Over a fixed finite field ${\bf F}_p$, families ofpolynomial equations$f_i(x_1, \dots, x_{n_N}) = 0$ for $i = 1, \dots, k_N$,that are uniformly determined by a parameter $N$, areconsidered. The notion of a uniform family is defined interms of first-order logic.A notion of an abstract Euler characteristic is used togive sense to a statement that the system has a solutionfor infinite $N$, and a statement linking the solvabilityof a linear system for infinite $N$ with its solvabilityfor finite $N$ is proved.This characterisation is used to formulate a criterionyielding 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 certainabelian varieties of $\operatorname{GL}_2$-type was established by Ribet (`Abelian varieties over $\mathbb{Q}$ and modular forms', {\em Proceedings ofthe 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 aspectsof $\mathbb{Q}$-curves with no complex multiplicationand 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 thefields where a $\mathbb{Q}$-curve and all the isogeniesbetween 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 ofWeil'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 abelianvarieties of $\operatorname{GL}_2$-type.Finally, Section 6 contains examples:we parametrize the $\mathbb{Q}$-curves coming fromrational 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 ofmodules $M$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 properrefinement.A regular component $\cal C$ in the Auslander--Reiten quiver $\Gamma (H)$ of $H$-modis 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 mostwild hereditary algebras have filtration-closedAuslander--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 theAuslander--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 orientedcohomology 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 thatthere 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. Westudy a number of topological cases in some detail. 1991 Mathematics Subject Classification: 14L05, 55N22, 55N91, 57R85.

Let $X=\operatorname{Spec} B$ be an affine varietyover a field of arbitrary characteristic, and supposethat there exists an action of a unipotent group(possibly neither smooth nor connected). Thefundamental 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 aGalois extension, $k\hookrightarrow B=K$, which iscyclic of degree $p=\text{ch} (k)$ (that is, such thattheunipotent group is $G={\Bbb Z}/p {\Bbb Z}$),an element of minimal degree becomes anArtin--Schreier radical, and the method for computing invariants gives, in particular, theexpression for any element of $K$ in terms of theseradicals, with an explicit formula. This replacesthe well-known formula of Lagrange (which is validonly when the degree of the extension and thecharacteristic are relatively prime) in the case of an extension of degree $p=\text{ch}(k)$. (2)In this paper we give an effectiveconstruction of a stable open subset where there is a quotient. In this sense we obtain an algebraic localcriterion for the existence of a quotient in aneighbourhood. It is proved (provided the variety isnormal) that, in the following cases, such an openset is the greatest one that admits a quotient:\begin{enumerate}\item[(a)] when the action is such that the orbits havedimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any actionof 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 $A_5$ withprobability $\frac{11}{30}$.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 elementsof $G$ 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 $2$. Thenthe group $G$ is called a group of Lie type$\mathcal{B}$ if it is generated by theroot subgroups corresponding to the rootsof 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-knowntheorems of Seitz and Curtis and Tits, are obtained. 1991 Mathematics Subject Classification:20G15, 20E42.

We study finite and infinite entangled graphsin the bond percolation process in three dimensionswith 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 logarithmfor 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.