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 2:00 to 3:30 PM in Room 3209. 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.

Fall 2015 Schedule:

October 23: Tom Tucker (University of Rochester)

Title: The dynamical gcd problem over function fields

Abstract: A classic result of Bugeaud, Corvaja, and Zannier shows that if a and b are multiplicatively independent integers and e > 0, then
gcd(a^n - 1, b^n -1) < exp(en) for all sufficiently large n.  Ailon and Rudnick later showed that something is even stronger over function fields in characteristic 0, namely that if a, b are nonconstant multiplicatively independent polynomials in C[x], then the degree of gcd(a^n -1 , b^n -1) is bounded by a constant.  We seek to treat this problem in a more general dynamical context.   We look at "two compositional" variants of this problem, one where multiplicative powers of a and b are replaced by compositional powers of a and b, and another where we replace x --> x^n with an arbitrary monic polynomial with coefficients in C[x].  If time permits, we will discuss some applications of the latter.  This represents joint work with Liang-Chung Hsia. 

October 30: Rachel Ollivier (University of British Columbia)

Title: Torsion pairs in the representation theory of  Iwahori-Hecke algebras

Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of representations of G and the category of H-modules. When the field of coefficients has characteristic zero, this link is well understood by work of Bernstein and Borel. When the field has characteristic p, this link is much more involved and is still quite mysterious. We develop an approach to this question by defining a canonical torsion pair in the category of H-modules. This is joint work with Peter Schneider.

November 6: Joe Kramer-Miller (CUNY)

Title: p-adic L-functions and the geometry of the eigencurve

Abstract: A major theme in the theory of p-adic deformations of automorphic forms is how p-adic L-functions over eigenvarieties relate to the geometry of these eigenvarieties.  In this talk we explain results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable p-adic L-functions varying over families can detect geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space.  Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special L-values and then p-adically interpolating congruences using formal models.  These methods should extend to the entire eigencurve.

November 13: Wade Hindes (CUNY)

Title: Finiteness theorems and averages in arithmetic dynamics


November 20: Moshe Adrian (CUNY Queens college)

Title: The Langlands parameter of a simple supercuspidal representation of a classical group

Abstract: The Langlands correspondence for classical groups has recently been established by Arthur.  As Arthur's arguments are global in nature, one might want to know explicitly what the local correspondence is.  In this talk we will review some of the basic theory of the local Langlands correspondence, and then we explicitly describe the correspondence for a simple supercuspidal representation of an odd orthogonal group.

December 18: Manish Patnaik (University of Alberta)

Title: Cuspidal Eisenstein Series on Loop Groups over Function Fields

Abstract: I will explain the construction of cuspidal Eisenstein series on loop groups over function fields starting from both ‘positive’ and ‘negative’ maximal parabolics. The positive variant results in an entire function, whereas the negative one— which was first introduced by A. Braverman and D. Kazhdan— produces a (conjecturally) meromorphic object.  A functional equation can also be proven that relates the positive and negative series and in which L-functions of cusp forms (on finite dimensional groups) appear.

This is joint work with H. Garland and S.D. Miller.