Department Home

# Tentative Schedule for 22nd Annual Workshop on Automorphic Forms and Related Topics

## Saturday March 8

8:30-9:00 Snacks and Registration

9:00-9:30 Matt Young (Texas A&M University)
"Mean values with cubic characters"

9:50-10:20 Eduardo Duenez (UT-San Antonio)
"Low critical zeros of modular (and automorphic) L-functions"
Abstract: We discuss two two instances of the interplay between the statistics of low-lying zeros of L-functions and properties of the associated modular (or automorphic) forms. The first (with SJ Miller) explains the efect on the zero statistics of Rankin-Selberg convolution of the L-functions. The second (in progress with DK Huynh, JP Keating, SJ Miller, NC Snaith) hints at a connection between central zero repulsion and excess rank in families of elliptic curves. In both cases random matrix theory ideas play an important motivational role

10:40-11:10 Daniel Garbin (CUNY Graduate Center)
"Spectral convergence of elliptically degenerating Riemann surfaces"

11:30-12:00 Andrew Knightly (University of Maine)
"Averages of modular L-values via trace formula"

Lunch 12:00-2:00

2:00-2:40 Cris Poor and Dave Yeun (Fordham University and Lakeforest College)
"Paramodular Cusp Forms and Abelian Surfaces"

3:00-3:20 Olav Richter (University of North Texas)
"Congruences of Jacobi forms"

3:40-4:10 Kimberly Hopkins (UT-Austin)
"Higher weight Heegner points"

4:30-5:30 Panel Discussion: Where is research on Automorphic Forms Heading?

## Sunday March 9

8:30-9:00 Snacks

9:00-9:50 Kathrin Bringmann (University of Minnesota)
"Weak Maass forms, mock theta functions, and applications"
Abstract: The modularity of the partition generating function has many important consequences, for example asymptotics and congruences for p (n). Recently it became clear that the rank, a partition statistic introduced by Dyson, is related to weak Maass forms, a class of functions which are related to modular forms. I want to speak about some of the applications and also discuss higher weight analogues.

10:10-10:30 Jenny Fuselier (US Military Academy)
"Traces of Hecke operators in level 1 and hypergeometric functions over F_p"

"Zeros of certain Drinfeld modular functions"

11:20-12:00 Panel Discussion: How to start and maintain mathematical collaboration.

Lunch Break 12:00-2:00

2:00-2:40 Richard Hill (University College London/University of Bristol)

2:50-3:10 Xian-Jin Li (Brigham Young University)
"A transformation of Hankel type on the field of $p$-adic numbers"

3:30-4:00 Wissam Raji (Temple University)
"Eichler Cohomology of Generalized Modular Forms"

4:15-4:55 Joel Mohler (Lehigh University)
"Multiple Dirchlet series over the rational function field"

5:10-5:30 Nathan Jones (University of Montreal)
"A refined version of the Lang-Trotter conjecture"

6:00- 9:00 Conference Pizza Party

## Monday March 10

8:30-9:00 Snacks

9:00-9:40 Tomoyoshi Ibukiyama (Osaka University)
"Dimensions of Siegel modular forms of small weight"

10:00-10:20 Nathan Ryan (Bucknell University)
"Expirements with genus 2 siegel modular forms"

10:40-11:10 Paul Jenkins (UCLA)
"Integral traces of singular values of Maass forms"

"Differential Operators, Weak Maass Forms, and Rationality"

Monday Afternoon is Free

## Tuesday March 11

8:30-9:00 Snacks

9:00-9:40 Mahesh Agrawal (McMaster University)
Abstract: Let p be an odd prime. In this talk we will construct a p-adic analog of a degree eight L-function L(s, F×f) where F is an ordinary holomorphic degree 2 Siegel eigen cusp form of level a power of p and f is an ordinary eigen cusp form of level a power of p. Our method makes use of the work of M. Furusawa which gives an integral representation for this L-function. By suitably interpreting this integral representation in the context of inner products of automorphic forms, we show that it p-adically interpolates the L-values as the forms F and f vary in ordinary families (with the weights varying p-adically). This interpolation is carried out by constructing an Eisenstein measure on a higher-rank unitary group and exploiting a pull-back formula of P. Garrett and G. Shimura.

10:00-10:30 Tomonori Moriyama (Osaka University)
"Rankin-Selberg convolutions for $GSp(2)\times GL(2)$ : archimedean theory and applications"

10:50-11:10 Keith Oullete (UCLA)
"On the Fourier inversion formula for reductive groups"
We demonstrate a new proof of the Fourier inversion formula for the continuous spectrum in L2(SL2(Z)\SL2(R)/SO2(R)). Although this proof assumes the meromorphic continuation of the Maass- Eisenstein series, it avoids a contour shift that increases in complexity when generalizing to reductive groups of higher rank.

"Bounded Gaps between Products of Primes with Applications to Elliptic Curves and Ideal Class Groups"
Abstract: In recent work, Goldston, Pintz, and Yildirim proved that $\liminf (p_{n + 1} - p_n) / \log p_n = 0$. (Here $p_n$ denotes the nth prime.) In follow-up work with S. Graham, they also proved the existence of bounded gaps between products of two primes. Generalizing their work, we will present a similar bound for products of $r$ primes, for any $r \geq 2$, which covers the case when these primes are restricted to any Chebotarev set. As a consequence, we will obtain "bounded gaps" results concerning class numbers, Fourier coefficients of modular forms, and ranks of elliptic curves.

Lunch Break 12:00-2:00

2:00-2:30 Lloyd Kilford (University of Bristol)
"Approximating the action of the U_p operator on overconvergent modular forms"

2:50-3:20 Ahmad El-Guindy (Texas A&M University)
"Rankin-Cohen brackets for classical and Drinfeld modular forms"

3:40-4:00 Ken McMurdy (Ramapo College)
"Explicit Verification of a Theorem of Shimura"

4:20-4:50 John Leo (UCLA)
"Fourier Coefficients of Triangle Functions"