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Collaborative Number Theory Seminar at the CUNY Graduate CenterCo-organizers: Gautam Chinta, Brooke Feigon, Maria Sabitova,
Lucien Szpiro. Spring 2013 Schedule:February 22: Yves Martin (Universidad de Chile) Title: On the analogue of Cohen's kernel in the case of Jacobi forms March 1: Ellen Eischen (University of North Carolina at Chapel Hill) Title:
p-adic families of vector-weight Eisenstein series Abstract: One approach to constructing certain p-adic L-functions relies on construction of a p-adic family of Eisenstein series. I will explain how to construct such p-adic families (including the case of vector-weight automorphic forms) for certain unitary groups. This builds on my earlier work on scalar- weight families. As part of the talk, I will explain how to p-adically interpolate certain values of both holomorphic and non-holomorphic Eisenstein series. I will also mention some applications to number theory and beyond. March 8: Matilde Lalin (Université de Montréal) Title: Mahler measure and special values of L-functions Abstract: The Mahler measure of a Laurent polynomial P is defined as the integral of log|P| over the unit torus with respect to the Haar measure. For multivariate polynomials, it often yields special values of L-functions. In this talk I will discuss some of these relationships and the meaning behind them.
March 15: Thomas
Tucker (University of Rochester)
Title: Preperiodic portraits modul primes Abstract:
Let F be a rational function of degree > 1
over a number field K and let z be a point that
is not preperiodic. For any prime p of good
reduction of F, the reduction of z is preperiodic
since the residue field of p is finite. Ingram
and Silverman conjecture that for all but finitely
many positive integers (m,n), there is a prime p such
that z has exact preperiodic m and exact period n (we
call this pair (m,n) the portrait of z modulo p).
We present a counterexample to this conjecture
and show that a generalized form of abc implies that
these are the only counterexamples for generic
rational functions. This represents joint work
with several authors. March 22: No meeting this week March 29: Spring break April 5: No meeting this week April 12: *Please note the special time* 1:30--2:45 pm: Ju-Lee Kim (MIT)
Title: On the characters of unipotent representations of a semisimple p-adic group Abstract: Let G be a semisimple almost simple algebraic group defined and split over a non-archimedean local field K and let V be a unipotent repre- sentation of G(K) (for example, an Iwahori-spherical representation). We calculate the character of V at compact very regular elements of G(K). This is a joint work with George Lusztig. April 19: Jim Brown (Clemson University) Title:
The Bloch-Kato conjecture for modular forms of odd
square-free level Abstract: Let f be a
newform of weight 2k-2 and level N with N
odd and square-free. In joint work
with Mahesh Agarwal we show roughly half of
the Bloch-Kato conjecture in this setting,
namely, the size of the Shafarevich-Tate
group of the Galois representation
associated to f is bounded below by an
appropriately normalized special value of
the L-function associated to f. We
accomplish this by studying congruences
among automorphic forms on GSp(4). We
present the theorem and discuss the
necessary hypotheses. We will also
present a conjecture about twisting special
values to ensure they are p-adic units as
well as numerical evidence for this
conjecture.
April 26: *Please note the special time* 1:30--2:45 pm: Amy DeCelles (University of St. Thomas) Title: Automorphic Spectral Identities from Differential Equations Abstract:
In hindsight, subconvexity results of Diaconu,
Garrett, and Goldfeld can be viewed as an application
of a spectral identity obtained from an automorphic
partial differential equation. In this talk I
will discuss examples of spectral identities
obtained in this way and the analytic framework
necessary for legitimizing their derivations and
applications. This approach offers the
possibility of operating under somewhat different
hypotheses than one would usually take when using
trace formula or relative trace formula methods: while
trace formulas work best with very smooth data, we use
a Poincare series whose data is not smooth nor
compactly supported.
May 3: *Please note the special time and room*
Abstract: This talk is based on joint work with David Ginzburg. Motivated by known integral representations Langlands L-functions, we study a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other. Fourier coefficients attached to unipotent classes, Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process will be discussed, with examples selected from the exceptional group F4. Two new Eulerian integrals are included among these examples. May 10: Mihran Papikian (Pennsylvania State University)
Title: Quotients of Mumford curves and component groups Abstract: We consider elliptic curves parametrized by Mumford curves and answer negatively a question of Ribet and Takahashi about the surjectivity of the induced maps on component groups. In case of modular curves this leads to some interesting questions about the Hecke algebra and Galois represen-tations. (This is a joint work with Joe Rabinoff.) May 17: *Please note the special time*
Title: On Patterson's conjecture: sums of exponential sums
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