Showing papers in "Duke Mathematical Journal in 1999"

Set-theoretical solutions to the quantum Yang-Baxter equation

Abstract: In 1992 V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions to the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set $X\times X$, where X is a fixed set. In this paper we study such solutions, which in addition satisfy the unitarity and nondegeneracy conditions. We discuss the geometric and algebraic interpretations of such solutions, introduce several constructions of them, and make first steps towards their classification.

The gross-kohnen-zagier theorem in higher dimensions

Abstract: The Gross-Kohnen-Zagier theorem describes Heegner points on a modular curve in terms of coefficients of modular forms. We give another proof of this theorem which generalizes to higher dimensions.

Positivity of dunkl's intertwining operator

Abstract: For a finite reflection group on $\b R^N,$ the associated Dunkl operators are parametrized first-order differential-difference operators which generalize the usual partial derivatives They generate a commutative algebra which is - under weak assumptions - intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials In this paper it is shown that for non-negative parameter values, this intertwining operator is positivity-preserving on polynomials and allows a positive integral representation on certain algebras of analytic functions This result in particular implies that the generalized exponential kernel of the Dunkl transform is positive-definite

A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups

Abstract: 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 1. A Clifford algebra criterion for (ν,Bg) to be of Lie type . . . . . . . . . . . . . . . . . . 455 2. The cubic Dirac operator ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 3. Tensoring with the spin representation and the emergence of d-multiplets . . . 475 4. Multiplets and the kernel of the Dirac operator ! . . . . . . . . . . . . . . . . . . . . . . . . . 483 5. Infinitesimal character values on multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 6. Multiplets and topological K-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

Holder Regularity and Dimension Bounds for Random Curves

Abstract: Random systems of curves exhibiting fluctuating features on arbitrarily small scales (δ) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply that typically all the realized curves admit Holder continuous parametrizations with a common exponent and a common random prefactor, which in the scaling limit (δ → 0) remains stochastically bounded. The regularity is used for the construction of scaling limits, formulated in terms of probability measures on the space of closed sets of curves. Under the hypotheses presented here the limiting measures are supported on sets of curves which are Holder continuous but not rectifiable, and have Hausdorff dimensions strictly greater than one. The hypotheses are known to be satisfied in certain two dimensional percolation models. Other potential applications are also mentioned.

On the decomposition matrices of the quantized Schur algebra

Abstract: We prove the decomposition conjecture of Leclerc and Thibon for the Schur algebra. We also give a new approach to the Lusztig conjecture for the dimension of the simple U(sl_k)-modules at roots of unity via canonical bases of the Hall algebra.

A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements

Abstract: 1. Introduction. Imagine a collection of books labeled 1 through n arranged in a row in some order. We reorganize the row of books by successively choosing a book at random: choosing book i with probability w i and moving it to the front of the row. This \" move-to-front rule \" determines an interesting Markov chain on the set of arrangements of the books. If σ and τ denote any two orderings of the books, then the probability of transition from σ to τ is w i if and only if τ is obtained from σ by moving book i to the front. This Markov chain is commonly called the Tsetlin library or move-to-front scheme. Due to its use in computer science as a standard scheme for dynamic file maintenance as well as cache maintenance (cf. [Do], [FHo], and [P]), the move-to-front rule is a very well-studied Markov chain. A primary resource for this problem is Fill's comprehensive paper [F], which derives the transition probabilities for any number of steps of the chain and the eigenvalues with corresponding idempotents and discusses the rate of convergence to stationarity. Its thorough bibliography contains a wealth of pointers to the relevant literature. Of particular interest is the spectrum of this Markov chain. In general, knowledge of the eigenvalues for the transition matrix of a Markov chain can give some indication of the rate at which the chain converges to its equilibrium distribution. In the case of the Tsetlin library, the eigenvalues have an elegant formula, discovered (independently) Theorem 1.1. The distinct eigenvalues for the move-to-front rule are indexed by subsets A ⊆ {1,. .. , n} and given by λ A = i∈A w i. The multiplicity of λ A is the number of derangements (permutations with no fixed points) of the set {1,. .. , n − |A|} In this paper we study a sequence of generalizations of the Tsetlin library, culminating in a generalization of the setting of central hyperplane arrangements. In each case we develop a formula analogous to Theorem 1.1 for the distinct eigenvalues and multiplicities for this more general class of Markov chains. Our first generalization comes from viewing move-to-front as the operation of moving the books in the subset {i} to the front and then moving the subset [n] − {i} behind {i} while retaining their relative order; this is all done with probability w i …

Kostant polynomials and the cohomology ring for $G/B$

Abstract: The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1–26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the Weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the Bruhat order than w. The main theorem given here is an explicit formula for these values. The matrix of orbit values can be used to determine the cup product for the cohomology ring for G/B, using only linear algebra or as described by Lascoux and Schutzenberger [Lascoux, A. & Schutzenberger, M.-P. (1982) C. R. Seances Acad. Sci. Ser. A 294, 447–450]. Complete proofs of all the theorems will appear in a forthcoming paper.

The Yamabe problem on manifolds with boundary: Existence and compactness results

Abstract: §0. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 §1. Reductions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 §2. The proof of Proposition 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 §3. The proof of Proposition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 §4. The proof of Proposition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 §5. The proof of Theorems 0.1 and 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 §6. The proof of Theorem 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

Counting rational curves on k3 surfaces

Abstract: The aim of these notes is to explain the remarkable formula found by Yau and Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (Fg)g≥1 ; a surface in Fg admits a gdimensional linear system of curves of genus g . A naive count of constants suggests that such a system will contain a positive number, say n(g) , of rational (highly singular) curves. The formula is∑

Constant scalar curvature metrics with isolated singularities

Abstract: We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen [12], but the proof we give here, based on the techniques of [6], is more direct, and provides more information about their geometry. When Λ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in [7] and [8]

Blow-up results for nonlinear parabolic equations on manifolds

Abstract: 1. Introduction. The aim of this paper is threefold. First, by a unified approach, we prove that several classical blow-up results obtained over the last three decades for semilinear and quasilinear parabolic problems in R n are valid on noncompact, complete Riemannian manifolds, which include those with nonnegative Ricci curvatures. Next, we remove some unnecessary a priori growth conditions on solutions of the quasilinear case, which are assumed in the existing literature. Finally, we demonstrate a new critical phenomenon for some inhomogeneous, quasilinear parabolic equations. We also hope that this paper serves as a link for the many other papers on this subject, which lie scattered in several journals over a period of three decades. Specifically, we study the blow-up properties of the following homogeneous and inhomogeneous, semilinear parabolic equations and of the porous medium equations with nonlinear source: u − ∂ t u + V (x)u p = 0 in M n × (0, ∞), u(x, 0) = u 0 (x) in M n , u ≥ 0, (1.1)

L2-boundedness of the Cauchy integral operator for continuous measures

Abstract: 1. Introduction. Let µ be a continuous (i.e., without atoms) positive Radon measure on the complex plane. The truncated Cauchy integral of a compactly supported function f in L p (µ), 1 ≤ p ≤ +∞, is defined by Ꮿ ε f (z) = |ξ −z|>ε f (ξ) ξ − z dµ(ξ), z ∈ C, ε > 0. In this paper, we consider the problem of describing in geometric terms those measures µ for which |Ꮿ ε f | 2 dµ ≤ C |f | 2 dµ, (1) for all (compactly supported) functions f ∈ L 2 (µ) and some constant C independent of ε > 0. If (1) holds, then we say, following David and Semmes [DS2, pp. 7–8], that the Cauchy integral is bounded on L 2 (µ). A special instance to which classical methods apply occurs when µ satisfies the doubling condition µ(2) ≤ Cµ((), for all discs centered at some point of spt(µ), where 2 is the disc concentric with of double radius. In this case, standard Calderón-Zygmund theory shows that (1) is equivalent to Ꮿ * f 2 dµ ≤ C |f | 2 dµ, (2) where Ꮿ * f (z) = sup ε>0 |Ꮿ ε f (z)|. If, moreover, one can find a dense subset of L 2 (µ) for which Ꮿf (z) = lim ε→0 Ꮿ ε f (z) (3) 269 270 XAVIER TOLSA exists a.e. (µ) (i.e., almost everywhere with respect to µ), then (2) implies the a.e. (µ) existence of (3), for any f ∈ L 2 (µ), and |Ꮿf | 2 dµ ≤ C |f | 2 dµ, for any function f ∈ L 2 (µ) and some constant C. For a general µ, we do not know if the limit in (3) exists for f ∈ L 2 (µ) and almost all (µ) z ∈ C. This is why we emphasize the role of the truncated operators Ꮿ ε. Proving (1) for particular choices of µ has been a relevant theme in classical analysis in the last thirty years. Calderón's paper [Ca] is devoted to the proof of (1) when µ is the arc length on a Lipschitz graph with small Lipschitz constant. The result for a general Lipschitz graph was obtained by Coifman, McIntosh, and Meyer in 1982 in the celebrated paper [CMM]. The rectifiable curves , for which (1) holds for the arc length measure µ on …

Canonical periods and congruence formulae

Abstract: The purpose of this article is to show how congruences between the Fourier coefficients of Hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated L-functions. This study was initiated by B. Mazur in his fundamental work on the Eisenstein ideal (see [Maz77] and [Maz79]) where it was made clear that congruences for analytic L-values were closely related to the integral structure of certain Hecke rings and cohomology groups. The results of [Maz79] also showed that congruences were useful in the study of nonvanishing of L-functions. This idea was then further developed by Stevens [Ste82] and Rubin-Wiles [RW82]. The work of Rubin and Wiles, in particular, used congruences to study the behavior of elliptic curves in towers of cyclotomic fields. A key ingredient here was a theorem of Washington, which states, roughly, that almost L-values in certain families are nonzero modulo p. This theme has recently been taken up again, in the work of Ono-Skinner [OSa], [OSb], James [Jam], and Kohnen [Koh97]. While the earlier history was primarily concerned with cyclotomic twists, the current emphasis is on families of twists by quadratic characters. Here one wants quantitative estimates for the number of quadratic twists of a given modular form, which have nonvanishing L-function at s = 1. We continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three. This generalizes a beautiful example due to Kevin James, and provides new evidence for a conjecture of Goldfeld [Gol79]. It should, however, be pointed out that even the study of quadratic twists may be traced back to Mazur: the reader is urged to look at pages 212–213 of [Maz79], and especially at the footnote at the bottom of page 213. The theorems of Davenport-Heilbronn [DH71] and Washington [Was78], which are crucial in this paper, are both mentioned in Mazur’s article. We want to begin by discussing the congruences that lie at the heart of this article. Thus let f = ∑ anq n be an elliptic modular cuspform of level M and weight k ≥ 2. Assume that f is a simultaneous eigenform for all the Hecke operators and that a1(f) = 1. The L-function associated to f is defined by the Dirichlet series L(s, f) = ∑ ann −s, which converges for the real part of s sufficiently large, and has analytic continuation to s ∈ C. A fundamental theorem of Shimura [Shi76] states that L(s, f) enjoys the following algebraicity property:

An Algebraic Characterization of the Affine Canonical Basis

Abstract: The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a lattice over $\bz[q^{-1}]$. This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of $\cl$. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the ``imaginary'' subalgebra which is fixed by the affine braid group. Once the basis is found we construct a PBW-type basis whose $\bz[q^{-1}]$-span reduces to a ``crystal'' basis at $q=\infty,$ with the imaginary component given by the Schur functions.

Internal lifshits tails for random perturbations of periodic schrodinger operators

Abstract: 0. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 1. Some preliminary considerations on periodic Schrödinger operators . . . . . . . . 340 1.1. The Floquet decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 1.2. Wannier basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 1.3. The density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 2. Lifshits tails behavior for random Schrödinger operators . . . . . . . . . . . . . . . . . . 343 2.1. The main theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 2.2. Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 3. The proof of Proposition 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 4. A reduction procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4.1. The reference operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4.2. The reduction to a discrete problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 5. The proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 5.1. The periodic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 5.2. Cutoff in energy for the periodic approximations . . . . . . . . . . . . . . . . . . . . . 362 6. The lower bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 7. The upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 7.1. Case (2) of assumption (H.2bis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 7.2. Case (1) of assumption (H.2bis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

Amenability, bilipschitz equivalence, and the von Neumann conjecture

Abstract: We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups theory which show that the sign of the Euler characteristic is not a coarse invariant. Finally we get some general results on uniformly we finite homology which we will apply to manifolds in a later paper. In his fundamental paper on amenability, Von Neumann conjectured that if Γ is a finitely generated, non-amenable group, then Γ has a subgroup which is free on two generators. While this conjecture proved to be false in general [O], it is true for many classes of groups. Given the recent interest in studying groups via geometric methods, it seems natural to ask for a geometric version of the conjecture. If Γ does contain a free subgroup then its cosets will partition Γ into copies of the free group. The free group is, geometrically, the regular 4-valent tree. Our \" geometric Von Neumann conjecture \" , is: Theorem 1 If X is a uniformly discrete space of bounded geometry (in particular, for X a finitely generated group or a net in a leaf of a foliation of a compact manifold), X is non-amenable iff it admits a partition with pieces uniformly bilipschitz equivalent to the regular 4-valent tree. The proof depends on constructing a bilipschitz eqivalence of X × {0, 1} and X near the projection map. Our main technical result is a general 1

Bloch invariants of hyperbolic 3-manifolds

Abstract: We define an invariant \beta(M) of a finite volume hyperbolic 3-manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. \beta(M) lies in a subgroup of \B(\C) of finite \Q-rank determined by the invariant trace field of M. Moreover, the Chern-Simons invariant of M is determined modulo rationals by \beta(M). This leads to a simplicial formula and rationality results for the Chern Simons invariant which appear elsewhere. Generalizations of \beta(M) are also described, as well as several interesting examples. An appendix describes a scissors congruence interpretation of B(C).

Motivic exponential integrals and a Motivic Thom-Sebastiani theorem

Abstract: We introduce motivic analogues of p-adic exponential integrals. We prove a basic multiplicativity property from which we deduce a motivic analogue of the Thom-Sebastiani Theorem. In particular, we obtain a new proof of the Thom-Sebastiani Theorem for the Hodge spectrum of (non isolated) singularities of functions.

Resolving mixed Hodge modules on configuration spaces

Abstract: Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of f. The construction is reminiscent of techniques from the theory of hyperplane arrangements, and relies on Arnold's calculation of the cohomology of the configuration space of the complex line. This resolution is S_n-equivariant. We apply it to the universal elliptic curve with complete level structure of level N>=3 over the modular curve Y(N), obtaining a formula for the S_n-equivariant Serre polynomial (Euler characteristic of H^*_c(V,Q) in the Grothendieck group of the category of mixed Hodge structures) of the moduli space M_{1,n}. In a sequel to this paper, this is applied in the calculation of the S_n-equivariant Hodge polynomial of the compactication \bar{M}_{1,n}.

Quasimodes and resonances: Sharp lower bounds

Abstract: We prove that, asymptotically, any cluster of quasimodes close to each other approximates at least the same number of resonances, counting multiplicities. As a consequence, we get that the counting function of the number of resonances close to the real axis is bounded from below essentially by the counting function of the quasimodes.

On the $\eta$-invariant of certain nonlocal boundary value problems

Abstract: Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formula for the eta invariant. Our class of boundary conditions contains as special cases the usual (nonlocal) Atiyah-Patodi-Singer boundary value problems as well as the (local) relative and absolute boundary conditions for the Gauss-Bonnet operator.

On quantum cohomology rings of partial flag varieties

Abstract: 0. Introduction. The main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces SLn(C)/P , where P is a parabolic subgroup. The quantum cohomology ring of a smooth projective variety, or more generally of a symplectic manifoldX, has been introduced by string theorists (see [Va] and [W1]). Roughly speaking, it is a deformation of the usual cohomology ring, with parameter space given byH ∗(X). The multiplicative structure of quantum cohomology encodes the enumerative geometry of rational curves on X. In the past few years, the highly nontrivial task of giving a rigorous mathematical treatment of quantum cohomology has been accomplished both in the realm of algebraic and symplectic geometry. In various degrees of generality, this can be found in [Beh], [BehM], [KM], [LiT1], [LiT2], [McS], and [RT], as well as in the surveys [FP] and [T]. If one restricts the parameter space to H 1,1(X), one gets the small quantum cohomology ring. This ring, in the case of partial flag varieties, is the object of the present paper. In order to state our main results, we first describe briefly the “classical” side of the story. We interpret the homogeneous space F := SLn(C)/P as the complex, projective variety, parametrizing flags of quotients of C of given ranks, say, nk > · · ·> n1. By a classical result of Ehresmann [E], the integral cohomology of F can be described geometrically as the free abelian group generated by the Schubert classes. These are the (Poincare duals of) fundamental classes of certain subvarieties w ⊂ F , one for each element of the subset S := S(n1, . . . ,nk) of the symmetric group Sn, consisting of permutations w with descents in {n1, . . . ,nk}. A description of the multiplicative structure is provided by yet another classical theorem, due to Borel [Bor], which gives a presentation forH ∗(F,Z). Specifically, let σ 1 1 , . . . ,σ 1 n1 ,σ 2 1 , . . . ,σ 2 n2−n1, . . . ,σ k+1 1 , . . . ,σ k+1 n−nk be n independent variables. Define An to be the block-diagonal matrix diag(D1,D2, . . . ,Dk+1), where

Volume growth, Green’s functions, and parabolicity of ends

Abstract: §

The analytic rank of J0(q) and zeros of automorphic L-functions

Abstract: 1. Introduction. This paper is motivated by the conjecture of Birch and Swinnerton-Dyer relating the rank of the Mordell-Weil group of an abelian variety defined over a number field with (in its crudest form) the order of vanishing of its Hasse-Weil L-function at the central critical point. Mestre [Mes] began the study of the implications of this conjecture towards providing upper bounds for the rank. He used \" explicit formulae \" similar to that of Riemann-Weil and assumed the analytic continuation and (perhaps more significantly) the Riemann hypothesis for those L-functions. Brumer [Br1] first studied the special case of the Jacobian variety J 0 (q) of the modular curve X 0 (q). This is an abelian variety defined over Q of dimension about q/12. Here analytic continuation is known, by the work of Eichler and Shimura [Sh1]. Assuming only the Riemann hypothesis for the L-functions of automorphic forms (of weight 2 and level q), Brumer proved rank a J 0 (q) 3 2 + o(1) dim J 0 (q) and conjectured that rank J 0 (q) = rank a J 0 (q) ∼ 1 2 dim J 0 (q) (based on the fact that the sign of the functional equation for the automorphic L-functions of weight 2 and level q is approximately half the time +1 and half the time −1). Other authors, notably Murty [Mur] (who first applied the Petersson formula in this context), considered the same problem. Most recently, Luo, Iwaniec, and Sar-nak [LIS], using the same assumptions, proved an estimate rank a J 0 (q) c + o(1) dim J 0 (q) for some (explicit) constant c < 1. This turns out to be quite significant in light of the general conjectures of Katz and Sarnak [KaS] on the distribution of zeros of families of L-functions. This paper approaches the same problem with a different emphasis: we wish to avoid all assumptions about the L-functions involved and obtain a bound of the correct order of magnitude. Indeed, we prove the following theorem. Theorem 1. There exists an absolute and effective constant C > 0 such that for any prime number q, rank a J 0 (q) C dim J 0 (q). If the Birch and Swinnerton-Dyer conjecture holds for J 0 (q), then rank J 0 (q) C dim J 0 (q). This theorem provides the first known unconditional bound for the analytic rank of a family …

The Frobenius and monodromy operators for curves and abelian varieties

Abstract: In this paper, we give explicit descriptions of Hyodo and Kato's Frobenius and Monodromy operators on the first $p$-adic de Rham cohomology groups of curves and Abelian varieties with semi-stable reduction over local fields of mixed characteristic. This paper was motivated by the first author's paper "A $p$-adic Shimura isomorphism and periods of modular forms," where conjectural definitions of these operators for curves with semi-stable reduction were given.

The distribution of spacings between quadratic residues

Abstract: We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.