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]).

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.

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 $H^p_c(S,A(P, V_{\min}))$.

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.

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.