Algebra and Number Theory Seminar, Spring 2004

Thursdays, 11:15-12:05, 115 McAllister.

January 16           Alice Silverberg, Ohio State Univ

Supersinugular abelian varieties and cryptography

Note:  Special Time and Place: Friday, 2:30pm, 113 McAllister.

January 22            Ahmad Elguindy, U. Wisconsin

Weierstrass Points on X_0(pM) and Supersingular j-invariants

Abstract: Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on X_0(pM) (for M such that the genus of X_0(M) is 0 and prime p not dividing M) and proved that the reduction of Weierstrass points on X_0(pM) is supersingular mod p. In this talk we show that, for square free M on the list, all supersingular j-invariants are uniformly covered this way. Furthermore, In most cases where M is prime we describe the explicit correspondence between Weierstrass points and supersingular j-invariants. Along the way we also generalize a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo p.

January 29            You-Chiang Yi, U. of Illinois at champaign-Urbana

Modularity of a Calabi-Yau variety

I will prove the modularity of a Calabi-Yau variety by using the method of Wiles.

February 5
            George Andrews, Penn State University

Ramanujan and Partial fractions

Ramanujan's Lost Notebook is filled with mysteries. A valuable project is the attempt to understand the underlying methods Ramanujan used for the unproved formulas that populate the Lost Notebook. In the early 1980's, I wrote a lengthy paper, The Mordell integrals and Ramanujan's "lost" notebook.(in Lecture Notes in Mathematics #899, Springer-Verlag,  NY, pp. 10-48). In that paper I proved a long list of formulas from the Lost Notebook using a cumbersome method based on the maximum modulus principle. Recent investigations have led me to the conclusion that an entirely different method based on partial fractions must be closer to what Ramanujan had in mind.   I hope to provide a historical account of these developments and to supply examples to contrast the first method with the second.

February 12           James Sellers, Penn State University

New Views of Binary Partition Functions

Mike Hirschhorn and I recently proved that the number of partitions of n into parts which are powers of 2 (i.e., the number of binary partitions of n) is equal to the number of partitions of n wherein the parts satisfy a certain system of inequalities. The proof is quite straightforward and will be given in the talk. In December 2003, Neil Sloane and I teamed up to work on problems involving "non-squashing" partitions (which I will define in the talk) and have found interesting connections between these "non-squashing" partitions and the binary partitions mentioned above. I will describe these connections and prove some related results.

February 19           Yuri Zarhin, Penn State University

Very simple representations and doubly transitive permutation groups: applications to abelian varieties

We discuss a certain class of absolutely irreducible group representations. Applications to constructions of generic abelian varieties will be given.

February 26           Dale Brownawell, PSU

Siegel's Theorem via the Subspace theorem

Recently, P. Corvaja and U. Zannier have used W.M. Schmidt's Subspace Theorem to give a new proof of a celebrated theorem of Siegel's on the finiteness of the set of integer points on a curve of positive genus.  I plan to sketch their approach and, if time permits (me to learn about it), some of their even more recent work.  

March 5              
  William Stein, Harvard University 

Visibility of Shafarevich-Tate Groups at Higher Level

Note Special Time and Place: Friday, 2:30 pm, 115 McAllister.

If A is an abelian subvariety of J_0(N), then some of its Shafarevich-Tate group might be visible in J_0(N).  However, there are many examples in which some of the Shafarevich-Tate group is not visible in J_0(N).  One can hope that such Sha is visiblit in J_0(N*M), for some integer M, but very little is known about what is actually true. In this talk I'll recall the basic motivation for modular abelian variety and visibility of Shafarevich-Tate groups, then discuss some current work on visibility in modular Jacobians, with a particular emphasis on the situation when an element of Sha is invisible in J_0(N), but visible at some higher level N*M.

March 18              Robert Perlis, Louisiana State University

Iwasawa Invariants of Arithmetically Equivalent Number Fields

Let p be a prime number.  In 1956 Iwasawa gave a celebrated formula for the p-parts of the class numbers of the fields K_n forming a Z_p tower of a number field K:  There are integers lambda, mu, and nu for which the p-part of the class number of K_n is p^e with e given by

   e =lambda times n  +  mu times p^n  +  nu

for all sufficiently large indices n.  The constants lambda, mu, and nu are the Iwasawa invariants of the Z_p tower.
    One particular Z_p tower of L is the cyclotomic Z_p tower obtained from appropriate subfields of the fields of p-power roots of unity over L.  Ralph Greenberg has asked:  If K and L are arithmetically equivalent (have identical Dedekind zeta functions), do the cyclotomic Z_p extensions of K and of L have the same Iwasawa invariants?  In 1983 K. Komatsu gave an affirmative answer, but his paper lacks a precise statement of hypotheses and omits crucial steps in the proof. 
   This talk is to discuss the problem and some generalizations, and to salvage at least large pieces of Komatsu's assertions.

March 26               Sudhir Ghorpade, Purdue and IIT, Bombay

Some Inequalities for the Number of Points of Varieties over Finite Fields

Note Special Time and Place: Friday, 2:30 pm, 115 Mcallister. 

I will describe some recent work with G. Lachaud where we prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes the classical Lang-Weil inequality. I will also describe an effective version of the Lang-Weil inequality for arbitrary affine as well as projective varieties. An attempt to explain and elucidate some related conjectural statements of Lang and Weil, and recent results concerning them, will also be made.

I will try to explain some of the background and keep the prerequisites at a minimum.

April  1                   Conference in honor of G. Andrews

April 8                    Gary Mullen, PSU

Irreducible polynomials over finite fields with prescribed coefficients

For q a prime power let F_q denote the finite field of order q.  We will discuss a variety of problems dealing with the existence, and number, of irreducible and primitive polynomials over finite fields.  In particular, we will discuss work involving the distribution of irreducible and primitive polynomials over $F_q$ with prescribed coefficients.

April 15                  Judy Walker,  U. Nebraska, Lincoln

Codes, Lattices, and Their Shadows

The connection between codes and lattices is well-established, and the study of both self-dual codes and self-dual lattices is greatly enhanced by considering the shadows of these objects.  We will review these connections and then discuss recent results on additive GF(4)-codes which are self-dual with respect to the trace inner product.

April 22                  Holly Swisher,  U. Wisconsin, Madison

The Andrews-Stanley Partition Function and p(n)

Let pi be a partition of n and pi' its conjugate.  Define O(pi)  to be the number of odd parts in the partition pi.  Work of R. Stanley has led to a new partition statistic, O(pi) - O(pi').  In a recent paper, G. E. Andrews examines partitions in terms of O(pi) and O(pi'), and obtains results about a new partition function t(n), which counts partitions pi for which O(pi) is congruent to O(pi') modulo 4.  Andrews' paper brings up the question "What is the relationship between t(n) and p(n)?"  In this talk I will examine two different aspects of this question.  First I will address the growth of t(n), proving an asymptotic formula relevant to that for p(n).  Then I will discuss the issue of congruence properties for t(n).

April 29                  Geoffrey Mason,  U C Santa Cruz

Vertex operators and arithmetic at genus 1

The theory of vertex operators offers a dramatic new perspective on various aspects of number theory. We discuss the most basic example (free bosons, aka Heisenberg algebra) and its relation to classical topics such as partitions, modular forms and elliptic functions.

April 30                    Vadim Vologodsky,  U. of Chicago

Note Special Time and Place: Friday, 2:30 pm, 115 McAllister - to be confirmed.

On the canonical coordinates on the moduli space of Calabi-Yau varieties

It is predicted by the Mirror Symmetry Conjecture that the power series expansion for the canonical coordinates on moduli space of   Calabi-Yau varieties near a boundary point with maximal unipotent local monodromy should have integral coefficients.  This is a higher-dimensional generalization of the classical fact that the Fourier coefficients of the j-invariant are integers.  I will explain a proof of this Integrality Conjecture.