Peter Hilton, Binghamton
University
Breaking highgrade German ciphers in WWII
The battle is not always won on the battlefield.
One of the greatest secrets of World War II was an obscure installation
at which German codes were broken. Churchill considered it one of
his most potent weapons, and without it, Britain might have fallen.
Peter Hilton was there. Peter was a member of the team of mathematicians
working on the German Naval Enigma machine and, subsequently, Hitler's
favored ciphering machine, the Geheimschreiber, at Bletchley Park
in World War II. He will describe the nature of the machines and
of the methods used to decipher messages encoded by these machines;
he will also give a picture of the daily life of the cryptanalysts.
Speaker abstracts - Plenary Lectures
Rational points on elliptic curves and
algebraic cycles
Henri Darmon
McGill University Mathematics Department The theory of CM (Complex
Multiplication) points on modular and Shimura curves provides one
of the most fruitful approaches to the Birch and Swinnerton-Dyer
conjecture, leading to the best known results for elliptic curves
of analytic rank one. In this survey I will discuss the possibility
of more general constructions in which, loosely speaking, CM points
on Shimura curves are replaced by higher dimensional (topological,
or algebraic) cycles on Shimura varieties.
--------------------------------------------------------------------------------
Pretentiousness in analytic number
theory
Andrew Granville (Montreal)
Inspired by the "rough classification" ideas from additive
combinatorics, Soundararajan and I have recently introduced the
notion of pretentiousness into analytic number theory. Besides giving
a more accessible description of the ideas behind the proofs of
several well-known difficult results of analytic number theory,
it has allowed us to strengthen several results, like the Polya-Vinogradov
inequality, the prime number theorem, etc. In this talk we will
introduce these ideas and gave some flavour of these developments.
--------------------------------------------------------------------------------
Zeros of p-adic Forms
Roger Heath-Brown, Oxford University
Artin conjectured that a system of p-adic forms, with degrees d_1,
..., d_r, should have a non-trivial common zero as soon as there
are more than d_1^2+...+d_r^2 variables.
This is known to be false.
The talk will review what is known, and look at some recent work
on quartic forms, and on systems of quadratic forms.
--------------------------------------------------------------------------------
L-series and Transcendence
Ram Murty, Queen's University
We will discuss transcendence of special values of L-series. More
precisely, we will survey several results that connect non-vanishing
of certain L-series to transcendence theorems. This will be a report
of joint work with various authors including, S. Gun, V. Kumar Murty,
P. Rath, and N. Saradha.
--------------------------------------------------------------------------------
Undecidability in number theory
Bjorn Poonen, MIT
Hilbert's Tenth Problem asked for an algorithm that, given a multivariable
polynomial equation with integer coefficients, would decide whether
there exists a solution in integers. Around 1970, Matiyasevich,
building on earlier work of Davis, Putnam, and Robinson, showed
that no such algorithm exists. But the answer to the analogous question
with Z replaced by Q is still unknown, and there is not even agreement
among experts as to what the answer should be. I will discuss this
and Hilbert's Tenth Problem over other rings of arithmetic interest.
--------------------------------------------------------------------------------
Counting fields
Carl Pomerance, Dartmouth College
Coauthors: Karim Belabas (University of Bordeaux 1), Manjul Bhargava
(Princeton University)
We know that the set of algebraic number fields contained in the
complex numbers is a countable set. One natural proof involves counting
polynomials, but this is not the only way to count fields. Since
there can be at most finitely many fields with a given discriminant
(to the integers), another natural way to count fields is through
their discriminants. And it makes sense to organize this by degree,
so one might count all degree-n fields with the absolute value of
their discriminants below a given bound. This count might be further
refined by the Galois group of the normal closure of the field,
by the signature of the field, by the way a particular prime splits,
and so on. A now classical result of Davenport and Heilbronn gets
asymptotics for S_3-cubic fields, and a recent result of Bhargava
does the same for S_4-quartic fields. This talk will discuss recent
results towards getting power-saving error estimates for these asymptotic
formulas.
--------------------------------------------------------------------------------
TBA
Alice Silverberg (University of California, Irvine)
--------------------------------------------------------------------------------
Meeting and beating the limits of the
circle method for systems of cubic diophantine equations
Trevor D. Wooley, University of Bristol
Coauthors: Joerg Bruedern
We consider the solubility of simultaneous diagonal cubic diophantine
equations. In many instances, it is now possible to establish the
Hasse Principle for such systems when the number of variables is
equal to the limit imposed by the 'square-root' barrier from the
circle method. The work from the past decade that has delivered
these sharp new conclusions combines an entertaining mix of ideas,
some elementary, some motivated by harmonic analysis, and some utilising
'complification'. We will sketch as many of these ideas as time
permits, including one or two that beat the 'square-root barrier'.
Abstracts - Special Session Talks
A set of binary sequences, unimodal
functions, beta-expansions and Diophantine approximation
Jean-Paul Allouche, CNRS, LRI, Orsay
We describe a set of binary sequences studied by M. Cosnard and
the author in 1982-1983 in the framework of iterations of continuous
unimodal functions of the unit interval. This set happens to occur
both in beta-expansions for "univoque´´ real numbers
(i.e., real numbers in (1, 2) for which 1 has a unique beta-expansion)
and in Diophantine approximation (the so-called Cassaigne spectrum
also studied by Bugeaud and Laurent).
--------------------------------------------------------------------------------
Some remarks on relative Lehmer.
Francesco Amoroso, University of Caen
Coauthors: Umberto Zannier (Scuola Normale Superiore - Pisa - Italy)
Let K be a number field and let L be an abelian extension of K.
Let x be a non-zero algebraic number in L which is not a root of
unity. As a a very special case of the main result of a previous
joint paper with U. Zannier, the Weil height of x is bounded from
below by a positive function of K. The proof gives a function which
naturally depends on the degree *and* on the discriminant of K.
Here we show that the height of x can be bounded from below by
a positive function depending *only* of the degree of K over the
rational field.
As a corollary, in a dihedral extension of the rational field,
the height of a non zero algebraic number which is not a root of
unity is bounded from below by an absolute positive constant.
--------------------------------------------------------------------------------
Vector heights on K3 surfaces
A. Baragar (University of Nevada Las Vegas)
Coauthors: Ronald van Luijk
In this talk, we discuss the notion of a vector height for K3 surfaces
- a height from a K3 surface to its Picard group tensored with R.
We discuss the relationship between vector heights and Weil heights;
the advantage of using vector heights; and the existence and non-existence
of canonical vector heights. We use vector heights to calculate
the asymptotic behavior of the number of points with bounded height
and in an orbit under the group of automorphisms on certain K3 surfaces.
--------------------------------------------------------------------------------
Sums of Hecke eigenvalues over quadratic
polynomials
Valentin Blomer, University of Toronto
Let a(n) be the normalized Fourier coefficients of a modular form
f ? Sk(N, c), and let q(x) = x2+sx+t be an integral quadratic polynomial.
It is shown that
å
n = X
a(q(n)) = cX + O(X6/7+e)
for some constant c = c(f, q). The constant vanishes in many, but
not all, cases.
--------------------------------------------------------------------------------
Computing automorphic forms
Andrew Booker, University of Bristol
Coauthors: Ce Bian, Andreas Strombergsson
I will describe some of the theory and practice of computing non-holomorphic
modular forms, for both GL(2) and GL(3). This will cover recent
joint work with Andreas Strombergsson and my student, Ce Bian.
--------------------------------------------------------------------------------
Rational points on cubic hypersurfaces
Tim Browning, University of Bristol
Given a cubic hypersurface defined over the rationals, the Hardy-Littlewood
method allows one to show that the rational points on it are Zariski
dense if the dimension is sufficiently large. Thanks to the work
of Davenport, and more recently of Heath-Brown, we can now handle
arbitrary hypersurfaces of dimension at least 12. In this talk I
show that one extend this to dimension 11, provided that the underlying
cubic form can be written as the sum of two forms that have no variables
in common.
--------------------------------------------------------------------------------
Approximations to Weyl sums
Jörg Brüdern, Universität Stuttgart
Coauthors: Dirk Daemen
In this talk, we discuss the error between a Weyl sum ?n = N e(an)k
and its standard approximation q-1 S(q, a) v(a-a/q) where S(q, a)
= ?x=1q e(axk/q) and v(b)=?0N e(btk)dt. This error is known to be
O(q1/2+e(1+Nk|a-a/q|)1/2, but it has been suggested that the error
is actually much smaller, and that perhaps the exponents 1/2 can
be reduced to 1/k. We shall show that this is not the case: the
known estimate is essentially optimal, pointwise and in various
quadratic means.
--------------------------------------------------------------------------------
Diophantine approximation and Cantor sets
Yann Bugeaud, Université Louis Pasteur, Strasbourg
We provide an explicit construction of elements of the middle third
Cantor set with any prescribed irrationality exponent. This answers
a question posed by Kurt Mahler. We also survey related results
and establish that there exist automatic real numbers with any prescribed
rational irrationality exponent
--------------------------------------------------------------------------------
On the equation x2 - 2 z6 = yp.
Imin Chen, Simon Fraser University
The modular method has been successfully applied to tackle a number
of classes of ternary diophantine equations of the form A xa + B
yb = C zc. Of interest sometimes are equations obtained by setting
one of the variables x, y, z to 1. The equation x2 - 2 = yp is an
example, but it has resisted attempts so far because the solution
(x, y) = (±1, -1) is present for every p and the associated
elliptic curves over Q from the modular method do not have complex
multiplication. By regarding this equation as a special case of
x2 - 2 z6 = yp, we show that it is possible to associate to a solution
a Q-curve completely defined over Q(v2, v3). The solution (±1,
-1) now luckily corresponds to an elliptic curve with complex multiplication
by the order of discriminant -24 and the modular method using Q-curves
then can be applied to obtain some partial results
--------------------------------------------------------------------------------
TBA
B. Conrey (American Institute of Mathematics)
--------------------------------------------------------------------------------
Frobenius Fields and Frobenius Rings
of Elliptic Curves
Chantal David, Concordia University
Coauthors: Jorge Jimenez Urroz (UPC, Barcelona) and Nathan Jones
(Montreal)
Let E be an elliptic curve over Q. For each prime p of good reduction,
E reduces to a curve over F_p, the Frobenius endormorphism of E/F_p
satisfies x^2 - a_p(E) x + p, and the Frobenius ring Z[sqrt(a_p^2-4p)]
is a subring of the endomorphism ring End(E/F_p). The distribution
of the endomorphism rings End(E/F_p) and the endormorphism fields
Q(sqrt(a_p^2-4p)) when p varies is a difficult problem. For example,
there are no known examples of an elliptic curve over Q where a
given quadratic imaginary field K happen infinitely often as the
field of endomorphisms Q(sqrt(a_p^2-4p)). We will give in this talk
some evidence for the conjectural distributions for the fields and
rings of endomorphisms, by showing that we can prove those conjectural
distributions averaging over all elliptic curves in a box.
--------------------------------------------------------------------------------
Remarks on the discretisation process
Christophe Delaunay
Université Claude Bernard Lyon 1
The discretisation process is an heuristic argument due to J. B.
Conrey, J. P. Keating, M. O. Rubinstein and N. C. Snaith that allows
to conjecture some asymptotics for the number of elliptic curves
with extra rank in a family of quadratic twists. The aim of this
talk is to discuss this process. Part is joint work with M. Watkins
--------------------------------------------------------------------------------
Approximation of complex algebraic
numbers by algebraic numbers of bounded degree
Jan-Hendrik Evertse
Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300
RA Leiden, The Netherlands
Coauthors: Yann Bugeaud (Strassbourg)
Define the height H(P) of P ? Z[X] to be the maximum of the absolute
values of the coefficients of P. Further, for an algebraic number
x, define its height H(x) to be the height of the minimal polynomial
of x. For x ? C and for a positive integer n, denote by wn(x) the
supremum of all reals w with the property that there are infinitely
many polynomials P ? Z[X] of degree at most n such that 0 < |P(x)|
= H(P)-w. Further, define wn*(x) to be the supremum of all reals
w* such that there are infinitely algebraic numbers a ? C such that
|x-a| = H(a)-w*-1. The functions wn and wn* were introduced by Mahler
and Koksma, respectively, and they play an important role in the
classification of transcendental numbers.
It is known that wn(x)=wn*(x)=min(n-1, d) for every real algebraic
number x of degree d. This result is a consequence of W.M. Schmidt's
celebrated Subspace Theorem from Diophantine approximation. As it
turns out, the problem to compute wn(x), wn*(x) for complex, non-real
algebraic numbers x is more complicated. In my talk, I discuss some
joint work with Yann Bugeaud in this direction, in which we determined
wn(x), wn*(x) for all complex, non-real algebraic numbers x of degree
= n+2 or = 2n-1.
-----------------------------------------------------------------------------
TBA
J. Ellenberg (University of Wisconsin Madison)
-----------------------------------------------------------------------------
Some calculations involving higher-rank
L-functions
David W. Farmer
American Institute of Mathematics
I will describe some explorations of L-functions of degree 3 and
higher.
--------------------------------------------------------------------------------
TBA
J. Friedlander (University of Toronto)
--------------------------------------------------------------------------------
TBA
Wee Teck Gan (University of California San Diego)
--------------------------------------------------------------------------------
TBA
S. Gonek (Rochester)
--------------------------------------------------------------------------------
The Canonical Subgroup
Eyal Goren
McGill University
Coauthors: Payman Kassaei (King's College)
The canonical subgroup plays a crucial role in defining and studying
the U operator on overconvergent p-adic modular forms. It has been
studied by Lubin and Katz and recently by Abbes-Mokrane, Andreatta-Gasbarri,
Conrad, and Kisin-Lai. After a short introduction to p-adic modular
forms and some motivation, I shall discuss joint work with P. Kassaei
(King's College) on the canonical subgroup for a general class of
Shimura varieties. Our approach is different from the methods used
by all the authors above in that it relies, in essence, only on
the underlying geometry and not its interpretation in terms of moduli
of abelian varieties; the test case we shall focus on is that of
Hilbert modular varieties.
--------------------------------------------------------------------------------
On the exceptional sets for the Waring-Goldbach
problem for cubes.
Koichi Kawada, Iwate University
It is known for s=5, 6, 7 and 8 that almost all natural numbers
satisfying certain necessary congruence conditions can be written
as the sum of s cubes of primes, and the currently best bounds for
the density of the exceptions are due to Kumchev. It shall be reported
in this talk that slightly sharper bounds are obtained by altering
the sieve procedure of Kumchev.
--------------------------------------------------------------------------------
Second moments of twisted Koecher-Maass
series
Henry H. Kim, University of Toronto
We will talk about the average version of the second moments in
t-aspect of twisted Koecher-Maass series for Siegel cusps of degree
2. If applied to Saito-Kurokawa lift, due to the result of Duke-Imamoglu,
it gives the average version of the second moments in t-aspect of
Rankin-Selberg L-function of the half-integral weight modular forms
and Maass forms of weight 1/2.
--------------------------------------------------------------------------------
On the pullback of arithmetic theta functions
Stephen Kudla, University of Toronto
Coauthors: Tonghai Yang
In a series of papers with M. Rapoport and T. Yang, we introduced
modular forms -which we call arithmetic theta functions -whose Fourier
coefficients are constructed from classes in the arithmetic Chow
groups of certain moduli spaces. In the simplest case, the arithmetic
theta function f_C has weight 1 and is a generating function for
the arithmetic degrees of a family of zero cycles on the moduli
space C of elliptic curves with CM by the maximal order in a fixed
imaginary quadratic field. In another case, the arithmetic theta
function f_S has weight 3/2 and is the generating function for the
arithmetic degrees of certain divisors on an integral model S of
a Shimura curve.
In this talk I will describe natural morphisms j: C --> S and
give a formula expressing the pullback j^* f_S of the arithmetic
theta function for cycles on S in terms of f_C's and theta functions
of weight 1/2. This formula is analogous to factorization formulas
for classical theta functions.
--------------------------------------------------------------------------------
Noncongruence modular forms and modularity
Wen-Ching Winnie Li
Pennsylvania State University
Unlike classical modular forms, the arithmetic of noncongruence
modular forms is not well-understood. In their pioneering work,
Atkin and Swinnerton-Dyer suggested very interesting congruence
relations on the Fourier coefficients of noncongruence forms. Scholl
associated l-adic Galois representations to such forms. In this
survey talk we shall review the recent progress on the arithmetic
properties of noncongruence forms, including congruence relations
between the Fourier coefficients of noncongruence and congruence
forms, the modularity of the Scholl representations, and the unbounded
denominator criterion for noncongruence forms.
--------------------------------------------------------------------------------
Density of rational points on diagonal
quartic surfaces
Ronald van Luijk, Warwick University
Coauthors: David McKinnon, Adam Logan
It is a wide open question whether the set of rational points on
a smooth quartic surface in projective three-space can be nonempty,
yet finite. In this talk I will treat the case of diagonal quartics
V, which are given by ax^4+by^4+cz^4+dw^4=0 for some nonzero rational
a, b, c, d. I will assume that the product abcd is a square and
that V contains at least one rational point P. I will prove that
if none of the coordinates of P is zero, and P is not contained
in one of the 48 lines on V, then the set of rational points on
V is dense. This is based on joint work with Adam Logan and David
McKinnon.
--------------------------------------------------------------------------------
Dimensions of Spaces of Newforms
Greg Martin, University of British Columbia
A formula for the dimension of the space of cuspidal modular forms
on G_0(N) of weight k (k=2 even) has been known for several decades.
More recent but still well-known is the Atkin-Lehner decomposition
of this space of cusp forms into subspaces corresponding to newforms
on G_0(d) of weight k, as d runs over the divisors of N. A recursive
algorithm for computing the dimensions of these spaces of newforms
follows from the combination of these two results, but it is desirable
to have a formula in closed form for these dimensions. In this talk
we describe such a closed-form formula, not only for these dimensions,
but also for the corresponding dimensions of spaces of newforms
on G_1(N) of weight k (k=2). This formula is much more amenable
to analysis and to computation. For example, we derive asymptotically
sharp upper and lower bounds for these dimensions, and we compute
their average orders; even for the dimensions of spaces of cusp
forms, these results are new. We also establish sharp inequalities
for the special case of weight-2 newforms on G_0(N), and we report
on extensive computations of these dimensions. For example, we can
find the complete list of all N such that the dimension of the space
of weight-2 newforms on G_0(N) is less than or equal to 100; previous
such results had only gone up to 3.
------------------------------------------------------------------------------
Barker sequences and flat polynomials
M. Mossinghoff (Davidson College)
Coauthors: Peter Borwein (Simon Fraser University), Erich Kaltofen
(North Carolina State University)
A Barker sequence is a finite sequence of integers a0, ..., an-1,
each ±1, for which |?j aj aj+k| = 1 for k ? 0. It has long
been conjectured that no Barker sequences exist with length n >
13. We describe some recent work connecting this problem to several
open questions posed by Littlewood, Mahler, Erdös, Newman,
Golay, and others about the existence of polynomials with ±1
coefficients that are "flat" in some sense over the unit
circle. We will also briefly describe some recent related work concerning
Lp norms and Mahler's measure of polynomials.
------------------------------------------------------------------------------
TBA
K. Murty (University of Toronto)
-------------------------------------------------------------------------------
Algebraic independence of periods
and logarithms of Drinfeld modules
Matthew Papanikolas, Texas A&M University
Coauthors: Chieh-Yu Chang
This talk will focus on transcendence theory over function fields
in positive characteristic. In particular, for Drinfeld modules
we relate periods and logarithms of algebraic points to special
values of solutions of certain Frobenius difference equations. By
way of a result that equates the dimension of the associated difference
Galois group to the transcendence degrees of the values, for certain
Drinfeld modules we determine all algebraic relations among their
periods and logarithms.
--------------------------------------------------------------------------------
Class numbers not divisible by 3
Michael Rosen, Brown University
Coauthors: Allison Pacelli, Williams College
Let k=F(T), where k is a field with q elements and 3 does not divide
q+1. In a paper published in 1988, H. Ichimura gave an explicit
construction of infinitley many quadratic extensions K/k such that
3 does not divide the class number of K. In this paper we give a
partial generalization of this result. Let m be a positive integer
not divisible by 3. Then, for a large class of finite fields, F,
we give an explicit construction of infinitely many extensions K/k,
of degree m, such that 3 does not divide the class number of K.
The expression "a large class of finite fields" will be
made precise.
--------------------------------------------------------------------------------
A small value estimate in dimension two
by
Damien Roy
Department of Mathematics, University of Ottawa
A small value estimate is a statement providing necessary conditions
for the existence of a sequence of non-zero polynomials with integers
coefficients taking small values at many points of an algebraic
group. Such statements are desirable for applications to transcendental
number theory, but only few instances of them are known at the moment.
The purpose of this talk is to present a small value estimate for
the product Ga x Gm of the additive group Ga by the multiplicative
group Gm. We will show that if a sequence of polynomials with integer
coefficients take small values at a point (xi, eta) together with
its first derivatives with respect to the invariant derivation d/dx+y(d/dy),
then both xi and eta are algebraic over Q. The precise statement
compares favorably with constructions coming from Dirichlet's box
principle.
---------------------------------------------------------------
TBA
R. Sharifi (McMaster University)
--------------------------------------------------------------------------------
Periodic points modulo p as p varies
Joseph H. Silverman, Brown University
Let F : V -> V be a morphism of a quasiprojective variety defined
over a number field K and let R be a point in V(K) having infinite
orbit under iteration of F. For each prime p of good reduction,
let M(p) denote the length of the orbit of R modulo p, i.e., the
number of distinct iterates F^i(R) mod p. Let e > 0. We sketch
a proof that for a set of primes of analytic density 1, the orbit
length M(p) is greater than (log Norm p)^(1-e). We also conjecture
a stronger estimate.
-------------------------------------------------
TBA
K. Soundararajan (Stanford University)
--------------------------------------------------------------------------------
The integers n dividing a^n-b^n
Chris Smyth, University of Edinburgh
We find the set of integers of the title, for fixed integers a
and b, as well as the related set of those n dividing a^n + b^n.
--------------------------------------------------------------------------------
Recent applications of random matrix
theory in number theory
Nina Snaith, University of Bristol
Over the past few years random matrix theory has proved useful
in suggesting answers to a surprisingly broad range of questions
in number theory. The talk will illustrate this with some examples
of recent successes.
--------------------------------------------------------------------------------
A Cubic Extension of the Lucas Functions
Hugh Williams, University of Calgary
Coauthors: Siguna Mueller (University of Wyoming) Eric Roettger
(University of Calgary)
From 1876 to 1878 Lucas developed his theory of the functions Vn
and Un, which now bear his name. He was particularly interested
in how these functions could be employed in proving the primality
of certain large integers, and as part of his investigations succeeded
in demonstrating that the Mersenne number 2^127-1 is a prime. Vn
and Un can be expressed in terms of the nth powers of the zeros
of a quadratic polynomial, and throughout his writings Lucas speculated
about the possible extension of these functions to those which could
be expressed in terms of the nth powers of the zeros of a cubic
polynomial. Indeed, at the end of his life he stated that “by
searching for the addition formulas of the numerical functions which
originate from recurrence sequences of the third or fourth degree,
and by studying in a general way the laws of the residues of these
functions for prime moduli… we would arrive at important new
properties of prime numbers.” We have very little idea of what
functions Lucas had in mind because he provided so little information
concerning this in his published and unpublished work.
In this talk we will discuss a pair of functions that are easily
expressed in terms of the zeros of a cubic function and show how
they do seem to provide an extension of Lucas’ theory. We will
do this by developing analogs involving these new functions for
all the principal results of Lucas’ theory concerning Vn and
Un
--------------------------------------------------------------------------------
A Tale of Two Motives : Gamma and Zeta
in positive characteristic
Jing Yu, National Tsing Hua University, TAIWANB
We consider the special geometric gamma values at proper rational
functions and the special Carlitz zeta values at positive integers.
Both sets of values are of interests to the arithmetic of function
field in a fixed positive characteristic. They are analogues of
the classical Euler Gamma values at proper fractions and Riemann
zeta values at integers greater than 2. We determine all the algebraic
relations among these special values. In particular the Gamma values
in question are all algebraically independent from those zeta values
at "odd" integers. This is a joint work with C.-Y. Chang
and M. Papanikolas. The tool used is the motivic transcendence theory
recently developed in positive characteristic. The algebraic independence
is explained by the two motives constructed for the purpose of accomodating
these special values, and their motivic Galois groups.
--------------------------------------------------------------------------------
A dyadic exercise in the construction of supercuspidal
representations and types
Jiu-Kang Yu, Purdue University
The theory of Weil representation is often used in the construction
of supercuspidal representations and types of p-adic groups when
the residual characteristic is not 2. In this talk a new and different
construction will be given which works for all residual characteristic.
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