Abstracts for Thursdays'

     Number Theory Seminars- Spring 2006

Speaker: Ben Kane (UW-Madison), Jan. 26
Title: On the boundary behavior of automorphic forms
Using the Rankin-Selberg Dirichlet series attached to a modular form f, it is
possible to show that although $lim_{y ? 0+} y^{k/2} f(x+iy)$ exists only for
a set of x in [0,1] of measure 0, the limit of the "average value" (in terms of
an integral) exists.

Speaker: Ameya Pitale (Ohio State Univ.), Feb. 2
Title:  Lifting of Cusp forms from fSL2 to GSpin(1, 4)
We construct liftings of cuspidal automorphic forms from the metaplectic group fSL2 to
GSpin(1, 4) using the Maa Converse Theorem. In order to prove the non-vanishing of
the lift we derive Waldspurger’s formula for Fourier coecients of half integer weight Maa
forms. We analyze the automorphic representation of the adelic spin group obtained from
the lift and show that it is CAP to the Saito-Kurokawa lift from fSL2 to GSp4(A).

Speaker: Jennifer Johnson-Leung (Brandeis U.), Feb. 9
Title: The equivariant Tamagawa number conjecture and Iwasawa's main conjecture
Abstract: The equivariant Tamagawa number conjecture (ETNC) on special values of
L-functions can be viewed as a sweeping generalization of the analytic
class number formula.  For this talk, I will consider the case of
abelian extensions of imaginary quadratic fields.  In this setting, I
have proved the conjecture at negative integral values of the
L-function, while Bley has proved the conjecture at zero.  Both of
these results have some restrictions on primes arising from Iwasawa's
main conjecture.  I will sketch the proof of my result and explain how
these two conjectures are intimately related.

Speaker: Jordan Ellenberg (UW-Madison)  Feb. 16
Title:  p-torsion of class groups of number fields

Abstract:  The usual way to bound the order of a class group is by
analytic means, via Dirichlet's class number formula; in particular one
can show that |Cl(K)| << D_K^{1/2} where Cl(K) is the class group and D_K
is the discriminant.  The analytic bound does not give very much insight
into the _structure_ of the class group; in particular, it is usually a
non-trivial problem to give a bound on the size of the p-torsion subgroup
Cl(K)[p].  We will discuss some cases in which the p-torsion can be
non-trivially bounded (genus theory and recent theorems of Pierce and
Helfgott-Venkatesh) some applications of such bounds (e.g.
Brumer-Silverman's work on the number of elliptic curves of a given
conductor), some conjectures, and a recent result with Venkatesh in the
case p=3.

Speaker: Harris Nover (UW-Madison) Feb 23
Title: Computation of the Galois Groups of Pro-p Extensions
  Abstract:  Let K be a number field and K' its maximal unramified pro-p extension. We are interested in computing G=Gal(K'/K), and in
particular in determining if it is finite or infinite.  We discuss a new algorithm, a generalization of previous efforts by
Boston and Leedham-Green and by Bush, that computes a list of candidates for G.  We use this algorithm to reveal the structure
of several previously unknown Galois groups.

Speaker: Chris Holden (UW-Madison) March 2
Title: Mod 4 Galois Representations and Elliptic Curves
Modular Galois Representations with cyclotomic determinant arise from the n-torsion of elliptic curves for n=2,3,5.   For n=4,
we show that not every such representation can be obtained in this manner
Speaker: Nadya Markin (UIUC)   March 9

Title: Realization of Nilpotent groups with Restricted Ramification

For a fixed number field K, let RamK : {Finite Solvable groups} ! N map a group
G to the minimal number of primes that are ramified in some extension L, such
that G = Gal(L/K). Geyer and Jarden showed that for a number field K not
containing l and an l-group G , RamK(G)  logl(|G|) + t(K), where t(K) is a
constant depending on K. We generalize their method to obtain a similar bound
for nilpotent groups. We realize a nilpotent group G by solving a series of central
embedding problems of its Sylow-l subgroups. By solving them separately but in a
compatible way, we achieve a ramification bound that is no bigger than the bound
for one of its Sylow-l subgroups. Namely, we show the following:
Theorem 1 Let K be a number field and {lj , 1  j  r} a set of primes such that
lj /2 K 8j, where N is a primitive Nth root of 1. Let G = Qr
j=1 Gj , be a nilpotent
group where each Gj is an Sylow-lj subgroup of G with |Gj | = lnj
j .
Then there exists a non-negative integer t that depends only on the ground field K,
and an extension L/K such that G = Gal(L/K) and |Ram(L/K)|  max{nj} + t,
where Ram(L/K) = number of primes of K ramified in L.
Preprint submitted to Elsevier Science 28 February 2006

Speaker: Shuichi Hayashida (U. Siegen),  March 23
Siegel modular forms of half-integral weight and plus space
of degree two (joint work with T.Ibukiyama).
Plus space is a certain subspace of Siegel modular forms
of half-integral weight. It is a generalization of the notion
of Kohnen plus space to higher degree. Because Plus space
corresponds to the space of Jacobi forms of index 1, we can
regard plus space as a space of level 1 in Siegel modular
forms of half-integral weight.
We determined the explicit structure of Plus space of degree two
as a certain module. Moreover, we give a conjecture of a lifting
from two elliptic modular forms to Plus space of degree two.
Some explicit examples of Euler-factors support this conjecture.

Speaker: Sidney Graham, (Central Michigan University), March 30.
Title: Small Gaps Between Products of Two Primes
The techniques that Goldston, Pintz, and Yldrm recently used to prove the
existence of short gaps between primes can be applied to other sequences. For
example, one can apply these techniques to the sequence of numbers that are products
of exactly two primes. Using this, we can prove that there are in nitely many
integers n such that at least two of the numbers n; n + 2; n + 6 are products of
exactly two primes. The same can be done for more general linear forms; e.g.,
there are in ntely many n such at least two of 42n + 1; 44n + 1; 45n + 1 are products
of exactly two primes. This in turn leads to simple proofs of Heath-Brown's
theorem that d(n) = d(n+1) in nitely often and of Schlage-Puchta's theorem that
!(n) = !(n+1) in nitely often. With other choices of linear forms, we can sharpen
this to d(n) = d(n + 1) = 24 and !(n) = !(n + 1) = 3 in nitely often.
This is joint work with D. Goldston, J. Pintz, and C. Yldrim.

Speaker: Zhi-Wei Sun (Nanking Univ./UC Irvine)
Abstract:In 1913 A. Fleck proved that if p is a prime, and n > 0 and r are integers
X kr (mod p)n
k(1)k  0 mod pb(n1)/(p1)c.
Only recently the significance of Fleck’s congruence was realized. It plays
fundamental role in Colmez’ and Wan’s investigation of the  -operator
related to Fontaine’s theory and p-adic Langlands correspondence. In this
talk we present a survey of the recent development of Fleck’s congruence
and its various extensions, as well as some important applications to
Stirling numbers of the second kind and homotopy exponents of special
unitary groups given by Davis and the speaker. Both number-theoretic
and combinatorial approaches will be introduced.

Speaker: Patrick Rault (UW-Madison), April 13.
Title:  The Heegner Point Algorithm
Abstract:  Let E be an alliptic curve defined over Q.  The Mordell-Weil Theorem
states that E(Q), the rational points on E, form a finitely generated abelian
group.  There is a nice algorithm which outputs a "Heegner point" in E(Q).  In
the case that E(Q) has rank 1, we can choose the point to be a nontorsion
point.  Unfortunately the method returns a torsion point when the rank is
greater than 1.  The algorithm relies heavily on the theory of Complex
Multiplication and the theory of Gross-Zagier.

Speaker: Matt Papanikolas (Texas A & M U.) April 20
Title: Galois groups of Frobenius difference equations and transcendence

Abstract: In this talk we will present recent results on algebraic independence over function fields.  By introducing a
Tannakian formalism for Drinfeld modules and relating it to the Galois theory of certain Frobenius semi-linear difference
equations, we determine the transcendence degrees of fields generated by periods of Drinfeld modules and more generally Anderson
t-modules.  More precisely, we show that the transcendence degree of the period matrix of a Drinfeld module is equal to the
dimension of its Galois group.  As one application, we prove that Carlitz logarithms of algebraic numbers that are linearly
independent over F_q(t) are algebraically independent.  We also will discuss recent work of Chang and Yu and work in progress.

Speaker:  Dongho Byeon,  Seoul National University and UW-Madison, April 27
Title: Rank-one quadratic twists of elliptic curves
Abstract:Using Heegner points on elliptic curves, we give a systematic way
to find elliptic curves E such that  for a positive proportion of fundamental
discriminants D, the analytic and algebraic ranks of the quadratic twists E(D)
of E are equal to 1.

Speaker: Jintai Ding (U. Cincinnatti) May 4
Title: Internal Perturbation of Multivariate Public Key Cryptosystems
Abstract: Public key cryptography is an indispensable part of our  modern communication
systems. However, quantum computers can break the most commonly-used public key cryptosystems
like RSA, which are based on ``hard" number theory problems.  Recently a great effort has been
put into the search for alternative public key cryptosystems.  Multivariate public key
cryptosystems (MKPC), whose public key is a set of multivariate polynomials over a finite field,
provide one such promising alternative.  The theoretical security assumption  comes from the fact
that solving a system of  polynomial equations over a finite field is  in
 general  NP-hard and  quantum computers are not yet shown to be effective in solving this problem.
Furthermore, computations in a finite field  can be more efficient.   There are a few such systems, 
for example,the Matsumoto-Imai, the Sflash, the HFE, the  HFEv, the Dragon, the Oil-Vinegar, the  TTM.

Recently  we proposed a new idea, `internal perturbation'
 to improve the security and therefore the efficiency of MPKC. The idea
comes from a similar idea in a continuous system where in order to
understand the structure of the system, one often perturbs the
 system in a controlled way to see how the system changes
 accordingly. Roughly speaking, the perturbation should be
 small-scale controlled ``noise''. More specifically, for
any MPKC, a small dimensional subspace of the message space $k^n$
is used to perform the small-scale perturbation. Here $k$ is a small finite field.
The dimension $r$ of this subspace is chosen to be very small compared with $n$ so that we
maintain control of the system. The first such system was broken by Fouque, Grouboulan and Stern at ENS using
differential analysis, but suck attack can be easily prevented by using the "Plus" method.

In this talk, we will first give an introduction of multivariate public key
cryptosystems, then we will present the internal perturbation of MKPCs, its differential analysis
attack and how to prevent it.

Speaker: Don Cartwright, University of Sydney and Rutgers University, May 5th
Title: Groups acting simply transitively on the vertices of a building.
  Let $F$ be a local field, and $n\ge2$ an integer. Associated to
> $G=PGL(n,F)$ there is a Bruhat Tits building, which is a homogeneous
> tree in the case $n=2$. This tree is (usually) the Cayley graph of a
> free group, which can be embedded in $G$.
> In this talk I discuss groups which play a role analogous to that of the
> free group in the case $n\ge3$. One way to describe this is as follows:
> $G$ acts transitively on the vertices of the building; we seek subgroups
> of $G$ which act simply transitively on these vertices.
> These groups have a presentation of a very special form. They are
> interesting for a number of reasons, including because they
> a) are very explicit co-compact lattice subgroups of $PGL(n,F)$;
> b) are new examples of automatic groups,
> c) have Kazhdan's property (T),
> d) sometimes define new, non-classical buildings;
> e) are generated by automata;
> f) have been used to construct "Ramanujan complexes"