A well-known conjecture of E. Artin [1] states that for any integers $a \ne {\pm}1$ and $a$ is not a perfect square, there are infinitely many prime integers $p$ for which $a$ is a primitive root $({\bmod}\, p)$. An analogue of this conjecture for function fields was attacked successfully by Bilharz [2] in 1937 using the Riemann hypothesis for curves over finite fields (subsequently proved by A. Weil). The original conjecture of Artin remains open, though it was shown to be true if one assumes the Generalized Riemann hypothesis by Hooley [7]. In recent years, this conjecture of Artin has also been formulated and studied for elliptic curves over global fields instead of just ${\rm G}_m$ (the original case) (see [11]).

It is known that Siegel's theorem on integral points is effective for Galois coverings of the projective line. In this paper we obtain a quantitative version of this result, giving an explicit upper bound for the heights of S-integral K-rational points in terms of the number field K, the set of places S and the defining equation of the curve. Our main tools are Baker's theory of linear forms in logarithms and the quantitative Eisenstein theorem due to Schmidt, Dwork and van der Poorten.

We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent $p$ over the cyclotomic field obtained by adjoining a primitive $p$-th root of 1 to $Q$. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian $p$-groups.

Let $K$ be a number field, and let $X \subseteq P^{s-1}_K$ be a smooth complete intersection defined over $K$. In this paper, weak approximation is shown to hold for $X$ provided $s$ exceeds some function of the degree and codimension of $X$. This is a corollary of a more general result about the number of integral points on certain affine varieties in homogeneously expanding regions. This general result is established via a suitable adaptation of the Hardy-Littlewood Circle Method.

Let $f(x) = \Sigma a_ix^i$ be a monic polynomial of degree $n$ whose coefficients are algebraically independent variables over a base field $k$ of characteristic 0. We say that a polynomial $g(x)$ is generating (for the symmetric group) if it can be obtained from $f(x)$ by a nondegenerate Tschirnhaus transformation. We show that the minimal number ${\rm d}_k(n)$ of algebraically independent coefficients of such a polynomial is at least $[n/2]$. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert's 13th problem.

Our approach to this question (and generalizations) is based on the idea of the ‘essential dimension’ of a finite group $G$: the smallest possible dimension of an algebraic $G$-variety over $k$ to which one can ‘compress’ a faithful linear representation of $G$. We show that ${\rm d}_k(n)$ is just the essential dimension of the symmetric group ${\rm S}_n$. We give results on the essential dimension of other groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.

In this article we introduce a natural ‘elliptic’ generalization of the classical polylogarithms, study the properties of these functions and their relations with Eisenstein series.

The discriminant function $\Delta$ is a certain rigid analytic modular form defined on Drinfeld's upper half-plane $\Omega$. Its absolute value $\vert \Delta\vert$ may be considered as a function on the associated Bruhat–Tits tree ${\cal T}$. We compare $\log \vert \Delta\vert$ with the conditionally convergent complex-valued Eisenstein series $E$ defined on ${\cal T}$ and thereby obtain results about the growth of $\vert \Delta$ and of some related modular forms. We further determine to what extent roots may be extracted of $\Delta(z)/\Delta(nz)$, regarded as a holomorphic function on $\Omega$. In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.

Let $G$ be a complex reductive linear algebraic group and $G_0 \subseteq G$ a real form. Suppose $P$ is a parabolic subgroup of $G$ and assume that $P$ has a Levi factor $L$ such that $G_0 \cap L = L_0$ is a real form of $L$. Using the minimal globalization $V_{\min}$ of a finite length admissible representation for $L_0$, one can define a homogeneous analytic vector bundle on the $G_0$ orbit $S$ of $P$ in the generalized flag manifold $Y = G/P$. Let $A(P, V_{\min})$ denote the corresponding sheaf of polarized sections. In this article we analyze the $G_0$ representations obtained on the compactly supported sheaf cohomology groups $H^p_c(S,A(P, V_{\min}))$.

Explicit formulas for $K$-types of a $({\frak g}; K)$ module, such as Blattner's formula, are well known. However, the formulas are often too complex to make the $K$-type structure transparent. In this note we make use of some ideas of Vogan linking the ‘edges’ of the set of $K$-types to a certain restriction map from ${\frak u}$ to ${\frak u} \cap {\frak t}$ cohomology. It is hoped that such ideas will lead to tighter control of the set of $K$-types.

Starting from a sheaf of associative algebras over a scheme we show that its deformation theory is described by cohomologies of a canonical object, called the cotangent complex, in the derived category of sheaves of bi-modules over this sheaf of algebras. The passage from deformations to cohomology is based on considering a site which is naturally constructed out of our sheaf of algebras. It turns out that on the one hand, cohomology of certain sheaves on this site control deformations, and on the other hand, they can be rewritten in terms of the category of sheaves of bi-modules.

In this paper we construct smooth irreducible space curves $C$ which link geometrically by surfaces of minimal degree containing $C$ to curves $\Gamma$ of generic embedding dimension three. This produces interesting behavior with respect to both $C$ and $\Gamma$. The curves $\Gamma$ link to smooth connected curves by surfaces of low degree but cannot link to smooth connected curves by surfaces of high degree. The curves $C$ give counterexamples to a conjecture of Martin-Deschamps and Perrin.

One-to-one correspondences are established between the set of all nondegenerate graded Jacobi operators of degree $-1$ defined on the graded algebra $\Omega(M)$ of differential forms on a smooth, oriented, Riemannian manifold $M$, the space of bundle isomorphisms $L{\A}TM{\to} TM$, and the space of nondegenerate derivations of degree $1$ having null square. Derivations with this property, and Jacobi structures of odd $\Bbb Z_2$-degree are also studied through the action of the automorphism group of $\Omega(M)$.

For a smooth projective variety $X$ of dimension $n$ in a projective space ${\bb P}^N$ defined over an algebraically closed field $k$, the Gauss map is a morphism from $X$ to the Grassmannian of $n$-plans in ${\bb P}^N$sending $x \in X$ to the embedded tangent space $T_xX \subset {\bb P}^N$. The purpose of this paper is to prove the generic injectivity of Gauss maps in positive characteristic for two cases; (1) weighted complete intersections of dimension $n \geqslant 3$ of general type; (2) surfaces or 3-folds with $\mu$-semistable tangent bundles; based on a criterion of Kaji by looking at the stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman-Piene is true in case $X$ is of general type of dimension $n \geqslant 3$. The second result is a generalization of the injectivity for curves.

A relation between the Euler characteristics of the Milnor fibres of a real analytic function is derived from a simple identity involving complex monodromy and complex conjugation. A corollary is the result of Coste and Kurdyka that the Euler characteristic of the local link of an irreducible algebraic subset of a real algebraic set is generically constant modulo 4. A similar relation for iterated Milnor fibres of ordered sets of functions is used to define topological invariants of ordered collections of algebraic subsets.

If $E$ is an elliptic curve over ${\bb Q}$, then let $E(D)$ denote the $D$-quadratic twist of $E$. It is conjectured that there are infinitely many primes $p$ for which $E(p)$ has rank 0, and that there are infinitely many primes $l$ for which $E(l)$ has positive rank. For some special curves $E$ we show that there is a set $S$ of primes $p$ with density $\frac{1}{3}$; for which if $D = \Pi_{pj}$ is a squarefree integer where $pj \in S$, then $E(D)$ has rank 0. In particular $E(p)$ has rank 0 for every $p \in S$. As an example let $E_1$ denote the curve \[ E_1: y^2 = x^3 + 44x^2 -19360x + 1682384. \]Then its associated set of primes $S_1$ consists of the prime 11 and the primes $p$ for which the order of the reduction of $X0(11)$ modulo $p$ is odd. To obtain the general result we show for primes $p \in S$ that the rational factor of $L(E(p),1)$ is nonzero which implies that $E(p)$ has rank 0. These special values are related to surjective ${\bb Z}/2{\bb Z}$ Galois representations that are attached to modular forms. Another example of this result is given, and we conclude with some remarks regarding the existence of positive rank prime twists via polynomial identities.

In this paper we solve the problem of desingularization of an absolutely isolated singularity of a differential equation, including the dicritical case. As an application, we prove the finiteness of the number of dicritical points in the blowing up tree of an absolutely isolated singularity.

In the thesis [Pe1] it was introduced, studied and applied a general theory of Weierstrass loci for vector bundles on a smooth curve. The results, proofs, background, examples and motivations of this thesis are contained in [Pe2]. We believe that this theory, at least in characteristic 0, is the ‘right’ one. The aim of this paper is to introduce and study an extension of [Pe1] to the case of higher dimensional varieties. At least two possible theories seem to be useful and natural; see the discussion just after Remark 1.1 and Section 4. We strongly prefer the ‘symmetric’ one (see Definition 1.5). In the first section we introduce the general theory and give the main general results. In the second section we study in details the case of $P^2$ for three reasons: it is nice; it shows how to use the general theory and what could be expected in more general situations and (last but not least) to convince the reader that it is technically easier and often more interesting to work in the ‘symmetric’ set up. Then in the third section we apply the method of Section 2 to a much more general situation (essentially, any variety $X$ as base of the vector bundle). In the fourth section we give the set up and start the analysis of specific examples of what happens near a specific point $P$ of the base variety $X$ (even when $X$ is singular at $P$). Here, except at the first step we are able to work only with the ‘symmetric’ definition.

Let $f:X\rightarrow Y$ be a morphism of noetherian schemes, generically smooth and equidimensional of dimension $d, \iota : X^{\prime} \rightarrow X$ a closed embedding such that $f \circ \iota : X^{\prime} \rightarrow Y$ is generically smooth and equidimensional of dimension $d^{\prime}$, and $X^{\prime}, X$ and $Y$ are excellent schemes without embedded components. We exhibit a concrete morphism\[{\rm Res}_{X^{\prime}/X}: {\rm det}\,{\cal N}_{X^{\prime}/X} \otimes_{{\cal O}_{X^{\prime}}} \iota^* \omega^d_{X/Y} \rightarrow \omega^{d^{\prime}}_{X^{\prime}/Y},\] which transforms the integral of $X/Y$ into the integral of$X^{\prime}/Y$. Here ${\cal N}_{X^{\prime}/X}$ denotes the normal sheaf of $X^{\prime}/X$ and$\omega^d_{X/Y}$ resp. $\omega^{d^{\prime}}_{X^{\prime}/Y}$ denotes the sheaf of regular differential forms of $X/Y$ resp. $X^{\prime}/Y$. Using generalized fractions we provide a canonical description of residual complexes and residue pairs of Cohen-Macaulay varieties, and obtain a very explicit description of fundamental classes and their traces.