Collaborative Number Theory Seminar at the CUNY Graduate Center

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

The seminar currently meets Fridays 4:00 to 5:30 PM in Room 4422. 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 2012 Schedule:

February 3: Rafael von Kanel (Institute for Advanced Study)

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.

February 10: No meeting this week.

February 17: Anders Södergren (Institute for Advanced Study)

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.

February 24: Cesar Valverde (Rutgers University, Newark)

Title: A relative trace formula between the metaplectic and general linear groups.

Abstract: I will sketch a geometric identity between relative trace formulae for the general linear group and the metaplectic group. From the spectral side of the identity, one expects to obtain information about the product of distinct Whittaker coefficients on the metaplectic group.

March 2: Cristina Ballantine (College of the Holy Cross)

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).

March 9: *Please note the special time*

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.

March 23: Peng Zhao (Princeton University)

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.

March 30: *Please note the double header and special times*

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!

April 6: Spring break

April 13: Spring break

April 20: Lisa Berger (Stony Brook University)

Title: Surfaces dominated by products of curves and ranks of elliptic curves.

Abstract: We will study a few constructions of surfaces, each admitting a dominant rational map from a product of curves. Such constructions lead to applications to the study of ranks of elliptic curves over function fields. We discuss some of these applications, including a recent elementary, but interesting, observation.

April 27: Zev Klagsbrun (University of Wisconsin, Madison)

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.

May 4: David Geraghty (Institute for Advanced Study)

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.

May 11: No meeting this week