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
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
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.