Over a fixed finite field ${\bf F}_p$, families of polynomial equations $f_i(x_1, \dots, x_{n_N}) = 0$ for $i = 1, \dots, k_N$, that are uniformly determined by a parameter $N$, are considered. The notion of a uniform family is defined in terms of first-order logic. A notion of an abstract Euler characteristic is used to give sense to a statement that the system has a solution for infinite $N$, and a statement linking the solvability of a linear system for infinite $N$ with its solvability for finite $N$ is proved. This characterisation is used to formulate a criterion yielding degree lower bounds for various ideal membership proof systems (for example, Nullstellensatz and the polynomial calculus). Further, several results about Euler structures (structures with an abstract Euler characteristic) are proved, and the case of fields, in particular, is investigated more closely. 1991 Mathematics Subject Classification: primary 03F20, 12L12, 15A06; secondary 03C99, 12E12, 68Q15, 13L05.

The relation between $\mathbb{Q}$-curves and certain abelian varieties of $\operatorname{GL}_2$-type was established by Ribet (`Abelian varieties over $\mathbb{Q}$ and modular forms', {\em Proceedings of the KAIST Mathematics Workshop} (1992) 53--79) and generalized to building blocks, the higher-dimensional analogues of $\mathbb{Q}$-curves, by Pyle in her PhD Thesis (University of California at Berkeley, 1995). In this paper we investigate some aspects of $\mathbb{Q}$-curves with no complex multiplication and the corresponding abelian varieties of $\operatorname{GL}_2$-type, for which we mainly use the ideas and techniques introduced by Ribet (op. cit. and `Fields of definition of abelian varieties with real multiplication', {\em Contemp.\ Math.} 174 (1994) 107--118). After the Introduction, in Sections 2 and 3 we obtain a characterization of the fields where a $\mathbb{Q}$-curve and all the isogenies between its Galois conjugates can be defined up to isogeny, and we apply it to certain fields of type $(2,\dots,2)$. In Section 4 we determine the endomorphism algebras of all the abelian varieties of $\operatorname{GL}_2$-type having as a quotient a given $\mathbb{Q}$-curve in easily computable terms. Section 5 is devoted to a particular case of Weil's restriction of scalars functor applied to a $\mathbb{Q}$-curve, in which the resulting abelian variety factors over $\mathbb{Q}$ up to isogeny as a product of abelian varieties of $\operatorname{GL}_2$-type. Finally, Section 6 contains examples: we parametrize the $\mathbb{Q}$-curves coming from rational points of the modular curves $X^*(N)$ having genus zero, and we apply the results of Sections 2--5 to some of the curves obtained. We also give results concerning the existence of quadratic $\mathbb{Q}$-curves. 1991 Mathematics Subject Classification: primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.

Let $H=k\cal Q$ be a finite-dimensional connected wild hereditary path algebra, over some field $k$. Denote by $H$-reg the category of finite-dimensional regular $H$-modules, that is, the category of modules $M$ with $\tau_H^{-m}(\tau_H^m M) \cong M$ for all integers $m$, where $\tau_H$ denotes the Auslander--Reiten translation. Call a filtration \begin{equation} M = M_0\supset M_1\supset\ldots\supset M_r \supset M_{r+1}=0 \tag{$*$} \end{equation} of a regular $H$-module $M$ a {\em regular filtration} if all subquotients $M_i/M_{i+1}$ are regular. Call a regular filtration $(*)$ a {\em regular composition series} if it is strictly decreasing and has no proper refinement. A regular component $\cal C$ in the Auslander--Reiten quiver $\Gamma (H)$ of $H$-mod is called {\em filtration closed} if, for each $M\in\text{add\,}\cal C$, the additive closure of $\cal C$, and each regular filtration $(*)$ of $M$, all the subquotients $M_i/M_{i+1}$ are also in $\text{add\,}\cal C$. We show that most wild hereditary algebras have filtration-closed Auslander--Reiten components. Moreover, we deduce from this that there are also {\em almost serial} components, that is regular components $\cal C$, such that any indecomposable $X\in\cal C$ has a unique regular composition series. This composition series coincides with the Auslander--Reiten filtration of $X$, given by the maximal chain of irreducible monos ending at $X$. 1991 Mathematics Subject Classification: 16G70, 16G20, 16G60, 16E30.

Motivated by complex oriented equivariant cohomology theories, we give a natural algebraic definition of an $A$-equivariant formal group law for any abelian compact Lie group $A$. The complex oriented cohomology of the classifying space for line bundles gives an example. We also show how the choice of a complete flag gives rise to a basis and a means of calculation. This allows us to deduce that there is a universal ring $L_A$ for $A$-equivariant formal group laws and that it is generated by the Euler classes and the coefficients of the coproduct of the orientation. We study a number of topological cases in some detail. 1991 Mathematics Subject Classification: 14L05, 55N22, 55N91, 57R85.

Let $X=\operatorname{Spec} B$ be an affine variety over a field of arbitrary characteristic, and suppose that there exists an action of a unipotent group (possibly neither smooth nor connected). The fundamental results are as follows. (1) An algorithm for computing invariants is given, by means of introducing a degree in the ring of functions of the variety, relative to the action. Therefore an algorithmic construction of the quotient, in a certain open set, is obtained. In the case of a Galois extension, $k\hookrightarrow B=K$, which is cyclic of degree $p=\text{ch} (k)$ (that is, such that the unipotent group is $G={\Bbb Z}/p {\Bbb Z}$), an element of minimal degree becomes an Artin--Schreier radical, and the method for computing invariants gives, in particular, the expression for any element of $K$ in terms of these radicals, with an explicit formula. This replaces the well-known formula of Lagrange (which is valid only when the degree of the extension and the characteristic are relatively prime) in the case of an extension of degree $p=\text{ch}(k)$. (2) In this paper we give an effective construction of a stable open subset where there is a quotient. In this sense we obtain an algebraic local criterion for the existence of a quotient in a neighbourhood. It is proved (provided the variety is normal) that, in the following cases, such an open set is the greatest one that admits a quotient: \begin{enumerate} \item[(a)] when the action is such that the orbits have dimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any action of the additive group $G_a$; \item[(b)] in characteristic 0, when the action is proper (obtained from the results of Fauntleroy) or the group is abelian. 1991 Mathematics Subject Classification: primary 14L30; secondary 14D25, 14D20.

Let $G$ be a finite group. It is proved that if the probability that two randomly chosen elements of $G$ generate a soluble group is greater than $\frac{11}{30}$ then $G$ itself is soluble. The bound is sharp, since two elements of the alternating group $A_5$ generate $A_5$ with probability $\frac{11}{30}$. Similar probabilistic statements are proved concerning nilpotency and the property of having odd order. It is also proved that there is a number $\kappa$, strictly between $0$ and $1$, with the following property. Let $\cal X$ be any class of finite groups which is closed for subgroups, quotient groups and extensions. If the probability that two randomly chosen elements of $G$ generate a group in $\cal X$ is greater than $\kappa$ then $G$ is in $\cal X$. The proofs use the classification of the finite simple groups and also some of the detailed information now available concerning maximal subgroups of finite almost simple groups. 1991 Mathematics Subject Classification: 20F16, 20D06, 20D08, 60B99.

Let $\mathcal{B}$ be an irreducible spherical Moufang building of rank at least $2$. Then the group $G$ is called a group of Lie type $\mathcal{B}$ if it is generated by the root subgroups corresponding to the roots of some apartments of $\mathcal{B}$. This notion includes: \begin{enumerate} \item[(1)] classical groups of finite rank, \item[(2)] simple algebraic groups over arbitrary fields, \item[(3)] the `mixed' groups of Tits. \end{enumerate} General structure theorems and a general presentation type theorem for such Lie-type groups, which in a way generalize well-known theorems of Seitz and Curtis and Tits, are obtained. 1991 Mathematics Subject Classification: 20G15, 20E42.

We study finite and infinite entangled graphs in the bond percolation process in three dimensions with density $p$. After a discussion of the relevant definitions, the entanglement critical probabilities are defined. The size of the maximal entangled graph at the origin is studied for small $p$, and it is shown that this graph has radius whose tail decays at least as fast as $\exp(-\alpha n/\log n)$; indeed, the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant $\alpha$. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when $p$ is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation. http://www.statslab.cam.ac.uk/$\sim$grg/ 1991 Mathematics Subject Classification: primary 60K35; secondary 05C10, 57M25, 82B41, 82B43, 82D60.