March  2, 2024

Canadian Number Theory Association X Meeting
to be held at the University of Waterloo

Public Lecture Abstract

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
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.

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

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

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.

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.


J. Friedlander
(University of Toronto)
Wee Teck Gan
(University of California San Diego)

S. Gonek

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.


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
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.

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.

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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