### Number Theory Events

Past Events:

August 16-21, 2003




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.