Showing papers in "Crelle's Journal in 2000"
312 citations
241 citations
TL;DR: In this article, the authors obtained a simple formula for 4F3(1)p, where p is the trivial character modulo p and φp is the Legendre symbol modulo φ p. For n > 2 the non-trivial values of n+1Fn(x)p have been difficult to obtain.
Abstract: If p is prime, then let φp denote the Legendre symbol modulo p and let p be the trivial character modulo p. As usual, let n+1Fn(x)p := n+1Fn „ φp, φp, . . . , φp p, . . . , p | x « p be the Gaussian hypergeometric series over Fp. For n > 2 the non-trivial values of n+1Fn(x)p have been difficult to obtain. Here we take the first step by obtaining a simple formula for 4F3(1)p. As a corollary we obtain a result describing the distribution of traces of Frobenius for certain families of elliptic curves. We also find that 4F3(1)p satisfies surprising congruences modulo 32 and 11. We then establish a mod p2 “supercongruence” between Apery numbers and the coefficients of a certain eta-product; this relationship was conjectured by Beukers in 1987. Finally, we obtain many new mod p congruences for generalized Apery numbers.
167 citations
111 citations
TL;DR: In this paper, the subgroups of the mapping class groups of Riemann surfaces, called "geometric" subgroups, corresponding to the inclusion of subsurfaces, are studied.
Abstract: This paper is a study of the subgroups of the mapping class groups of Riemann surfaces, called "geometric" subgroups, corresponding to the inclusion of subsurfaces. Our analysis includes surfaces with boundary and with punctures. The centres of all the mapping class groups are calculated. We determine the kernel of inclusion-induced maps of the mapping class group of a subsurface, and give necessary and sufficient conditions for injectivity. In the injective case, we show that the commensurability class of a geometric subgroup completely determines up to isotopy the defining subsurface, and we characterize centralizers, normalizers, and commensurators of geometric subgroups.
102 citations
TL;DR: In this article, a sheaf homology theory for etale groupoids is introduced and proved invariance under Morita equivalence as well as Verdier duality between Haeiger cohomology and this homology.
Abstract: Etale groupoids arise naturally as models for leaf spaces of foliations for orbifolds and for orbit spaces of discrete group actions In this paper we introduce a sheaf homology theory for etale groupoids We prove its invariance under Morita equivalence as well as Verdier duality between Haeiger cohomology and this homology We also discuss the relation to the cyclic and Hochschild homologies of Connes convolution algebra of the groupoid and derive some spectral sequences which serve as a tool for the computation of these homologies
96 citations
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TL;DR: In this article, the authors study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed generalized linear subspaces of H.
Abstract: We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H. As an application, we derive recursive formulas for the number of such curves when the number is finite. These recursive formulas require as ``seed data'' only one input: there is one line in P^1 through two points. These numbers can be seen as top intersection products of various cycles on the Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give information about the Chow ring (and hence the topology) of these objects. The formula can also be interpreted as an equality in the Chow ring (not necessarily at the top level) of the appropriate Hilbert scheme or space of stable maps. In particular, this gives an algorithm for counting rational and elliptic curves in P^n intersecting various fixed general linear spaces. (The genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for n=3.)
87 citations
TL;DR: In this paper, a new category of "rigid spaces with overconvergent structure sheaf" called dagger spaces is introduced, which is the correct category in which de Rham cohomology in rigid analysis should be studied.
Abstract: We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces --- this is the correct category in which de Rham cohomology in rigid analysis should be studied. We compare it with the (usual) category of rigid spaces, give Serre and Poincare duality theorems and explain the relation with Berthelot's rigid cohomology.
TL;DR: In this article, the authors propose a modular $L$-functions with mollification and nonvanishing results for the central value of high derivatives of a Jacobian variety.
Abstract: Keywords: modular $L$-functions ; mollification ; nonvanishing ; results ; central value of high derivatives ; analytic ; rank ; Jacobian variety Reference TAN-ARTICLE-2000-004doi:10.1515/crll.2000.074 Record created on 2008-11-14, modified on 2017-05-12
TL;DR: In this article, the authors prove the existence of classical solutions for a Bernoulli-type free boundary problem, with the p-Laplacian as the governing operator under convexity assumptions for the data.
Abstract: In this paper we prove, under convexity assumptions for the data, the existence of classical solutions for a Bernoulli-type free boundary problem, with the p-Laplacian as the governing operator. The method employed here originates from a pioneering work of A. Beurling where he proves the existence for the harmonic case in the plane, though with no geometrical restrictions.
TL;DR: In this paper, the Yamabe invariant of any simply connected smooth manifold of dimension n greater than four is shown to be non-negative, and the infimum of the L n/2 norm of the scalar curvature over the space of all Riemannian metrics on the manifold is zero.
Abstract: We prove that the Yamabe invariant of any simply connected smooth manifold of dimension n greater than four is non-negative. Equivalently that the infimum of the L^{n/2} norm of the scalar curvature, over the space of all Riemannian metrics on the manifold, is zero.
TL;DR: In this article, a new kind of non-holomorphic Eisenstein series, first introduced by Goldfeld, is studied and a functional equation is obtained relating the values at s to those at 1−s.
Abstract: In this work a new kind of non-holomorphic Eisenstein series, first introduced by Goldfeld, is studied. For an arbitrary Fuchsian group of the first kind we fix a holomorphic cusp form and consider Eisenstein series constructed with the modular symbol associated with this cusp form. We develop the theory analogously with that of the usual Eisenstein series starting with its meromorphic continuation to the entire complex plane. A functional equation is then obtained relating the values at s to those at 1− s. Introduction Let H = {z ∈ C : Im z > 0} be the upper half plane and let Γ ⊂ SL2(R) be a fixed non co-compact Fuchsian group of the first kind, (for example Γ(N), Γ0(N)), acting on H. For simplicity assume that Γ has a unique cusp at infinity with stability group Γ∞ = { ± ( 1 m 0 1 ) ,m ∈ Z } . For each γ in Γ we shall label its matrix elements ( γa γb γc γd ) . Let f(z) be an element of S2(Γ), the space holomorphic cusp forms of weight 2 for Γ. Following [Go] we define a modified Eisenstein series (0.1) E∗(z, s) = ∑ γ∈Γ∞\Γ 〈 γ, f 〉Im(γz)s, z ∈ H, where for γ ∈ Γ the modular symbol is given by 〈 γ, f 〉 = −2πi ∫ γw w f(τ) dτ, the definition being independent of w ∈ H. Note that since 〈 γ1γ2, f 〉 = 〈 γ1, f 〉+ 〈 γ2, f 〉 the series is not automorphic. The transformation rule is E∗(γz, s) = E∗(z, s)− 〈 γ, f 〉E(z, s), for all γ ∈ Γ where E(z, s) is the usual Eisenstein series for Γ. This new type of non-holomorphic Eisenstein series was introduced by Goldfeld in order to study the distribution properties of modular symbols 〈 γ, f 〉 as γ ranges over the group Γ. The series (0.1) converges for Re(s) > 2 and Goldfeld hypothesised that it should have an analytic continuation and a functional equation. In this paper Selberg’s method, (described in [Iw], [He]), is extended to establish the following results. Typeset by AMS-TEX 1
TL;DR: The main purpose of as mentioned in this paper is the generalization of the well-known Eichler-Shimura congruence relations for modular curves to Shimura varieties of PEL type whose Shimura group G is split over a fixed prime p.
Abstract: The main purpose of this paper is the generalization of the well-known Eichler-Shimura congruence relations for modular curves to Shimura varieties of PELtype whose Shimura group G is split over a fixed prime p. In particular we prove a conjecture of Blasius and Rogawski for these Shimura varieties. Further we give a formula for the cycle which is the reduction to positive characteristic of the Hecke operator Tp in terms of the root data of G and calculate this cycle in two examples. MSC Classification: 14K10 14L05, 20G25