Speaker: Ben Kane (UW-Madison), Jan. 26

Title: On the boundary behavior of automorphic forms

Abstract:

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)

Abstract:

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

Abstract:

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

Abstract:

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

Title:

Siegel modular forms of half-integral weight and plus space

of degree two (joint work with T.Ibukiyama).

Abstract:

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

Abstract

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 innitely 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 inntely 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) innitely often and of Schlage-Puchta's theorem that

!(n) = !(n+1) innitely often. With other choices of linear forms, we can sharpen

this to d(n) = d(n + 1) = 24 and !(n) = !(n + 1) = 3 innitely often.

This is joint work with D. Goldston, J. Pintz, and C. Yldrim.

Speaker: Zhi-Wei Sun (Nanking Univ./UC Irvine)

Title: RECENT PROGRESS ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS

Abstract:In 1913 A. Fleck proved that if p is a prime, and n > 0 and r are integers

then

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.

Abstract:

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"

>