hecke characters, as well as associated results on explicit uniform Weyl laws and limit multiplicity theorems. This work is joint with Farrell Brumley.
Steven Miller
Title: One-level density for holomorphic cusp forms of arbitrary level
Abstract: In 2000 Iwaniec, Luo, and Sarnak proved for certain families of $L$-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milicevic, which is of use for other problems as well.
Andreea Mocanu
Title: Linear operators for Jacobi forms of lattice index
Abstract: Jacobi forms are a mix of modular forms and abelian functions and they were introduced in the 80s by Eichler and Zagier. They arise in the most natural way as functions of lattices, such as the well known Jacobi theta functions and they enjoy much richer properties than elliptic modular forms, due to the extra abelian variable. Jacobi forms of lattice index in particular play an important role in the theory of orthogonal modular forms and mirror symmetry for K3 surfaces. In this talk, we introduce Hecke operators, level raising operators and operators arising from the action of the orthogonal group of the lattice. We discuss some of their properties and possible applications. Grant Molnar
Title: Weakly holomorphic modular forms of level 11
Abstract: In analogy with earlier work by Duke and Jenkins, El-Guindy, and others, we present generators for spaces of weakly holomorphic modular forms of level 11. We also construct canonical bases for certain subspaces of level 11 weakly holomorphic modular forms in arbitrary weight, and provide generating functions for these bases, which are obtained by means of duality relationships between their coefficients. We also prove some divisibility results on the coefficients of weight 0 forms.
Thomas Morrill
Title: Two Families of Frobenius Representations of Overpartitions
Abstract: We generalize two overpartition rank generating functions using a hypergeometric series transformation due to Andrews, and obtain $k$-fold variants of each. We give a combinatorial interpretation of the $k$-fold functions by introducing two families of \emph{buffered Frobenius representations} of overpartitions, which extend the first and second Frobenius representations of overpartitions studied by Lovejoy.
Michael Neururer
Title: Degree 2 quotients of automorphic L-functions and a converse theorem for Maass forms
I will discuss ongoing work with Tom Oliver where we prove that certain quotients of automorphic L-functions whose degrees differ by 2 have infinitely many poles. The main ingredient is a converse theorem for Maass forms that is analogous to Weil's original converse theorem but does not require that the L-functions one studies are entire.
Abhishek Parab
Title: Continuity of Arthur's Twisted Trace Formula
Abstract: We will show that distributions occurring in the geometric and spectral sides of Arthur's twisted trace formula extend to non-compactly supported test functions, unconditionally for $\GL(n)$ and modulo a hypothesis in root systems for other groups. This extends the work of Finis-Lapid (and Muller, spectral side) in the non-twisted setting.
Krzysztof Pawelec
Title: Value distribution of \frac{L'}{L}(\rho,1+it)
Abstract: For $\rho$, a cuspidal automorphic representation of $\GL_m( \mathbb{A}_{\mathbb{Q}})$, there is an associated $L$-function, $L(\rho, s)$. We study value distribution of its logarithmic derivative on 1-line, $\frac{L’}{L}(\rho, 1+it)$. We are able to prove that in some sense $\frac{L’}{L}(\rho, 1+it)$ has almost normal distribution with mean 0 and variance $\frac{(\log\log(t))^2}{(\exp((\log\log(t))^2)}$. An essential ingredient of the proof is the fact that our function of interest can be approximated by Dirichlet polynomial with coefficients supported on prime powers.
Neha Prabhu
Title: Fluctuations in the distribution of Hecke eigenvalues
Abstract: A famous conjecture of Sato and Tate (now a celebrated theorem of Taylor, et. al.) predicts that the normalised $p$-th Fourier coefficients of a non-CM Hecke eigenform follow the semicircle distribution as we vary the primes $p$. In 1997, Serre obtained a distribution law for the vertical analogue of the Sato-Tate family, where one fixes a prime $p$ and considers the family of $p$-th coefficients of Hecke eigenforms. In this talk, we address a situation in which we vary the primes as well as families of Hecke eigenforms. In 2006, Nagoshi obtained distribution measures for Fourier coefficients of Hecke eigenforms in these families. We consider another quantity, namely the number of primes $p$ for which the $p$-th Fourier coefficient of a Hecke eigenform lies in a fixed interval $I$. On averaging over families of Hecke eigenforms, we obtain a conditional central limit theorem for this quantity. This is joint work with Kaneenika Sinha.
Sudhir Pujahari
Title: Distinguishing Hecke Eigenforms
Abstract: In this talk, we will see that for two normalized Hecke eigenforms of weight $k_1$ and $k_2$ and level $N_1$ and $N_2$ such that one of them is not of $CM$ type. If the set of primes $\mathcal{P}$ such that the $p$-th coefficients of $f_1$ and $f_2$ matches has positive upper density, then $f_1$ is a Dirichlet character twist of $f_2$. This is joint work with Prof. M. Ram Murty.
Wissam Raji
Title: Non-vanishing of L-functions associated to cusp forms of half-integer weight on the plus space
Abstract: We show a non-vanishing result for L-functions associated to cuspidal Hecke eigenforms of half-integral weight in the plus space. (joint work with Winfried Kohnen)
Daniel Reiss
Title: Action of Paramodular Hecke Operators on Fourier Coefficients of Siegel Paramodular Forms
Abstract:In this talk we look at the explicit action of the Paramodular Hecke Operators $T(p)$ and $T_1(p^2 )$ ($p$ prime) on a Siegel paramodular form of squarefree level M. We also present some motivation for studying these Hecke operators by looking at certain types of Dirichlet series.
Angelo Rendina
Title: Nearly holomorphic modular forms: formulae for the Ramanujan tau function
Abstract: Nearly holomorphic modular forms were introduced by Shimura as a generalization of classical modular forms. I will give a brief definition and present the Shimura-Maass differential operator and the holomorphic projection, which allow to move between the space of nearly holomorphic forms and the space of classical ones. In particular, by comparing non-holomorphic functions with the Delta cusp form, we can obtain formulae for the tau function (which is conjectured to never vanish).
Manami Roy
Title: Level of Siegel modular forms constructed via. \Sym^3 map
Abstract: Ramakrishnan and Shahidi proved a lifting from an elliptic (non-CM) modular form f of weight 2 and level N to a degree 2 Siegel modular form F of weight 3. Our goal is to obtain precise results for the level of the Siegel modular form coming from this lifting under different congruence subgroups, like the principal congruence subgroup, the paramodular group, and the Siegel congruence subgroup.
Malcolm Rupert
Title: An explicit theta lift from Hilbert modular forms to Siegel paramodular forms
Abstract: Let E/Q be a real quadratic extension for which 2 is unramified and let \pi_0 be an irreducible, cuspidal automorphic representation of gl(2,A_E). I provide a formula, locally at every prime, for an explicit theta lift which produces a Siegel paramodular newform from the data of \pi_0. After showing the connection to curves by modularity results, I will outline the steps necessary to produce this formula and sketch a proof that a given test vector satisfies the required invariance properties to produce a nonzero Siegel paramodular newform.
Gunja Sachdeva
Title: Special values of L-functions for GL(3)×GL(1)
Abstract: In the talk I will prove an algebraicity result for all critical values of certain Rankin-Selberg $L$-functions for $\GL_3\times\GL_1$ over a totally real field $F$, which derives from the theory of $L$-functions attached to a pair of automorphic representations on $\GL_3\times\GL_2$. This is a generalization and refinement of the results of Mahnkopf. Such rationality results are the automorphic analog of the conjecture of Deligne regarding motivic $L$-functions. This is joint work with Professor A. Raghuram. To end the talk, I will give a glimpse of the same rationality result over a CM field.
Adrienne Sands
Title: Dirac operators and pseudo-Hamiltonians on spaces of automorphic forms
Abstract: We use the spectral theory of unbounded self-adjoint operators to apply quantum mechanical Hamiltonians to spaces of automorphic forms. Just as $-\Delta$ does not have compact resolvent on $L^2(\mathbb{R})$, where it is reasonable to perturb $-\Delta$ to the Hamiltonian $-\Delta+x^2$ obtaining a self-adjoint operator with discrete spectrum, we add a confining potential $q$ to discretize the spectrum of the invariant $-\Delta$ on $L^2(\SL_2(\mathbb{Z})\backslash\mathfrak{h})$. We first note that the pseudo-Hamiltonian $S=-\Delta+q$ has purely discrete spectrum $q>>y^{\epsilon}$, for every $\epsilon>0$. To factor $S$ in terms of the corresponding invariant Dirac operator $\Gamma\backslash\mathfrak{h}$, we take $q$ to be related to the zero-order part of the Laurent expansion of the Eisenstein series $E_s$ at $s=1$, as given by the Kronecker limit formula. We use techniques in semiclassical analysis to understand the spectrum of $S_h=-\Delta+hq$ as $h\rightarrow 0^+$.
Alok Shukla
Title: Co-dimensions of the spaces of cusp forms for Siegel congruence subgroups
Abstract:We give a computational algorithm for describing the one-dimensional cusps
of the Satake compactifications for the Siegel congruence subgroups in the case of degree
two for arbitrary levels. As an application of the results thus obtained, we calculate
the co-dimensions of the spaces of cusp forms in the spaces of modular forms of degree
two with respect to Siegel congruence subgroups of levels not divisible by 8. We also
construct a linearly independent set of Klingen Eisenstein series with respect to the Siegel
congruence subgroup of an arbitrary level.
Saurabh Singh
Title: On double shifted convolution sums SL_2(Z} Hecke eigenforms
Abstract:
Fredrik Stromberg
Title: Dimension formulas for vector-valued Hilbert modular forms
Abstract: I will present results from joint work with Nils Skoruppa on explicit dimension formulas for vector-valued Hilbert modular forms.
Siddhesh Wagh
Title: Maass space for liftings from SL(2,R) to GL(2,B) over a division quaternion algebra
The Saito-Kurokawa lifts provide counter examples to GRC. A paper by Muto, Narita and Pitale constructs similar lifts from SL(2,R) to GL(2,B). This talk tries to exactly characterize these lifts.
Jordan Wiebe
Title: Arithmetic in Quaternion Algebras
Abstract: In this talk we'll describe general quaternion algebras and construct orders of a given level, and outline applications to modular form computations via Pizer's algorithm using Brandt matrices.
Ka Lun Wong
Title: Farkas' identities with quartic characters
Abstract: In 2004, H. Farkas introduced an arithmetic function and found an identity involving this function. In 2009, P. Guerzhoy and W. Raji generalized Farkas' function for primes that are congruent to 3 modulo 4 by introducing a quadratic Dirichlet character, found another similar identity, and demonstrated that, in a certain sense, there are no more exact analogs of Farkas' identity for these primes. Consideration of primes congruent to 5 mod 8 now yields a theorem similar to that of Guerzhoy and Raji which involves quartic Dirichlet characters.
Shaoyun Yi
Title: Dimensions of Kl(p^2)-invariant vectors for GSp(4, F)
Abstract: In this talk, at first I will review the result of Parahoric-invariant vectors for Iwahori-spherical representations of $\GSp(4, F)$. Then I would like to talk about the Kl($p^2$)-invariant case for all the irreducible non-supercuspidal representations of $\GSp(4, F)$, which I am still working on.