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Showing papers in "Annals of Mathematics in 1975"



Journal ArticleDOI
TL;DR: It is proved that the monadic theory of the real order is undecidable, which means that all known results in a unified way are proved.
Abstract: We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of the real order is undecidable. Our methods are modeltheoretic, and we do not use automaton theory.

336 citations


Journal ArticleDOI
TL;DR: This paper gives a unified proof that the obvious sign condition demanded by the Gauss Bonnet Theorem, namely, is given.
Abstract: In previous work [11], [13] we have considered the problem of describing the set of Gaussian curvatures (if dim M = 2) and scalar curvatures (if dim M > 3) on a compact, connected, but not necessarily orientable manifold M (see also [12] for the case of open manifolds). We refer the reader to the introductions of [11] and [13] for relevant background and literature. In this paper we give a unified proof that the obvious sign condition demanded by the Gauss Bonnet Theorem, namely,

325 citations




Book ChapterDOI
TL;DR: In this paper, it was shown that the Poincare-Bendixson Theorem can be generalized to foliations of codimension one provided that the leaves of the foliation satisfy a growth condition.
Abstract: It has been known for some time that qualitative results concerning flows in the plane do not fully generalize, either to flows on arbitrary manifolds or to foliations of codimension one. On the other hand, in [11] it was shown that the Poincare-Bendixson Theorem can be generalized to foliations of codimension one provided that the leaves of the foliation satisfy a growth condition. In the classical situation of flows in the plane this condition is trivially satisfied. If a foliation comes from a finite dimensional Lie group action then the growth condition may be interpreted in terms of the ergodic properties of the group. Of course, for arbitrary foliations we do not necessarily have such a group action, but we do have a pseudogroup, called the holonomy pseudogroup, which acts transversely to the leaves. Growth conditions on the leaves can be interpreted as conditions on the holonomy pseudogroup which, in some cases, imply the existence of a measure which is invariant under the action of the pseudogroup. It is such invariant measures that capture the essence of the classical qualitative theory of flows on surfaces. The basic idea of using invariant measures to study qualitative aspects of foliations goes back to the rotation numbers of Poincare. A modern treatment of these notions for flows on metric spaces which clearly brings out the role of invariant measures is in Schwartzman [22]. This last work is based on Kryloff-Bogoliuboff [7] in which the space of invariant measures of a dynamical system is studied at length and is recognized to be a topological invariant of the dynamical system itself. Invariant measures have also appeared earlier in the qualitative study of foliations. Sacksteder [21] uses the notion of an invariant measure to describe the structure of smooth codimension one foliations having trivial holonomy groups, and Sinai [23] considers invariant measures for the foliations invariant under a transitive Anosov diffeomorphism. In both cases the invariant measures involved were "positive on open sets." Hirsch and Thurston [4] consider more general invariant measures for foliated bundles

306 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that Br(X) (non-p) is a subgroup of elements of order prime to p, and has the order predicted by (A-T).
Abstract: In [20] it is shown that if (T) is assumed to hold for X/k, then it follows that Br(X) (non-p), the subgroup of Br(X) of elements of order prime to p, is finite and has the order predicted by (A-T). The proof of this makes heavy use of the known properties of l-adic etale cohomology. In attempting to extend this result to include the p-part, three essential difficulties arise. Firstly, the different components of the formula in (A-T) are described by two different cohomology theories; viz., Br(X) and NS(X) are described by the flat cohomology and P2(t) by the crystalline cohomology. Secondly, neither the flat nor the crystalline cohomology is as well understood as the etale cohomology. Thirdly, the p-part of (A-T) seems genuinely to have

195 citations


Journal ArticleDOI
TL;DR: In this paper, the authors define a contravariant functor F3@ from the category of algebraic schemes over k to the class of sets, in the following way: if S is an algebraic scheme, let '0B,(S) be the set of equivalence classes of families of vector bundles on X over S, such that for each s e S the induced sheaf on P`'(s) is torsion-free.
Abstract: The object of this paper is to derive separation and completeness properties for the families of vector bundles on a nonsingular projective variety X over a fixed algebraically closed field k. By a vector bundle on X we mean a torsion-free coherent sheaf on X. If Xis a curve, such a sheaf must be locally free; thus this definition corresponds with the usual notion of vector bundle. On higher dimensional varieties, it appears that the category of locally free sheaves is too restrictive; for example, in this category, bundles do not in general have complete flags, whereas in the category of torsion-free sheaves, complete flags always exist (Proposition 1). Let S be an algebraic scheme over k. We define a family of vector bundles on X over S to be a coherent sheaf E on X x S, flat over S, such that for each s e S the induced sheaf E. on P`'(s) is torsion-free. We consider two such families E and E' to be equivalent if there is an invertible sheaf L on S such that E _ E' ( p*(L). Starting from the concept of a family of vector bundles, we are led to define a contravariant functor 'F3@ from the category of algebraic schemes over k to the category of sets, in the following way: If S is an algebraic scheme, let '0B,(S) be the set of equivalence classes of families of vector bundles on X over S. Then if T > S is a morphism of algebraic schemes and E a family of vector bundles over S, (g x X)*(E) will be a family of vector bundles over T. It is natural to ask whether the functor F3SX is representable; that is, is there an algebraic scheme V and a family of vector bundles E over V such that Hom (S, V) -= B,(S) for all S? Such a V would be a natural parameter space for the bundles on X, and E would be a "universal family". It turns out that this is too much to expect. First of all, there are too many bundles to be parameterized by an algebraic scheme: we must break the functor up into separate parts corresponding to certain natural invariants, for example, the Hilbert polynomial. Next, we must throw away some

190 citations


Journal ArticleDOI
TL;DR: In this article, the problem of proving that the ring of invariants, RG, is finitely generated over fields of positive characteristic has been studied and a weaker version of complete reducibility has been conjectured.
Abstract: Let G be a semi-simple algebraic group over an algebraically closed field, k. Let G act rationally by automorphisms on the finitely generated k-algebra, R. The problem of proving that the ring of invariants, RG, is finitely generated originates with the invariant theorists of the nineteenth century. When k = C, the complex numbers, and G GL (n, C) the question is answered affirmatively by Hilbert's "fundamental theorem of invariant theory". The proof involved constructing a G equivariant projection from R to RG and then using it to prove the result algebraically. When k is of characteristic 0 and G is any semi-simple group, by a theorem of H. Weyl, every finite dimensional representation of G is completely reducible. In the 1950's D. Mumford and others (Cartier, Iwahori, Nagata) applied Weyl's theorem to construct a projection from R to RG for any semi-simple group. This made it possible to generalize Hilbert's proof to an arbitrary semi-simple group. Certain geometric applications, particularly to the theory of moduli, made a generalization to groups over fields of positive characteristic highly desirable. In positive characteristic, complete reducibility definitely fails. Hence attempts were made to replace complete reducibility with a weaker condition which would at once hold for all semi-simple groups and make a proof of finite generation of RG possible. The weakest way to state complete reducibility is the following. If V is a finite dimensional G-module containing a G-stable sub-space of co-dimension one, VT, then there is a G-stable line LC V such that V0 E L = V. Mumford conjectured a weaker version of this statement by seeking a complement only in a higher symmetric power of V, SI( V). This is the conjecture as it is stated in the preface to [16]:

175 citations



Journal ArticleDOI
TL;DR: In this article, the equivariant Witt ring of a finite group w over a Dedekind domain R is studied, and it is shown that-modulo the prime 2-GW(r, Z) equals the character ring of real representations of w and GW(w, R) equals GW w, Z ( W(R).
Abstract: The equivariant Witt ring GW(, R) of a finite group w over a Dedekind domain R is studied. It is shown that-modulo the prime 2-GW(r, Z) equals the character ring of real representations of w and GW(w, R) equals GW(w, Z) ( W(R). From this, induction theorems a la E. Artin and R. Brauer are derived for GW(-, R) and it is shown how these can be applied towards the computation of L-groups.

Journal ArticleDOI
TL;DR: In this article, it was shown that a certain nonholomorphic derivative of a Hilbert modular form with algebraic Fourier coefficients, divided by another modular form, takes an algebraic value at every point with complex multiplication.
Abstract: The purpose of this paper is twofold. We first present a principle that a certain non-holomorphic derivative of a Hilbert modular form with algebraic Fourier coefficients, divided by another modular form, takes an algebraic value at every point with 'complex multiplication'. Second, we investigate the Hilbert or Siegel modular forms with cyclotomic Fourier coefficients in the framework of canonical models as developed in our previous papers. The first principle, even specialized to the one-dimensional case, seems new and is as follows. Let f (z) and g(z) be modular forms of weight r and r + 2n, respectively, with respect to a congruence subgroup r of SL2(Z), and put

Journal ArticleDOI
TL;DR: Recently Linch as discussed by the authors showed that TeichmUller space does not have negative curvature and that every finite subgroup of the Teichmiiller modular group has a fixed point.
Abstract: The study of the geometry of the classical Teichmiiller spaces was begun in 1959 by Kravetz [9]. The starting point was the classical theorem of TeichmUller on extremal quasiconformal maps between compact Riemann surfaces. The TeichmUller theorem was used to argue that with respect to the Teichmiiller metric, TeichmUller space is straight and that it has negative curvature. In turn, negative curvature was used to show that every finite subgroup of the Teichmiiller modular group has a fixed point. This latter statement is known to be equivalent to the Nielsen Realization Problem which conjectures that every finite subgroup of the mapping class group of a surface can be realized by a finite subgroup of the group of homeomorphisms. Recently Linch [10] found a mistake in Kravetz's curvature arguments so that problem and consequently also the fixed point problem were reopened. The main result in this paper is that Teichmiiller space does not have negative curvature. This result was announced in [14]. TeichmUller's theorem exhibits a close relationship between extremal quasiconformal maps and quadratic differentials. This in turn leads to the characterization of a geodesic through a point in TeichmUller space as all extremal maps determined by a fixed quadratic differential on the underlying surface. An attempt here is made again to study the geometry of Teichmiiller space this time using the class of geodesic rays determined by quadratic differentials with closed horizontal trajectories. Strebel has studied these particular differentials extensively, and two of his results are crucial for this paper. The first describes how the critical trajectories of a differential with closed trajectories partition the Riemann surface into ringdomains each of which is swept out by freely homotopic closed trajectories, the trajectories in different ringdomains not being freely homotopic. The corresponding






Journal ArticleDOI
TL;DR: In this paper, it was shown that all endomorphisms of a semi-stable abelian variety over Q are rational whenever the variety has a certain endomorphism structure (similar to that of [4, Section 5]).
Abstract: (End Jo(N)) (0 Q of all endomorphisms of Jo(N) is equal to the algebra generated by the Hecke operators, regarded as Q-endomorphisms of J0(N). Our initial proof that this is indeed the case combined P. Deligne's theorem that Jo(N) is semi-stable over Q (in the sense of [6]) with a rather elaborate argument involving l-adic representations. This argument gave a proof that all endomorphisms of a semi-stable abelian variety over Q are rational whenever the variety has a certain endomorphism structure (similar to that of [4, Section 5]). With Shimura's help we were able to simplify the argument and at the same time make it more general. Eventually the principle emerged that all endomorphisms of a semi-stable variety are unramifted. (See (1.1) and (1.3) for precise statements.) Since Q has no unramified extensions, it follows that every endomorphism of a semi-stable abelian variety over Q is rational (i.e., defined over Q). We apply these results to Jo(N) in Section 3 and to certain other modular varieties in Section 4. The applications require an auxiliary result (based on an idea of W. Casselman) concerning the endomorphism algebra of an abelian variety with many real endomorphisms; the result is proved in Section 2.

Journal ArticleDOI
TL;DR: In this article, it was shown that under certain assumptions about the holonomy homomorphism of a foliation, there are strong restrictions on the homomorphisms induced by the bundle projection in real homology and cohomology.
Abstract: Let e = (p, E, M) be a smooth (= C1) bundle. A foliation of e is a foliation i of E where leaves are transverse to the fibres and of complementary dimension. The leaves are required to be smooth, and their tangent planes must vary continuously on M but the foliation may be merely C0. Such foliated bundles occur in several geometrical situations. Suppose a manifold M has an affine connection of zero curvature, i.e., M is a flat manifold; then its tangent sphere bundle has a foliation. Or if a compact Riemannian manifold has negative sectional curvature, then its tangent sphere bundle is foliated. The normal sphere bundle of a leaf of a foliation is foliated. The purpose of this paper is to show that under certain assumptions about the holonomy homomorphism of a foliated bundle, there are strong restrictions on the homomorphisms induced by the bundle projection in real homology and cohomology. In some cases it follows that the bundle has a section. A typical application of our results is:


Journal ArticleDOI
TL;DR: In this article, it was shown that if GCH holds below, then it holds at 8 [Lo2lo]wr for some p < 03, where Wo is an ordinal power; in the rest of this paper, only cardinal exponentiation is used.
Abstract: Silver [7] recently proved that, if GCH holds below , then it holds at 8 [Lo2lo]wr for some p < 03. (Here (wo, is an ordinal power; in the rest of this paper, only cardinal exponentiation is used.) We thank Prikry and Silver for communicating their results to us.



Journal ArticleDOI
TL;DR: Theorem (1.3) of the highest weight theorem was proved in this paper for all discrete series representations of a semisimple Lie group, and the results of this paper are a generalization of the results in this paper.
Abstract: In the theory of irreducible representations of a compact Lie group, the formula for the multiplicity of a weight and the so-called theorem of the highest weight are among the most important results. At least conjecturally, both of these statements have analogues for the discrete series of representations of a semisimple Lie group. Let G be a connected, semisimple Lie group, K c G a maximal compact subgroup, and suppose that rk K = rk G. Exactly in this situation, G has a non-empty discrete series [8]. Blattner's conjecture predicts how a given discrete series representation should break up under the action of K; precise statements can be found in [10], [15], [16]. Formally, the conjectured multiplicity formula looks just like the formula for the multiplicity of a weight. Partial results toward the conjecture have been proved in [10], [15]. More recently, the full conjecture was established for those linear groups G, whose quotient G/K admits a Hermitian symmetric structure [16]. As this paper was being completed, H. Hecht and I succeeded in proving Blattner's conjecture for all linear groups, by extending the arguments of [16]. According to Blattner's conjecture, any particular discrete series representation 7t contains a distinguished irreducible K-module V, with multiplicity one; moreover, w contains no irreducible K-module with a highest weight which is lower, in the appropriate sense, than that of V,:. For "most" discrete series representations, it was known that these two properties characterize at, up to infinitesimal equivalence, among all irreducible representations of G [10], [15]. In this paper, I shall give an infinitesimal characterization, by lowest K-type, for all discrete series representations. The result, which is stated as Theorem (1.3) below, closely resembles the theorem of the highest weight. I shall also draw a number of conclusions from it. The methods of this paper have some further, less immediate consequences, which will be taken up elsewhere. For the remainder of the introduction, I assume that G is a linear group.

Journal ArticleDOI
TL;DR: In this paper, a theorem of strongly closed 2-subgroups of finite groups is presented, where S is a strongly closed subgroup of G if NSg (S) ⊆ S for all g ∈ G.
Abstract: Publisher Summary This chapter describes and presents a theorem of strongly closed 2-subgroups of finite group. S is a strongly closed subgroup of G if NSg (S) ⊆ S for all g ∈ G. If S is a p-group, the condition is equivalent to Sg ⋂ P ⊆ S for all g ∈ G, where S ⊆ P ∈ Sylp (G). The following theorem can be obtained. Suppose the finite group G contains a direct product of two strongly closed 2-subgroups S1 × S2, then