# Tuesdays' Number Theory Seminars--Spring 2006

Jan. 17
Speaker: Jorge Jimenez-Urroz (University of Barcelona)
Title: Small gaps between primes and almost primes; Recent work
of Goldston, Pintz and Yildirim.

Abstract. Recently Goldston, Pintz and Yildirim have proven
one of the most important open problems on the distribution of
primes. Namely that the difference between consecutive primes
can be arbitrarily smaller than the average. The talk will serve to
describe a little bit the proof of this important theorem.

Speaker: Jeremy Rouse (UW-Madison) Jan. 31 (regular seminar)
Title: Almost-Square Primes

Abstract:
An old but difficult problem in number theory is to determine if there
are polynomials f(x) that represent infinitely many prime numbers.
Dirichlet settled the degree 1 case in 1837, but no substantial progress
has been made on the degree 2 case. In particular, it is not known if
there are infinitely many n so that n^2 + 1 is prime. We consider primes
of the form p = a^2 + b^2 where b is small (less than C*log(p)). Assuming
GRH, we can show there are at least (3/4)*sqrt(x) such between x and 2x
(provided x is big enough), and as a corollary there are two such primes
that are very close ( less than C^2*log^2 x). This talk will be quite
expository and will be accessible to students with minimal background.

Title: Some properties of partitions  by Atkin and Swinnerton-Dyer (1953)
Abstract: In this paper, the authors prove Dyson's conjecture that the partition
rank proves the Ramanujan congruences modulo 5 and 7 by decomposing the
partitions into classes of equal size.  They achieve this by proving
identities for the generating functions, largely through the application
of a striking vanishing criterion for analytic functions with a certain
transformation property.

Speaker: Masataka Chida (Tohoku Univ. and UW-Madison) Feb. 14
Title: "Indivisibility of orders of Selmer groups for modular forms".
Abstract:
For a prime p, James, Kohnen and Ono considerd the p-indivisibility of orders
of Selmer (or Tate-Shafarevich) groups of elliptic curves over Q.
For any elliptic curve E, they proved that the number of
{D: |D|<X, Sel_p(E(D))={0}} is greater than c\sqrt X/log(X) for all $X$ sufficiently large,
where Sel_p(E(D)) is the p-primary part of Selmer group of the D-quadratic twist of E.
We will discuss the generalization of this result to higher weight modular forms.

Speaker: Harris Nover (UW-Madison) Feb. 21
Title: Introduction to Class Field Theory, Infinite Galois Theory and Pro-p Groups
Abstract:   Class field theory concerns itself with, among other things, the classification of abelian extensions of global fields.  As
well as being one of the major results of early 20th century number theory, it is a key tool in nearly all branches of modern
number theory.  Meanwhile, the study of profinite extensions of number fields, such as the algebraic closure of Q, began in
earnest in the latter half of the last century, with the study of pro-p extensions and the corresponding pro-p Galois groups
playing a major role.     In this talk, we will provide a brief overview of these topics at a very introductory level.

Speaker: Sharon Garthwaite  (UW-Madison) Feb. 28
Title: Fourier Expansion of vector valued Poincare serie
Abstract: Modular forms appear quite often in our Number Theory Seminar.  (Recall recent talks by Jeremy Rouse, Ben Kane, and Karl
Mahlburg, to name a few.)
Here we look at some related types of functions that also appear on occasion, namely Maass forms and Poincare series.  In
particular, we will look at relationships that occur between these functions when we extend their definitions to vector-valued
versions.

For reference, see the papers:
"On vector-valued modular forms and their Fourier coefficients" (2003)
and "Vector-valued modular forms and Poincare series" (2004)
by Marvin Knopp and Geoffrey Mason (2003)

Speaker: Mike Woodbury (UW-Madison)  March 7
Title:
"Non-Vanishing of L-functions, Waldspurger's Theorem and Ranks of
Elliptic Curves"

Abstract:
I will first discuss some results about the group of rational points
on elliptic curves, particularly ones relating to the rank.  Then,
motivated by the Birch Swinnerton-Dyer conjecture, I will discuss the
role of L-functions and the use of modular forms and Waldspurger's
theorem in Ono and Skinner's work on the vanishing of L-functions to
get results on the ranks of elliptic curves in families of quadratic
twists.

Speaker: Frank Thorne (UW-Madison) March 29
Title: Introduction to Sieve Methods

Abstract: We will give an overview of some elementary sieve methods and some
of their applications in number theory. In particular we will discuss the small sieve method of Selberg, and give a few general
results as well as a couple of applications. We will conclude with a brief overview of how the Selberg sieve figures in
Goldston, Graham, Pintz, and Yildirim's recent results in the direction of proving the twin prime conjecture, and hopefully
provide some background for Prof. Graham's talk on 3/31.

Speaker: Zhi-Wei Sun (Nanking Univ/UC-Irvine) April 4
Title: SOME CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS
Abstract: In this talk we tell the story how the developments of some curious
identities concerning Bernoulli (and Euler) polynomials finally led to the
following unified symmetric relation (of Z. W. Sun and H. Pan): If n is a
positive integer, r + s + t = n and x + y + z = 1, then we have
r s t
x y n
+ s t r
y z n
+ t r s
z x n
= 0
where
s t
x y n
:=
n Xk=0
(1)ks
k t
n kBnk(x)Bk(y).
It is interesting to compare this with the easy identity
0 =
r s t
r s t
z x y

= r
s t
x y + s
t r
y z + t
r s
z x .
We will also talk about some explicit congruences for Bernoulli and
Euler polynomials, and a q-analogue of Stern~{!/~}s result concerning Euler
numbers.

Speaker: Adrian Diaconu (U. Minnesota), April 18
Title: INTEGRAL MOMENTS AND BOUNDS FOR AUTOMORPHIC L-FUNCTIONS

Abstract. Motivated by the Lindel\"of Hypothesis, integral moments of
automorphic L-functions were subject of intensive study over the past 90
years. In 1918, Hardy and Littlewood obtained an asymptotic formula for the
second integral moment of the Riemann zeta-function, and about 8 years later,
Ingham obtained a similar result for the fourth integral moment of the same
function. Subsequently, Good, Heath-Brown, Ivic, Iwaniec, Jutila,  Motohashi
and others made important contributions to the subject, notably, asymptotic
formulae with good error terms were obtained for the second integral moment
of all GL_2 automorphic L-functions defined over \Bbb{Q}.

The main objective in this talk is to present a new method
of obtaining asymptotic formulae for the second integral moments
of GL_2 automorphic L-functions. The approach being adelic, it has the
advantage of working over any number field F. While for F = \Bbb{Q}, one
recovers the classical results, the asymptotic
formulae I am going to present are new for any other number
field F\ne \Bbb{Q}. Finally, I will indicate how to obtain good error terms
in the asymptotics, this allowing to break convexity bounds in t aspect. This
is joint work with Paul Garrett.

Speaker: Matt Darnall (UW-Madison) April 25
Title: Roth's Theorem
Abstract:  Following Hindry and Silverman, I will outline the proof of Roth's
theorem about the approximation of algebraic numbers by rational numbers.

Speaker: Edray Goins (Purdue U.)
Title:  Extending the Serre-Faltings Method for $\mathbb Q$-Curves
Abstract:  In this talk, we consider a method for calculating modular forms associated to elliptic curves with a rational point
of order $\ell = 2, \, 3$.  We discuss a variant of the Serre-Faltings method which considers the symmetric square
representations.  As an application, we show that certain $\mathbb Q$-curves with reducible mod 3 representations are modular.