## Collaborative Number Theory Seminar at the CUNY Graduate CenterCo-organizers: Gautam Chinta, Brooke Feigon, Maria Sabitova,
Lucien Szpiro. ## Spring 2012 Schedule:
Title: Inequalities relating invariants of elliptic curves. Abstract: We present explicit inequalities which relate heights, discriminants and conductors of elliptic curves over number fields and we discuss Diophantine applications.
Title: Poisson statistics and the value distribution of the Epstein zeta function. Abstract: We will discuss the value distribution of the Epstein zeta function E_n(L,cn) for real c and a random lattice L of covolume 1 and large dimension n. Important ingredients in our study will be the distribution of vector lengths in a random lattice as the dimension n tends to infinity and a new bound on the remainder term in the generalized circle problem. We will also present some applications of our results to for example the distribution of heights of flat tori.
Title:
A relative trace formula between the metaplectic and
general linear groups.
Title: Ramanujan Bigraphs. Abstract: Expander graphs are well-connected yet sparse graphs. The expansion property of a regular graph is governed by the second largest eigenvalue of the adjacency matrix. One can consider quotients of the Bruhat-Tits building of GL(n), n=2,3, over a p-adic field and view them as graphs. In this context the relationship between regular expander graphs and the Ramanujan Conjecture is well understood and has led to the definition and construction of asymptotically optimal regular expanders called Ramanujan graphs. The notion of Ramanujan graph can be extended to bigraphs (i.e.,biregular, bipartite graphs). In this talk I will use the representation theory of SU(3) over a p-adic field to investigate whether certain quotients of the associated Bruhat-Tits tree are Ramanujan bigraphs. I will show that a quotient of the Bruhat-Tits tree associated with a quasisplit form G of SU(3) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. (This is joint work with Dаn Ciubotaru).
2--3:30 pm: Jay Jorgenson (The City College of New York)
Title: Constructing elliptic and hyperbolic Eisenstein series using the wave distribution. This is joint work with Anna von Pippich and Lejla Smajlovic. March 16: Yu
Yasufuku (Nihon University, Tokyo, Japan) Title: Relative Sizes of Coordinates of Orbits Points under a Morphism on P^N. Abstract: Let phi be a
morphism on the projective space, and we denote by
phi^n the n-fold composition of phi. We discuss
a sufficient condition on phi so that for any rational
point P, the number of digits of coordinates of phi^n
(P) to become comparable as n goes to infinity. This
uses ingredients from Diophantine approximation,
specifically (a weak version of) Vojta's conjecture in
general, but I will discuss some cases where we can
prove the result unconditionally. I will also show
some examples which demonstrate that our condition is
not yet optimal, i.e., it is not a necessary
condition.
Title: The Quantum Variance of PSL(2,Z)\PSL(2,R) Abstract:
We discuss the quantum variance, which is introduced
by Zelditch and describes the fluctuations of a
quantum observable, on the phase space of modular
surface. We asymptotically evaluate the quantum
variance and show that it is equal to the classical
variance of the geodesic flow on the phase space after
inserting the "correction factor" of certain
L-function's central value on each irreducible
subspace. It is also very close to the arithmetic
variance studied by Luo-Rudnick-Sarnak. This talk is
based on a joint work with Peter Sarnak.
4-5 pm: John A. G. Roberts (The University of New South Wales, Sydney, Australia) Title: The dynamics
of birational maps over finite fields and their
signatures. 5:15-6:15 pm: Yiannis Sakellaridis (Rutgers University, Newark) Title: Non-standard comparison of relative trace formulas. Abstract:
The standard paradigm of endoscopy involves an
orbit-by-orbit comparison of the geometric sides of
two trace formulas. I will present a different way to
compare trace formulas, using certain integral
transforms between the pertinent spaces of orbital
integrals. More precisely, I will compare the
Kuznetsov trace formula to Jacquet's relative trace
formula for torus periods on GL(2), obtaining a new
proof of a well-known result of Waldspurger (also
proven by Jacquet). The global argument involves a
Poisson summation formula for functions defined on a
meager subset of the adeles!
Title: Surfaces dominated by products of curves and ranks of elliptic curves. Abstract:
Title: New results concerning the distribution of 2-Selmer ranks within the quadratic twist family of an elliptic curve. Abstract:
Given an elliptic curve E defined over a number field
K, we can ask what proportion of quadratic twists of E
have 2-Selmer rank r for any non-negative integer r. I
will present new results obtained by Mazur, Rubin, and
myself about this distribution, including some
surprising results relating to parity that have
implications regarding Goldfeld's conjecture over
number fields as well as some of my own results in the
special case when E(Q)[2] = Z/2 that conflict with the
conjectured distribution arising from the Delaunay
heuristics on the Tate-Shafaravich group.
Title: Modularity lifting in non-regular weight. Abstract:
Modularity lifting theorems were introduced by Taylor
and Wiles and formed a key part of the proof of
Fermat's Last Theorem. Their method has been
generalized successfully by a number authors but
always with the restriction that the Galois
representations and automorphic representations in
questions have regular weight. I will describe a
method to overcome this restriction in certain cases.
I will focus mainly on the case of weight 1 elliptic
modular forms. This is joint work with Frank Calegari. |