The Global Mckay-Ruan Correspondence Via Motivic Integration

The purpose of this paper is to show how the methods of motivicintegration of Kontsevich, Denef–Loeser (Invent. Math.135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290)and Looijenga (Astérisque 276 (2002) 267–297) canbe adapted to prove the McKay–Ruan correspondence, a generalizationof the McKay–Reid correspondence to orbifolds that arenot necessarily global quotients. 2000 Mathematics Subject Classification14A20, 14E15, 14F43.     

Zariski Pairs of Index 19 and Mordell-Weil Groups of K3 Surfaces

We find Zariski pairs of sextics with simple singularities havingmaximal total Milnornumber. We also relate them to the existenceof distinct Mordell–Weil groups of extremal elliptic K3surfaces with a fixed set of semistable singular fibres. 1991Mathematics Subject Classification: 14F45, 14F27, 14F28.     

Q-Curves and Abelian Varieties of GL2-Type

The relation between Q-curves and certain abelian varietiesof GL₂-type was established by Ribet (‘Abelian varietiesover Q and modular forms’, Proceedings of the KAIST MathematicsWorkshop (1992) 53–79) and generalized to building blocks,the higher-dimensional analogues of Q-curves, by Pyle in herPhD Thesis (University of California at Berkeley, 1995). Inthis paper we investigate some aspects of Q-curves with no complexmultiplication and the corresponding abelian varieties of GL₂-type,for which we mainly use the ideas and techniques introducedby Ribet (op. cit. and ‘Fields of definition of abelianvarieties with real multiplication’, Contemp.\ Math. 174(1994) 107–118). After the Introduction, in Sections 2and 3 we obtain a characterization of the fields where a Q-curveand all the isogenies between its Galois conjugates can be definedup to isogeny, and we apply it to certain fields of type (2,...,2).In Section 4 we determine the endomorphism algebras of all theabelian varieties of GL₂-type having as a quotient a given Q-curvein easily computable terms. Section 5 is devoted to a particularcase of Weil's restriction of scalars functor applied to a Q-curve,in which the resulting abelian variety factors over Q up toisogeny as a product of abelian varieties of GL₂-type. Finally,Section 6 contains examples: we parametrize the Q-curves comingfrom rational points of the modular curves X*N having genuszero, and we apply the results of Sections 2–5 to someof the curves obtained. We also give results concerning theexistence of quadratic Q-curves. 1991 Mathematics Subject Classification:primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.     

Holomorphic Almost Modular Forms

Holomorphic almost modular forms are holomorphic functions ofthe complex upper half plane that can be approximated arbitrarilywell (in a suitable sense) by modular forms of congruence subgroupsof large index in SL(2,Z). It is proved that such functionshave a rotation-invariant limit distribution when the argumentapproaches the real axis. An example of a holomorphic almostmodular form is the logarithm of . The paper is motivated by the author's previous studies [Int.Math. Res. Not. 39 (2003) 2131–2151] on the connectionbetween almost modular functions and the distribution of thesequence n²x modulo one. 2000 Mathematics Subject Classification11F11 (primary), 11F06, 11J71 (secondary).     

On the Hecke Equivariance of the Borcherds Isomorphism

The Borcherds isomorphism is proved to be Hecke equivariantif one considers multiplicative Hecke operators acting on theintegral weight meromorphic modular forms. This answers a partof a question of Borcherds (see ‘Automorphic forms onO_{s+2, 2}(R) and infinite products’, Invent. Math. 120 (1995)161–213, 17.10), using his suggestion to define the multiplicativeHecke operators. 2000 Mathematics Subject Classification 11F37.     

Another Proof for the Presentation of the Quantum Cohomology of the Moduli of Bundles over a Riemann Surface

The presentation of the quantum cohomology of the moduli spaceof stable vector bundles of rank two and odd degree with fixeddeterminant over a Riemann surface of genus g > 2 is obtained.The argument avoids the use of gauge theory, providing an alternativeproof to that given by the author in Duke Math. J. 98 (1999)525–540. 2000 Mathematics Subject Classification 14N35(primary); 14H60, 53D45 (secondary).     

Un Theoreme De Finitude Dans Le Spectre Automorphe Pour Les Formes Interieures De GLn Sur Un Corps Global

A proof is given to show that for an inner form of GL_n overa global field of zero characteristic, there exist only a finitenumber of automorphic representations with fixed local factor(up to equivalence) at almost every place. What is new in comparisonto earlier work (see A. I. Badulescu and P. Broussous, ‘Unthéorème de finitude’, Compositio Math.132 (2002) 177–190) is the case when the local factorsare not fixed at the infinite places, as well as the statementof the result for the automorphic spectrum, rather than thecuspidal one. 2000 Mathematics Subject Classification 11F70.     

Non-Complex Symplectic 4-Manifolds with Formula

Simply connected closed symplectic 4-manifolds with and K² = 0 are investigated. As aresult, it is confirmed that most of homotopy elliptic surfaces{E(1)_k|K is a fibred knot in S³} constructed by R. Fintusheland R. Stern in Invent. Math. 134 (1998) 363–400 are simplyconnected closed minimal symplectic 4-manifolds that do notadmit a complex structure. 2000 Mathematics Subject Classification57R17, 57R57 (primary), 14J26 (secondary).     

Equivalences of Triangulated Categories and Fourier-Mukai Transforms

We give a condition for an exact functor between triangulatedcategories to be an equivalence. Applications to Fourier–Mukaitransforms are discussed. In particular, we obtain a large numberof such transforms for K3 surfaces. 1991 Mathematics SubjectClassification 18E30, 14J28.     

Deformation of Complete Intersections in the Plane

We study deformations of zero-dimensional complete intersectionsin the plane, and prove the following results. (1) Two complexnon-singular curves intersecting at r points with multiplicitiesd₁,...,d_r can be deformed into curves intersecting (at somepoints) with multiplicities d'₁,...,d'_s which are arbitraryprescribed partitions of the numbers d₁,...,d_r. (2) Two realcurves intersecting with multiplicity at most 2 at each of theirreal common points can be deformed so that all real multipleintersection points split into real simple intersection points.1991 Mathematics Subject Classification 14M10, 14P05.     

Maximal Functions Associated with The Jacobi Transform

In this paper, we define a Hardy–Littlewood maximal functionoperator M associated with the Jacobi transform, and prove thatM is of weak (1, 1) and strong (p, p) (p > 1) type. 1991Mathematics Subject Classification 42B25, 44A15.     

On Atkin-Lehner Quotients of Curves

We study the erednik–Drinfeld p-adic uniformization ofcertain Atkin–Lehner quotients of Shimura curves overQ. We use it to determine over which local fields they haverational points and divisors of a given degree. Using a criterionof Poonen and Stoll, we show that the Shafarevich–Tategroup of their jacobians is not of square order for infinitelymany cases. 1991 Mathematics Subject Classification 11G18, 11G20,14G20, 14G35, 14H40.     

A Characterization of Strongly Continuous Groups of Linear Operators on a Hilbert Space

It is proved that the infinitesimal generator A of a stronglycontinuous semigroup of linear operators on a Hilbert spacealso generates a strongly continuous group if and only if theresolvent of –A, ( + A)^–1, is also a bounded functionon some right-hand-side half plane of complex numbers, and convergesstrongly to zero as the real part of tends to infinity. Anapplication to a partial differential equation is given. 1991Mathematics Subject Classification 47D03.     

Morita Equivalence for Locally C*-Algebras

The concept of Morita equivalence is generalized to the contextof locally C^*-algebras. This generalizes a well-known theoremof Brown, Green and Rieffel, Pacific J. Math. 71 (1977) 349–363.2000 Mathematics Subject Classification 46L08, 46L05.     

On Manifolds Whose Tangent Bundle is Big and 1-Ample

A line bundle over a complex projective variety is called bigand 1-ample if a large multiple of it is generated by globalsections and a morphism induced by the evaluation of the spanningsections is generically finite and has at most 1-dimensionalfibers. A vector bundle is called big and 1-ample if the relativehyperplane line bundle over its projectivisation is big and1-ample. The main theorem of the present paper asserts that any complexprojective manifold of dimension 4 or more, whose tangent bundleis big and 1-ample, is equal either to a projective space orto a smooth quadric. Since big and 1-ample bundles are ‘almost’ample, the present result is yet another extension of the celebratedMori paper ‘Projective manifolds with ample tangent bundles’(Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractionsof complex symplectic manifolds and of manifolds whose tangentbundles are numerically effective. In the appendix we re-provethese results. 2000 Mathematics Subject Classification 14E30,14J40, 14J45, 14J50.     

Tameness and complexity of finite group schemes

Using a representation-theoretic interpretation of support varietiesdue to Friedlander and Pevtsova (Amer. J. Math. 127 (2005) 379–420;Erratum Amer. J. Math. 128 (2006) 1067–1068, we show thatthe complexity of tame blocks of finite group schemes is boundedby 2. In this context, our result salvages a theorem by Rickard(Bull. London Math. Soc. 22 (1990) 540–546), the proofof which is flawed.     

Descent Calculations for the Elliptic Curves of Conductor 11

Let A be any one of the three elliptic curves over Q with conductor11. We show that A has Mordell–Weil rank zero over itsfield of 5-division points. In each case we also compute the5-primary part of the Tate–Shafarevich group. Our calculationsmake use of the Galois equivariance of the Cassels–Tatepairing. 2000 Mathematics Subject Classification 11G05, 11Y40,11R23.     

The rationality of vector valued modular forms associated with the Weil representation

 In a recent paper [Duke Math. J., 97, 219–233], Borcherds asks whether or not the spaces of vector valued modular forms associated to the Weil representation have bases of modular forms whose Fourier expansions have only integer coefficients. We give an affirmative answer to Borcherds' question. This strengthens and simplifies Borcherds' main theorem which is a generalization of a theorem of Gross, Kohnen, and Zagier [Math. Ann., 278, 497–562]. Received: 27 September 2001 / Revised version: 22 July 2002 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 11F30; 11F27     

Topologie Des (M - 2)-Courbes Reelles Symetriques

Let X be a non-singular real algebraic curve in CP² of evendegree. In this paper a refinement is proved of a theorem ofKharlamov about (M – 2)-curves that are invariants underthe projective involution. In particular, if the (M –2)-symmetric curve X satisfies the Arnold congruence, then eitherX or its twin is a separating curve. 2000 Mathematics SubjectClassification 14P25.