# SCIENTIFIC PROGRAMS AND ACTIVITIES

December 18, 2014

## June 27-29, 2011 CONFERENCE IN NUMBER THEORY to be held at Carleton University, Ottawa

Organizers:
Saban Alaca (Carleton University),
Abdellah Sebbar (University of Ottawa)
Hugh C. Williams (Carleton University and University of Calgary)
Kenneth S. Williams (Carleton University)

#### INVITED SPEAKER ABSTRACTS

Stephen Choi, Simon Fraser University
On the Norms of Littlewood Polynomials

In this talk, we will discuss the norms over the unit circles of Littlewood polynomials, that is, polynomials with coefficients +1 or -1. In particular, we are interested in the $L_4$ norm and Mahler measure of Littlewood polynomials. Some current results and related conjectures, such as Barker sequences conjecture will be discussed in the talk.

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Todd Cochrane, Kansas State University
Waring's Problem over Finite Fields

For prime p and positive integer k we define Waring's number g(k,p) to be the minimal s such that every integer is a sum of s k-th powers (mod p). Equivalently, letting A denote the set of k-th powers in the finite field F in p elements, g(k,p) is the minimal s such that sA=F. We discuss various estimates for g(k,p) and the variety of methods used to obtain them including the estimation of exponential sums, additive combinatorics and the sum-product phenomenon, the geometry of numbers, and heights and zeros of integer polynomials. The strength of the method depends on the size of A.

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Karl Dilcher, Dalhousie University
A mod p^3 analogues of a theorem of Gauss on binomial coefficients

The theorem of Gauss that gives a modulo p evaluation of a certain central binomial coefficient was extended modulo p^2 by Chowla, Dwork, and Evans. In this talk I present a further extension to a congruence modulo p^3, with a similar extension of a theorem of Jacobi. This is done by first obtaining congruences to arbitrarly high powers of p for certain quotients resembling binomial coefficients and related to the p-adic gamma function. These congruences are of a very simple form and involve Catalan numbers as coefficients. As another consequence we obtain complete p-adic expansions for certain Jacobi sums. (Joint work with John B. Cosgrave).

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Matthew Greenberg, University of Calgary
Computing with automorphic forms

Like all of scientic research, the study of automorphic forms was fundamentally altered by the development of the computer. Landmarks in the computational theory of automorphic forms are Cremona's systematic enumeration of elliptic curves via their associated modular forms and Stein's sophisticated software packages for computing with modular forms. Many students and researchers use these now ubiquitous resources on a daily basis. In this talk, I will discuss the evolution of the computational methods for automorphic forms, current developments in the field, and prospects for future development.

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Matilde Lalín, Université de Montréal
Higher Mahler measure

The classical Mahler measure of an $n$-variable nonzero polynomial P is the integral of $\log |P|$ over the $n$-dimensional unit torus $T^n$ with the Haar measure. We consider, more generally, the integral of $\log^k |P|$. Specific examples yield special values of zeta functions, Dirichlet L-series, and polylogarithms. Moreover, one can ask the equivalent to Lehmer's question, and explore what happens at cyclotomic polynomials. This talk includes joint work with N. Kurokawa
and H. Ochiai and with K. Sinha.

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Hugh L. Montgomery, University of Michigan
Families of polynomials

This talk falls into two parts. Recently Mauduit and Rivat solved a problem posed by Gelfond in 1968, which in particular asserts that the sum of the binary digits of a prime is odd asymptotically half the time. Their work depends on estimates of moments of an associated exponential polynomial. We find that as the order increases, the sequence of moments satisfies a linear recurrence, arising from polynomials with interesting properties.
Let f_n denote the density of the sum of n independent uniformly distributed random variables. A formula for this density was derived independently by Lagrange and Stirling, roughly 250 years ago. It is piecewise polynomial, and at its transition points is proportional to eulerian numbers. The eulerian numbers satisfy a Pascal-like recurrence.
We find that f_n satisfies such a recurrence not just at the transition points, but for all arguments. This allows us to define a one-parameter family of polynomials of which the eulerian polynomials are a special case.

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Kumar Murty, University of Toronto
Lifting elliptic curves to characteristic zero

Let S be a set of primes and for each prime p in S, suppose we are given an elliptic curve E(p) over the field of p elements. Under what conditions does there exist an elliptic curve E over the rationals such that for each p in S, the reduction of E mod p is E(p)? If S is a finite set, the existence of E follows from the Chinese remainder theorem. If the complement of S is finite, we will give a criterion in terms of "minimal lifting conductors". This is joint work with Sanoli Gun.

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Damien Roy, University of Ottawa
Rational approximation to real points on algebraic curves

Let C be a projective algebraic curve defined over Q. Assume that the set of real points of C with Q-linearly independent coordinates is infinite, and define lambda(C) to be the supremum of the uniform exponents of approximation of those points by rational points. Although Dirichlet's box principle simply shows that lambda(C) is at least 1/n, it is tempting to conjecture that it is always greater, i.e. that there always exist such points which are constantly much better approximated by rational points then expected from the box principle. At the moment, the only curves for which this is known to hold is the curve defined by the polynomial xz - y^2 and those which derive from it by a linear automorphism. For most curves, we only have an upper bound for lambda(C). In the talk, we will discuss this and also the case of the curve defined by the polynomial x^2 z - y^3 for which recent joint work with Stéphane Lozier has shown that lambda(C) is at most 107/151.

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Renate Scheidler, University of Calgary
Infrastructure of Function Fields

The infrastructure of a global field has been used for a variety of important applications, including computing the regulator and the class number of a global field, and even for cryptography. Originally proposed by D. Shanks in 1972 in the setting of real quadratic fields, the concept has since been generalized to number fields of higher degree and to function fields. Strictly speaking, every field extension has multiple infrastructures, one for each ideal class, each containing a certain finite subset of "small" ideals in that class. Of particular interest is the infrastructure belonging to the principal class. Geometrically, this infrastructure is a torus whose dimension is the unit rank of the field extension. It is possible to equip this torus with a binary operation that is akin to multiplication and is called a giant step. The resulting structure behaves "almost" like an Abelian monoid -- and in quadratic extensions even almost like an Abelian group -- failing only associativity, and just barely. In the unit rank one case, a second addition-like operation, called a baby step, imposes an ordering on the infrastructure ideals according to a natural distance which is "almost" additive under giant steps. We present the baby step giant step arithmetic framework of the infrastructure and explain what it means to "just barely" fail associativity.

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Edlyn Teske-Wilson, University of Waterloo
Homomorphic Cryptosystems

Over the past few years, several solutions have been proposed that address the problem of homomorphic encryption and homomorphic signing. This talk highlights selected features of homomorphic cryptosystems.

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Gary Walsh, University of Ottawa and the Tutte Institute
Rational and integral points on families of elliptic curves

Though not exhaustive, the goal of this lecture is to present some recent developments on the computation and existence of rational and integral points on ellptic curves. This will be exhibited by way of focussing on certain specific families of elliptic curves, thereby, hopefully, providing some illuminating examples of the methods that have been developed for these purposes.

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Lawrence C. Washington, University of Maryland
Class numbers of real cyclotomic fields

Computation of class numbers of real subfields of cyclotomic fields by standard methods is very difficult because of the presence of units. Over the past several years, alternative methods have been developed. This talk will describe some of these methods and discuss results and conjectures that have arisen from some of this work.

#### CONTRIBUTED TALKS

Timothy Caley, University of Waterloo
The Prouhet-Tarry-Escott problem for Gaussian integers

The Prouhet-Tarry-Escott (PTE) problem is a classical number theoretic problem which asks for integer solutions to sums of equal powers. Solutions to the PTE problem give improved bounds for the "Easier" Waring problem, but they are difficult to find using conventional methods.

We will describe how solutions can be found by connecting the problem to finding rational points on elliptic curves. There will also be a statement of open questions relating to the PTE problem.

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Michael Dewar, Queen's University
The image and kernel of Atkin's $U_p$ operator modulo $p$

We determine the image of Atkin's $U_p$ operator acting on $\pmod p$ reduced modular forms. In 1972, Serre showed that for level 1 modular forms, $U_p$ was contractionary (i.e. the image has lower weight than the preimage.) We determine the exact weight of the space of images and generalize to all levels not divisible by $p$. As a consequence, we determine the dimension of the kernel of $U_p \pmod p$ for large weights. This contrasts with the situation for small weights, which is stil confoundingly mysterious.

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Greg Doyle
, Carleton University
A Recursive Formula for the Convolution Sum of Divisor Functions

For fixed arbitrary positive integers $a$ and $b$, we are interested in determining the value of the convolution sum $\sum_{m=1}^{n-1}\sigma_a(m)\sigma_b(n-m)$, for an arbitrary positive integer $n$, where $\sigma_a(n) = \sum_{d \mid n} d^a$. Using an identity given by Alaca, Alaca, McAfee and Williams, we derive a recursive formula for this convolution sum for all pairs of odd positive integers $a$ and $b$. In a similar fashion, we discuss how we might derive a similar recursive formula for the twisted convolution sum $\sum_{m<n/k}\sigma_a(m)\sigma_b(n-km)$.

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On the behaviour of the Liouville function on polynomials with integer coefficients

Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that {\bf Conjecture (Chowla).} $\sum_{n\le x} \lambda (f(n)) = o(x)$ for any polynomial $f(x)$ with integer coefficients, not in the form of $bg^2(x)$, where $b$ is a constant. Chowla's conjecture is proved for linear functions but for the degree greater than 1, the conjecture seems to be extremely hard and still remains wide open. One can consider a weaker form of Chowla's conjecture, namely, {\bf Conjecture 1 (Cassaigne, et al).} If $f(x) \in \Z [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\in \Z[x]$ and constant $b$, then $\lambda (f(n))$ changes signs infinitely often. Although it is weaker, Conjecture 1 is still wide open for polynomials of degree $>1$. In this talk, I will describe some recent progress made while studying Conjecture 1 for the quadratic polynomials. This is joint work with Peter Borwein and Stephen Choi.

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Hester Graves
, Queen's University
Euclidean Ideal Classes and Hilbert Class Fields

Lenstra generalized the concept of the Euclidean algorithm to Euclidean ideals. If a domain has a Euclidean ideal, then its class group is cyclic and the Euclidean ideal's class generates the class group. Lenstra showed that, assuming GRH, that for every finite extension of Q that is not imaginary quadratic, said field's class group is cyclic if and only if every generator is a Euclidean ideal class. In this talk, we will remove prove Lenstra's result for certain classes of fields without assuming GRH.

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The Number of Irreducible Polynomials of Even Degree $n$ over $\F_2$ with Four Given Coefficients

The problem of estimating the number of irreducible polynomials of degree $n$ over the finite field $F_q$ with some prescribed coefficients has been largely studied. This is a study of the number of irreducible polynomials of even degree $n$ over the finite field $\F_2$ where the coefficients of the terms $x^{n-1}\cdots,x^{n-r}$ are given, for $r \ge 4$. This number is represented by $N(n,t_1,\cdots,t_r)$. In this paper an approximation for these numbers is given and also experimentally is shown how good is the approximation. [Joint work with D. Panario]

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Rob Noble, Dalhousie University
Asymptotics of the weighted Delannoy numbers

The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this talk we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.

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Fabien Pazuki, University Bordeaux 1
Lower bounds on heights and applications.

Abstract : Let k be a number field and A/k an abelian variety. We will explain how, from a lower bound on the Neron-Tate height of the rationnal points P on A, one can give precise information about the arithmetic of the variety itself or about diophantine aspects of algebraic curves inside the variety. We will describe recent results in the case of abelian surfaces.

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Ethan Smith, CRM
Average Frobenius distribution for the degree two primes of a number field

Let $K$ be a fixed number field and $E$ an elliptic curve defined over $K$. Given a fixed integer $t$, we consider the problem of counting the number of degree two prime ideals $\mathfrak p$ of $K$ such that the trace of Frobenius of $E$ at $\mathfrak p$ is equal to $t$.
Under some assumptions on the number field, we show that on average, over the elliptic curves defined over $K$, the number of such prime with norm less than or equal to $x$ satisfies an asymptotic that is in accordance with standard heuristics. This is joint work with Kevin James.

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David Thomson
, Carleton University
Swan-like results for low-weight polynomials over finite fields of odd characteristic.
Joint with B. Hanson (Toronto) and D. Panario (Carleton)

The study of low-weight polynomials is important for implementations of fast finite field arithmetic using a polynomial basis. Swan (1962) applies a theory of Stickelberger to determine the parity of the number of irreducible factors of trinomials (polynomials with exactly three nonzero terms) over the binary field. Vishne (1992) extends Swan's work to all finite fields of even characteristic. In this work, we give the parity of the number of irreducible factors of all binomials and on a number of classes of trinomials over finite fields of odd characteristic. We indicate the bottleneck of this method in the trinomial case, which depends on additive properties of the quadratic character over finite fields.

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Colin Weir, University of Calgary
A method for constructing cubic function fields of fixed discriminant
This is joint work with R. Scheidler (Calgary).

Research into the construction of certain low degree function fields has surged in recent years, in part due to the cryptographic significance of elliptic and hyperelliptic curves. However there is comparatively little data available for higher degree function fields, leaving open many questions about the number of non-conjugate function fields of fixed degree and given discriminant. We will present an algorithm for tabulating a complete list of cubic function fields over a fixed finite field of bounded discriminant degree. Our methods are based on those of H. Cohen (1999) who tabulates cubic extensions of arbitrary number fields. The algorithm uses the tools of Kummer Theory and Class Field Theory, allowing for a natural transition to the function field setting. However, we are able to utilize the additional automorphisms of algebraic function fields to make significant improvements. Moreover, the algorithm is constructive in nature, allowing us to easily generate function fields of a specified discriminant as well. We will present the algorithm, our improvements, and a summary of the data we able able to generate.

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