Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains $\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there is a unique normalised generator of the field of $G$-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal $q$ series using modular polynomials.

Let $X^5 + aX + b \in Z[X]$ have Galois group $D_5$. Let $\theta$ be a root of $X^5 + aX + b$. An explicit formula is given for the discriminant of $Q(\theta)$.

In this paper we find a formula for the Alexander polynomial $\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with $(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and $p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7} (x)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.

Let $F$ be a totally real number field and let $\GL_{n}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

We find a lower bound on the number of imaginary quadratic extensions of the function field $\F_q(T)$ whose class groups have an element of a fixed order. More precisely, let $q \geq 5$ be a power of an odd prime and let $g$ be a fixed positive integer $\geq 3$. There are $\gg q^{\ell (\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D) \leq \ell$ such that the class groups of the quadratic extensions $\F_q(T,\sqrt{D})$ have an element of order~$g$.

Jacobi-like forms for a discrete subgroup $\G \subset \SL(2,\mbb R)$ are formal power series with coefficients in the space of functions on the Poincar\'e upper half plane satisfying a certain functional equation, and they correspond to sequences of certain modular forms. We introduce Hecke operators acting on the space of Jacobi-like forms and obtain an explicit formula for such an action in terms of modular forms. We also prove that those Hecke operator actions on Jacobi-like forms are compatible with the usual Hecke operator actions on modular forms.

We give an explicit recipe for the determination of the images associated to the Galois action on $p$-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over $\QQ$ without complex multiplication with conductor less than 200 and for each prime number~$p$.

James Arthur was awarded the Canada Gold Medal of the National Science and Engineering Research Council in 1999. This introduction to his work is an attempt to explain his methods and his goals to the mathematical community at large.

The zeta function of a nonsingular pair of quadratic forms defined over a finite field, $k$, of arbitrary characteristic is calculated. A.~Weil made this computation when $\rmchar k \neq 2$. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is

In this note the multiplicities of binary recurrences over algebraic number fields are investigated under some natural assumptions.

An asymptotic formula is obtained for the number of cyclic quartic fields over $Q$ with discriminant $\leq x$.

The generating degree $\gdeg (A)$ of a topological commutative ring $A$ with $\Char A = 0$ is the cardinality of the smallest subset $M$ of $A$ for which the subring $\Z[M]$ is dense in $A$. For a prime number $p$, $\C_p$ denotes the topological completion of an algebraic closure of the field $\Q_p$ of $p$-adic numbers. We prove that $\gdeg (\C_p) = 1$, \ie, there exists $t$ in $\C_p$ such that $\Z[t]$ is dense in $\C_p$. We also compute $\gdeg \bigl( A(U) \bigr)$ where $A(U)$ is the ring of rigid analytic functions defined on a ball $U$ in $\C_p$. If $U$ is a closed ball then $\gdeg \bigl( A(U) \bigr) = 2$ while if $U$ is an open ball then $\gdeg \bigl( A(U) \bigr)$ is infinite. We show more generally that $\gdeg \bigl( A(U) \bigr)$ is finite for any {\it affinoid} $U$ in $\PP^1 (\C_p)$ and $\gdeg \bigl( A(U) \bigr)$ is infinite for any {\it wide open} subset $U$ of $\PP^1 (\C_p)$.

Precise, elegant evaluations are given for Gauss sums of orders six and twelve.

Let $p$ be prime and let $\vartheta\in\Z^*_p$ be of multiplicative order $t$ modulo $p$. We consider exponential sums of the form $$ S(a) = \sum_{x =1}^{t} \exp(2\pi i a \vartheta^{x^2}/p) $$ and prove that for any $\varepsilon > 0$ $$ \max_{\gcd(a,p) = 1} |S(a)| = O( t^{5/6 + \varepsilon}p^{1/8}) . $$

The purpose of this paper is to show the limitations of the conjectures of algebraic approximation. For this, we construct points of $\bC^m$ which do not admit good algebraic approximations of bounded degree and height, when the bounds on the degree and the height are taken from specific sequences. The coordinates of these points are Liouville numbers.

This paper is concerned with the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an $M$-Cantorval.

Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos \frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place au-dessus de $p$. Lorsque l'image de $(2 - 2\cos \frac{2\pi}{r})$ dans $\F$ n'est pas un carr\'e, nous donnons une construction g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3, sec.~6.3], et son existence est pr\'edite par le crit\`ere de rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction s'obtient en tordant la representation galoisienne associ\'ee aux points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien. Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que l'on retrouve quand $r = 2$ et $r = 3$. Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos \frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place above $p$. When the image of $(2 - 2\cos \frac{2\pi}{r})$ in $\F$ is not a square, we describe a geometric construction of a regular extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$'' in the sense of [3, sec.~6.3], and its existence is predicted by the rigidity criterion of Belyi, Fried, Thompson and Matzat. Its construction is obtained by twisting the mod $p$ galois representation attached to a family of abelian varieties with real multiplications by $K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian varieties are defined in general over a quadratic field, and are isogenous to their galois conjugate. Our construction generalises a method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.

We examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group's action on the division points of an appropriate Drinfeld module.

It is shown that given a local $L$-function defined by Langlands-Shahidi method, there exists a highly ramified character of the group which when is twisted with the original representation leads to a trivial $L$-function.

The results herein continue observations on norm form equations and continued fractions begun and continued in the works \cite{chows}--\cite{mol}, and \cite{mvdpw}--\cite{schinz}.

We prove that $\{(n^p-n)/p\}_p \in \prod_p \mathbf{F}_p$, with $p$ ranging over all primes, is independent of $1$ over the integers, assuming a conjecture in elementary number theory generalizing the infinitude of Mersenne primes. This answers a question of Buium. We also prove a generalization.

For an integer $n$, let $d(n)$ denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum $$ S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2). $$ It is proved in the paper that, as $x \to \infty$, $$ S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 + \epsilon}), $$ where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any fixed positive real number. The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error $O \bigl( x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.

An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group $\PSL(2,{\sf Z})$ of level $6$. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function {\it syst\/} which associates to every Riemann surface $M$ the length of a systole, which is a shortest closed geodesic of $M$.

A new infinite product $t_n$ was introduced by S.~Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about $t_n$ by establishing new connections between the modular $j$-invariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, $t_n$ generates the Hilbert class field of $\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class invariant according to H.~Weber's definition of class invariants.

A sequence of positive rationals generates a subgroup of finite index in the multiplicative positive rationals, and group product representations by the sequence need only a bounded number of terms, if and only if certain related sequences have densities uniformly bounded from below.