We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over ${\bb Q}$ is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.

Let $F$ be a $p$-adic field, let $L$ be the completion of a maximal unramified extension of $F$, and let $\sigma$ be the Frobenius automorphism of $L$ over $F$. For any connected reductive group $G$ over $F$ one denotes by $B(G)$ the set of $\sigma$-conjugacy classes in $G(L)$ (elements $x,y$ in $G(L)$ are said to be $\sigma$-conjugate if there exists $g$ in $G(L)$ such that $g^-1 \kappa \sigma(g)=y$. One of the main results of this paper is a concrete description of the set $B(G)$ (previously this was known only in the quasi-split case).

Proper homogeneous $G$-spaces (where $G$ is semisimple algebraic group) over positive characteristic fields $k$ can be divided into two classes, the first one being the flag varieties $G/P$ and the second one consisting of varieties of unseparated flags (proper homogeneous spaces not isomorphic to flag varieties as algebraic varieties). In this paper we compute the Chow ring of varieites of unseparated flags, show that the Hodge cohomology of usual flag varieties extends to the general setting of proper homogeneous spaces and give an example showing (by geometric means) that the $D$-affinity of Beilinson and Bernstein fails for varieties of unseparated flags.

The cohomology ring of the moduli space $M(n,d)$ of semistable bundles of coprime rank $n$ and degree $d$ over a Riemann surface $M$of genus $g \geqslant 2$ has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when $n>2$. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these ‘dual’ Mumford relations form a complete set for the rational cohomology ring of $M(3,d)$ and show (Sect. 5) that the Pontryagin ring vanishes in degree$12g-8$ and above.

We investigate the relationship between the usual and general Hodge conjectures for abelian varieties. For certain abelian varieties $A$, we show that the usual Hodge conjecture for all powers of $A$ implies the general Hodge conjecture for $A$.

In this paper we take up the problem of discussing $C R$ manifolds of arbitrary $C R$ codimension. We closely follow the general method of N. Tanaka, while concentrating our attention to the case of manifolds endowed with partial complex structures. This study required a deeper understanding of the structure of the Levi–Tanaka algebras, which are the canonical prolongation of pseudocomplex fundamental graded Lie algebras. These algebras enjoy special properties, the understanding of which provided also a way to build up several different examples and points to a rich field of investigations. Here we restrained further our consideration to the homogeneous models.

In classical number theory, one has the famous theorem of Kummer-Sinnott giving the index of the cyclotomic units in the total unit group. Using the division values of sgn-normalized rank one Drinfeld modules, we construct the group of extended cyclotomic units for an abelian extension of a global function field and calculate its index in the whole unit group. The result could be regarded as an analogue and a generalization of Kummer-Sinnott's.

Let $G$ a compact group of isometries of a closed riemannian manifold $(X,m)$. Sunada proved that if $\Gamma_1$ and $\Gamma_2$ are two finite almost-conjugated subgroups of $G$, then $\Gamma_1\setminus X,m)$ and $\Gamma_2\setminus X,m)$ are isospectral. We prove that if $G$ is finite, there exists an open dense set in the set of $G$-invariant metrics for which the converse of this resukt is true. If $G$ is infinite, the situations is more complicated and we obtain some partial results.

Let $k$ be a perfect field of a positive characteristic $p, K$ – the fraction field of the ring of Witt vectors $W(k)$. Let $X$ be a smooth and proper scheme over $W(k)$. We present a candidate for a cohomology theory with coefficients in crystalline local systems: $p$-adic étale local systems on $X_K$ characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber $X_k$. We show that this cohomology satysfies a duality theorem.

The representation theory of polynomial growth strongly simply connected finite dimensional algebras over an algebraically closed field is established.