Number Theory Events 2001

Number Theory Events

Past Events:

Pacific Northwest Number Theory Seminar

The Seventh Pacific North West Number Theory Conference, April 5-6, 2003

BIRS Workshop on “The Many aspects of Mahler’s measure” April 26 - May 01, 2003

PIMS Summer Program in Number Theory, SFU, June 2003

Current Trends in Arithmetic Geometry and Number Theory, August 16-21, 2003

PIMS Number Theory CRG Number Theory Day, December 5, 2003, SFU Harbour Center

CMS Winter 2003 Meeting (Number Theory Session), December 6-8, 2003, SFU Harbour Center

20-21 April 2002, The Sixth Pacific North West Number Theory Conference, SFU, Vancouver

Friday 7 December 2001, Number Theory Study Seminar, SFU, 14:30-16:00
Thursday 6 December 2001, Number Theory Seminar, UBC, 15:30-16:30
Friday 30 November 2001, Number Theory Study Seminar, SFU, 14:30-16:00
Thursday 29 November 2001, Number Theory Seminar, UBC, 15:30-16:30

Friday 7 December 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Chapter 7: Will Galway

Thursday 6 December 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Kevin O'Bryant, Department of Mathematics, University of Illinois

"The algebraic life of a combinatorial object arising in the analytic

theory of Diophantine approximation"

Friday 2 November 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Chapter 6 and 7: Will Galway

Thursday 29 November 2001

Number Theory Seminar

UBC

Math Annex 1102 (Note change in room)

15:30-16:30 Michael Bennett, Department of Mathematics, UBC

"Cubic Thue equations"

Friday 2 November 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Chapter 5: Ron Ferguson

Thursday 22 November 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Ron Ferguson

"A complete description of Golay pairs for lengths less than 100"

Saturday 17 November 2001

Pacific Northwest Number Theory Seminar

Western Washington University

Bond 106

11:30-12:30 Robert Pollack (University of Washington)

"p-adic L-functions of elliptic curves at supersingular primes"

Abstract: The p-adic L-function of an elliptic curve at an ordinary

prime has finitely many zeroes all encoded in a single polynomial.

This polynomial has great conjectural arithmetic importance via the

Main Conjecture.  The supersingular case is very different as the

L-series is known to have infinitely zeroes.  This talk will attempt

to shed some light on the arithmetic nature of these zeroes.

14:30-15:30 John Friedlander (University of Toronto)

"The subconvexity problem for Artin L-functions"

(joint work with W. Duke and H. Iwaniec)

Abstract: Given an Artin $L$-function over the rationals having

conductor $D$ it is known to be meromorphic and to satisfy a

functional equation of standard type. The Artin conjecture predicts

that those $L$ not containing the trivial representation are

entire. In such a case there follows from the functional equation and

the Phragmen--Lindelof principle the "convexity" bound

$$L(s) \ll D^{{1\over 4}+ \ve} $$

on the central line $\Re s = \half$, where the implied constant is

allowed to depend on $s$ and $\ve$.

The subconxexity problem at issue is to improve the exponent 1/4.

Artin $L$-functions of degree one over the rationals are just the

$L$-functions of Dirichlet and for these Burgess proved such a bound

with exponent $ 3/16 + \ve$ about forty years ago. His result has had

numerous applications.

After Artin, Hecke, Langlands, and Tunnell, the Artin conjecture is

now known in the case of degree two apart from the "icosahedral"

representations.  For these degree two cases where the conjecture is

known we are able, by using the $GL_2$ theory, to provide a

subconvexity bound. Along the way we incidentally prove similar

subconvexity bounds for the $L$-functions attached to the Hecke--Maass

forms which intrude in an essential way into the argument.

We also give some applications to the class groups of quadratic

fields, both real and imaginary.

Friday 2 November 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Chapter 4: Idris Mercer

Saturday 13 October 2001

Pacific Northwest Number Theory Seminar

Western Washington University

Bond 227

11:30-12:30 Glenn Stevens (Boston University)

"The eigencurve and $p$-adic L-functions"

Abstract: The ``eigencurve'' was constructed by Coleman and Mazur as a

tool for organizing and understanding $p$-adic analytic families of

modular forms. Indeed, the eigencurve is a $p$-adic rigid analytic

curve $D$ whose points parametrize all overconvergent modular

eigenforms $f$ of finite slope and tame level $1$. As such, the

eigencurve may be regarded as the universal $p$-adic analytic family

of such forms.

In this talk we will connect the theory of the eigencurve to the

theory of $p$-adic $L$-functions. We will construct a coherent sheaf

$\cal H$, locally free of rank $2$ over the eigencurve whose fiber at

any classical parabolic eigenform $f$ (of small slope) is canonically

isomorphic to the subspace of the parabolic cohomology cut out by $f$.

Moreover, if $\Phi$ is a local section of $\cal H$ over an admissible

open $X$, then we define a $p$-adic $L$-function $L_p(\Phi, x, s)$

which is a rigid analytic function on $X\times {\cal X}$ and which

interpolates the Amice-Velu $p$-adic $L$-functions at points of $X$

corresponding to classical modular eigenforms.

14:30-15:30 Will Galway (Simon Fraser University)

"The density of Pseudoprimes with Two Prime Factors"

Abstract: Let $P_2(x)$ denote the number of pseudoprimes $n\leq x$ of

the form $n=p q$, where $p$, $q$ are distinct primes. We conjecture

$P_2(x)$\sim C\sqrt{x}/\ln^2(x)$, where $C$ is an explicit, although

difficult to compute, constant. Our conjecture is closely related to a

similar conjecture of Granville and Pomerance on the density of

Carmichael numbers with $k$ prime factors.  We present pretty

pictures, a heuristic argument, and computational evidence to support

our conjecture.

Thursday 1 November 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Hugh Edgar

"1/2 PINT"

Friday 2 November 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Chapter 3: Alan Meischner

Thursday 1 November 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Imin Chen (SFU)

"Rational points on a certain modular curve of level p^2"

Friday 19 October 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Chapter 2: Keshav Mukunda

Thursday 25 October 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Izabella Laba (UBC)

"A characterization of finite sets that tile the integers"

Friday 19 October 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Chapter 1: Samantha Carruthers

Thursday 18 October 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Greg Martin (UBC)

"Egyptian fractions with lots and lots and lots of terms"

Friday 12 October 2001

Number Theory Study Seminar

SFU

K9509

14:30-16:00 Organizational meeting

Thursday 11 October 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Nike Vatsal (UBC)

"Uniform distribution of Heegner points"

Thursday 4 October 2001

Number Theory Seminar

UBC

WMAX 216

15:30-16:30 Chris Smyth (University of Edinburgh)

"Polylogs and Mahler measures"

Thursday 27 September 2001

Number Theory Seminar

UBC

WMAX 216 (note permanent change of location)

15:30-16:30 Nils Bruin (PIMS, SFU, UBC)

"Walking around a local-global obstruction for elliptic curves"

Thursday 20 September 2001

Number Theory Seminar

UBC

Math Annex 1102

15:30-16:30 Michael Bennett (UBC)

"Variants of Fermat's last theorem, d'apres Wiles"

Sunday 16 September 2001

Number Theory Hike

Grouse Mountain

Guest Services

11:00-17:00 Grouse Mountain -- Goat Mountain -- Goat Ridge -- Return

Pictures

Thursday 13 September 2001

Number Theory Seminar

UBC

Math Annex 1102

15:30-16:30 David Boyd (UBC)

"Mahler measure and unusual models for elliptic curves"

Thursday 5 July 2001

A Day of Number Theory at Simon Fraser

PIMS/MITACS at SFU

East Academic Annex Room 120

10-11 Doug Bowman (University of Illinois)

"Zeta Values: From Leibniz to Today"

11:15-12 David Bradley (University of Maine)

"Research Update on Multiple Polylogarithms"

13:30-14:30 Edlyn Teske (University of Waterloo)

"Factoring N= pq^2 with the Elliptic Curve Method"

14:45-15:45 Nils Bruin (Simon Fraser University)

"Skolem-Mahler-Lech and Chabauty-Coleman"

Saturday 28 April 2001

Fifth Pacific Northwest Number Theory Conference

Microsoft Research, Redmond

Cedar Court, Building 113, Room 1021

DigiPen Institute of Technology, Redmond

10-11 Ed Schaefer (Santa Clara University)

"How to compute the p-Selmer group of an elliptic curve for an odd prime p"

In this talk, we will give an algorithm for the problem given in the title.

p-Selmer groups are used to bound Mordell-Weil ranks when 2-descent fails and

to study the p-part of the Shafarevich-Tate group. They are also of interest in

Iwasawa theory.

11-12 Trevor Wooley (University of Michigan)

"Slim exceptional sets in Waring's problem"

Oftentimes in the additive theory of numbers, one encounters situations in

which current technology lacks the power to establish that all large integers

are represented in some prescribed manner, yet it can be shown that almost all

positive integers are thus represented. For example, while it remains only a

conjecture that all large integers are represented as the sum of four positive

integral cubes, a celebrated theorem of Davenport (1939) shows that almost all

positive integers are represented in such a manner. Equipped with the technology

available hitherto, quantitative versions of such statements improve little with

the addition of extra variables. We present a method for better exploiting such

additional variables, especially exotic variables, and thereby slim down the available

estimates for associated exceptional sets in various problems of Waring type. Both

methods and results will be accessible to non-specialists.

15-16 Audrey Terras (UC San Diego)

"Comparison of Selberg's Trace Formula with its Discrete Analogues"

We compare the original trace formula of Selberg for discrete groups acting on

the upper half plane with that for trees and finite upper half planes.

16-17 Nike Vatsal (University of British Columbia)

"Ergodic theory and Heegner points"

We discuss the applications of ergodic theory to the resolution of certain conjectures

of Mazur on the growth of the Mordell-Weil group of an elliptic curve, over an

anticyclotomic extension of number fields. The main input is a theorem of M Ratner

on the ergodicity of certain uniptent flows on p-adic Lie groups.

Saturday 24 February 2001

University of Washington, Seattle

Padelford C-36

11:30-13:00 Haruzo Hida (UCLA)

"Arithmetic of $p$-adic Hecke $L$-functions"

In this talk I will describe the relation between the power series

expansion of the $p$-adic Hecke $L$-function constructed by Katz and the

$q$-expansion of an Eisenstein series. In some favourable cases, one can

prove the vanishing of the $\mu$-invariant.

15:00-16:00 Nike Vatsal (UBC)

"Elliptic curves over anticyclotomic fields"

We will describe recent results, due to the speaker and C.Cornut, on

Mazur's conjectures about the structure of the Mordell-Weil

group of an elliptic curve over an anticyclotomic tower of number fields.

As an application, we discuss Nekovar's unconditional proof of the parity

conjecture of Birch and Swinnerton-Dyer.

Saturday 27 January 2001

Simon Fraser University, Vancouver

Harbour Center, 1325 Axa Pacific Lecture Room

11-12 Nils Bruin (PIMS, SFU, UBC)

"Generalised Fermat equations"

The famous equation x^n+y^n=z^n can de be seen as a special case of a more

general equation x^r+y^s=z^t. The latter has become known as the

"Generalised Fermat equation". For given exponents r,s,t, the structure of

the set of primitive solutions (integral solution with coprime x,y,z)

depends heavily on the sum of the inverses of the exponents, 1/r+1/s+1/t.

Especially, if r,s,t are large enough, then there are only finitely many

primitive solutions. If we assume the ABC-conjecture, then there should be

only trivial solutions for large r,s,t.

However, some surprisingly large solutions are known. For instance

43^8+96222^3=30042907^2.

In general, the structure of the solution set has close ties to the

structure of the set of rational points on an algebraic curve. In this

talk, I will explain this relation and I will sketch how one can try to

describe the solution sets for specific exponent triples.

The first part of the talk is accessible to a general mathematical

audience. At the end, we will hit on current areas of research.

13:30-14:30 Adrian Iovita (University of Washington)

"Explicit description of the local Galois representations attached to modular forms"

(joint work with R.Coleman) We give an explicit description, using p-adic

integration, of the log-crystalline cohomology of semistable curves

over p-adic fields with values in F-isocrystals. This provides, via

Fontaine's theory, an explicit description of the Galois representations

arising as etale cohomology on the respective semistable curves.

We have nice applications of this to p-adic L-functions of modular forms.

15-16 Stephen Choi (Simon Fraser University)

"A Problem of Cohn on Classifying Characters"

Let $p$ be a prime and $F$ be a finite field with $q=p^s$

elements. It is well-known that for any nontrivial

multiplicative character $f$ of $F$,

\[

\sum_{b\in F}f(b)\overline{f(b+a)}=

\begin{cases}

q-1 & \text{if $a=0$;} \\

-1 & \text{if $a\neq 0.$}

\end{cases}

\]

H. Cohn asked whether the converse is true. For the case $p$

is odd and $s=1$, A. Bir\'{o} gives a partial answer to

Cohn's problem. In this talk, we give a negative answer to

Cohn's problem when $q > 4$ and $s > 1$.

Saturday 13 January 2001

University of Washington, Seattle

Padelford C-36

11-12:30 Michael Spiess (University of Nottingham)

"Logarithmic differential forms on p-adic symmetric spaces"

P-adic symmetric spaces are certain infinite hyperplane

arrangements over p-adic fields. We describe explicitly their de Rham

cohomology in terms of logarithmic differential forms. As an

application one obtains a Hodge type decomposition of the de Rham

cohomology of algebraic varieties over p-adic fields which are

quotients of p-adic symmetric spaces.

14:30-16 Imin Chen (Simon Fraser University)

"On relations between induced representations for GL_2(Z/p^2) and

applications to modular curves"

(joint with B. deSmit and M. Grabitz) We will exhibit a

relation between induced representations on GL_2(Z/p^2) and discuss

possible applications to studying mod(p^2) representations of elliptic

curves.