## Collaborative Number Theory Seminar at the CUNY Graduate Center

Co-organizers: Gautam Chinta, Brooke Feigon, Krzysztof Klosin, Maria Sabitova, Lucien Szpiro.

The seminar currently meets Fridays 3:00 to 4:30 PM in Room 3212. The CUNY Graduate Center is located on Fifth Avenue, on the east side of the street, between 34th and 35th Streets in midtown Manhattan. For further information, please contact Maria Sabitova.

## Spring 2017 Schedule:

February 24: Mihran Papikian (Pennsylvania State University)

Title: Graph laplacians and Drinfeld modular curves

Abstract: The relationship between combinatorial laplacians and automorphic forms is an active area of current research with applications to a variety of problems arising in number theory, group theory, and coding theory. I will discuss certain combinatorial laplacians arising in the theory of Drinfeld modular curves, and their applications to estimating congruences between automorphic forms.

March 3
:
Joe Kramer-Miller (University College London)

Title: Genus stability in ordinary p-adic towers of curves

Abstract: The topic of this talk is genus growth in $\mathbb Z_p$-towers of curves in characteristic $p$. For example, by work of Katz and Mazur we know that the genus of the $p^n$-th Igusa curve is given by a quadratic in $p^n$. This quadratic genus growth property is known as genus stability.  We show that any tower arising from the monodromy of a family of ordinary varieties is genus stable.  This is the first step towards the geometric Iwasawa theory program devised by Daqing Wan.

March 10
:
Hisa-aki Kawamura (Hiroshima University)

Title: The semi-ordinary p-stabilization of Siegel Eisenstein series for symplectic groups and unitary groups

Abstract: For each prime number p, we introduce a certain kind of p-stabilization of holomorphic Siegel Eisenstein series for the symplectic group GSp(2n) defined over the field of rational numbers, and for the unitary group U(n,n) defined over an imaginary quadratic field such that the resulting automorphic forms are assembled into the so-called "semi-ordinary" p-adic analytic families, respectively. If time permits, we’ll also show some applications of the above result, for instance, to construct the lifting of p-adic analytic families from GL(2) to GSp(2n) and U(n,n).

April 21
:
Kimball Martin (University of Oklahoma)

Title: Atkin-Lehner signs and congruences mod 2

Abstract: In the first part of the talk, I will explain some things about the distribution of Atkin-Lehner signs for modular forms fixed level and weight. In the second part of the talk, I will explain how to prove the existence of many congruences mod 2 within a fixed space of modular forms, and how this is related to the first part.

May 5
: Catherine Hsu (University of Oregon)

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 $\frac\left\{\varphi\left(N\right)\right\}\left\{24\right\}$. We then show that if $N$ has at least three prime factors and some prime $p\geq 5$ divides $\varphi\left(N\right)$, the Eisenstein ideal is not locally principal. We will conclude by illustrating these results with explicit computations and  give an interesting commutative algebra application related to Hilbert-Samuel multiplicities.

May 12
: David Goldberg (Purdue University)

Title: A survey of R-groups and reducibility of induced representations for reductive p-adic groups

Abstract: Parabolic induction has played a crucial role in the classification of reductive groups over local fields.  The techniques developed by Knapp-Stein, Harish-Chandra, and others, for reductive Lie groups was extended to the setting of p-adic groups by Silberger. The Langlands-Shahidi method shows that understanding this aspect of the harmonic analysis on these groups has deep arithmetic consequences, particularly in terms of understanding local L-functions. The theory of the Knapp-Stein R-group, gives a combinatorial algorithm for understanding the structure of induced representations, and these R-groups (and their construction on the dual side by Arthur et al) have played a key role in trace formula methods. We’ll give an overview of this program, including the known results for quasi-split groups. We’ll conclude by describing our joint work with Choiy on inner forms, as well as developing work with ban and Choiy on Spin groups.

May 19
: Tobias Berger (University of Sheffield)

Title:  Deformations of Saito-Kurokawa type

Abstract: I will report on work in progress with Kris Klosin on the modularity of 4-dimensional p-adic representations whose reductions modulo p are of Saito-Kurokawa type. I will explain, in particular, how this can be used in certain cases to verify Brumer and Kramer's paramodular conjecture for abelian surfaces over Q with a rational torsion point of order p.

Seminar schedule in past semesters:

Fall 2016
Spring 2016
Fall 2015

Spring 2015

Fall 2014

Spring 2014
Fall 2013

Spring 2013

Fall 2012
Spring 2012

Fall 2011

Spring 2011
2010
Fall 2009
Spring 2009
Spring 2007
Fall 2006
Spring 2006