The Arithmetic Seminar

TOPICS: Arithmetics in the broadest sense that includes Number Theory, Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Arithmetic Dynamics, etc.

PLACE and TIME: The seminar meets either on Tuesdays at 4:15 p.m. or on Fridays at 3:30 p.m. in LN 2205. Before the talks, there will be refreshments in the Anderson Reading Room at 4:00 p.m.(Tuesday) and 3:30 p.m. (Friday).

ORGANIZERS: Dikran Karagueuzian, Marcin Mazur, Adrian Vasiu, Joel Robert Dodge, Daniel Vallieres, Ding Ding and Jinghao Li.

Number theory at Binghamton University: Presently consists of two faculties (Marcin Mazur and Adrian Vasiu), four Ph.D. students (Ding Ding, Michael Fink, Jinghao Li, Daniel Williams), and two post-docs Joel Dodge, Daniel Vallières.

Past Ph.D. students in number theory related topics that graduated from Binghamton University: Ilir Snopce (Dec. 2009), Xiao Xiao (May 2011).

Spring 2013

• Spring 2013: 21 talks=9 seminar and colloquium talks and 1 conference consisting in 8 plenary talks and 4 short talks in 3 parallel sessions.


• February 1 : Daniel Vallières (Binghamton University)
  Title: The Brumer-Stark conjecture and complex multiplication of abelian varieties.
  Abstract: We will present one proof of a weaker statement of the Brumer-Stark conjecture when the base field is Q which do not appeal to Gauss sums, but rather to certain abelian varieties with complex multiplication. (This idea goes back to Shimura and Lang and was pushed further by Schmidt.) If we have enough time, we will explain how to perform numerical computations for other base fields to see if something similar happens.



• March 1 : Adrian Vasiu (Binghamton University)
  Title: Cohomological invariants of projective varieties in positive characteristic.
  Abstract: Let X be a projective smooth variety over an algebraically closed field k. If k has characteristic zero, then the singular (Betti) cohomology groups of X are finitely generated abelian groups and therefore all the invariants associated to them are discrete and in fact do not change under good deformations. If k has positive characteristic, then the crystalline cohomology groups of X have a much richer structure and are called F-crystals over k. In particular, one can associate to them many subtle invariants which do vary a lot under good deformations and which could be of either discrete or continuous nature. We present an accessible survey on the classification of F-crystals over k via subtle invariants with an emphasis on the recent results obtain by us, by Gabber and us, and by Lau, Nicole, and us. The talk will be quite a lot in a colloquium style.



• March 8 : Mahdi Asgari (Oklahoma State University and Cornell University)
  Title: The Rankin-Selberg Method
  Abstract: I will give an introduction to the Rankin-Selberg method of studying analytic properties of automorphic L-functions, starting with the simplest examples. While very powerful when it works, this method is often more of an art than science! There is a long list of results treating various cases where this method has been successful. I will survey some of those results, both old and new, at the end.



• April 4 (CROSS LISTING WITH THE COLLOQUIUM; SPECIAL DAY THURSDAY and TIME 4:30): Farshid Hajir (UMass Amherst)
  Title: From D=B^2-4AC to non-abelian Cohen-Lenstra heuristics
  Abstract: In his 1801 masterpiece, Disquisitiones Arithmeticae, Gauss laid the foundations not only of the arithmetic theory of quadratic forms, but also of algebraic number fields in general. Most of the conjectures he made there, about the frequency of occurrence of various groups as class groups of binary quadratic forms, are still open. But in the 1980s, Cohen and Lenstra formulated a simple heuristic which "explains" Gauss's observations in terms of the theory of abelian groups. In recent joint work with Nigel Boston and Michael Bush, we study the variation of pro-p fundamental groups, generalizing the Cohen-Lenstra heuristics to a non-abelian setting. In this expository talk for a general audience, I will sketch the outlines of this work.



• April 11 (CROSS LISTING WITH THE COLLOQUIUM --Dean's Speaker Series in Geometry/Topology--; SPECIAL DAY THURSDAY and TIME 4:30pm): Yuri Zarhin (Penn State)
  Title: One-dimensional polynomial maps, periodic orbits and multipliers
  Abstract: We study the map that sends a monic degree n complex polynomial f(x) without multiple roots to the collection of n values of its derivative at the roots of f(x). We give an answer to a question posed by Ju.S. Ilyashenko.



• April 12 (Dean's Speaker Series in Geometry/Topology): Yuri Zarhin (Penn State)
  Title: Abelian varieties with homotheties.
  Abstract: We discuss variants of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields.



• April 19 : Joel Dodge (Binghamton University)
  Title: An introduction to the Carlitz module and explicit class field theory for F_q(T).
  Abstract: The study of the Carlitz module can be used to provide an explicit class field theory for the rational function field k=F_q(T) in the sense that: (1) it provides an explicit collection of polynomials whose roots generate the maximal abelian extension of k and (2) it gives an explicit description of the action of the Galois group of k^ab/k on the roots of these polynomials. This is in perfect analogy with the construction of the maximal abelian extension of Q via cyclotomic extensions.
  I will introduce the basic notions of the Carlitz module and state some of the main theorems. This will already go a long way towards developing the analogy between extensions of k built from the Carlitz module and the cyclotomic extensions of Q. In the end, I will state the analog of the Kronecker-Weber theorem for k.



• April 26 - April 28: Third Annual Upstate New York Number Theory Conference
  Conference web page available home. Schedule of the 8 plenary talks and of the 4 sessions of 3 parallel short talks see schedule.



• May 9 : Daniel Williams (Binghamton University)
  Title: Serre-Tate Ordinary Theory and Generalizations (Part 1)
  Abstract: In the late 60's Serre and Tate developed a theory of 'ordinary' abelian varieties and p-divisible groups. We will discuss results of this theory as well as generalizations found in recent work of Vasiu. To make these talks somewhat self contained, material on Witt vectors and Dieudonn\'e theory will be included.



• May 10 : Daniel Williams (Binghamton University)
  Title: Serre-Tate Ordinary Theory and Generalizations (Part 2)
  Abstract: See above.