On the Non-Vanishing of Elliptic Curve L-Functions at the Central Point

We show that a large number of elliptic curve L-functions donot vanish at the central point, conditionally on the GeneralizedRiemann Hypothesis and on an assumption on the regular distributionof the root number. 2000 Mathematics Subject Classification11G05, 11G40, 11M41.     

Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number S_n G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationx^m=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof S_n G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function S_n G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary).     

The Weierstrass Subgroup of a Curve has Maximal Rank

We show that the Weierstrass points of the generic curve ofgenus g over an algebraically closed field of characteristic0 generate a group of maximal rank in the Jacobian. 2000 MathematicsSubject Classification 11G30, 14H40, 14H10, 14H55, 14Q05.     

Mackey formula in type A

Let G be a connected reductive group defined over a finite fieldwith q elements and let F: G G be the corresponding Frobeniusendomorphism. We prove that the Mackey formula for Lusztig inductionand restriction holds in G^F if all the simple components ofG are of type A. Our result holds for every Frobenius endomorphism(split or non-split) and for every q. 1991 Mathematics SubjectClassification: 20G05, 20G40.     

Concealed-Canonical Algebras and Separating Tubular Families

We characterise those concealed-canonical algebras which ariseas endomorphism rings of tilting modules, all of whose indecomposablesummands have strictly positive rank, as those artin algebraswhose module categories have a separating exact subcategory(that is, a separating tubular family of standard tubes). This paper develops further the technique of shift automorphismswhich arises from the tubular structure. It is related to the characterisation of hereditary noetheriancategories with a tilting object as the categories of coherentsheaves on a weighted projective line. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.     

Modular Branching Rules and the Mullineux Map for Hecke Algebras of Type A

We prove the quantum version - for Hecke algebras H A_n of typeA at roots of unity - of Kleshchev's modular branching rulefor symmetric groups. This result describes the socle of therestriction of an irreducible H A_n-module to the subalgebraH A_n–1. As a consequence, we describe the quantum versionof the Mullineux involution describing the irreducible moduleobtained on twisting an irreducible module with the sign representation.1991 Mathematics Subject Classification: 20C05, 20G05.     

The (Q, q)-Schur Algebra

In this paper we use the Hecke algebra of type B to define anew algebra S which is an analogue of the q-Schur algebra. Weshow that S has ‘generic’ basis which is independentof the choice of ring and the parameters q and Q. We then constructWeyl modules for S and obtain, as factor modules, a family ofirreducible S-modules defined over any field. 1991 MathematicsSubject Classification: 16G99, 20C20, 20G05.     

Cm Elliptic Curves and Bicolorings of Steiner Triple Systems

It is shown that a necessary condition for the existence ofa bicolored Steiner triple system of order n is that n can bewritten in the form A²+3B² for integers A and B. In the casewhen n=q is either a prime congruent to 1 mod 3, or the squareof a prime congruent to 2 mod 3, it is shown that the numbersof colored vertices in the triple system would be unique, andare given by the number of points on specific twists of theCM elliptic curve y²=x³–1 over the finite field F_q. 2000Mathematics Subject Classification 05B07, 11G20, 14G15 (primary);11G15, 14K22 (secondary).     

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.     

Tilting Cotorsion Pairs

Let R be a ring and T a 1-tilting right R-module. Then T isof countable type. Moreover, T is of finite type in the casewhere R is a Prüfer domain. 2000 Mathematics Subject Classification16D90, 16D30, 13G05 (primary), 03E75, 20K40, 16G99 (secondary).     

Root Numbers of Non-Abelian Twists of Elliptic Curves

We study the global root number of the complex L-function oftwists of elliptic curves over Q by real Artin representations.We obtain examples of elliptic curves over Q which, while nothaving any rational points of infinite order, conjecturallymust have points of infinite order over the fields for every cube-free m > 1. We describe analogousphenomena for elliptic curves over the fields , and in the towers and , where r 3 is prime.2000 Mathematics Subject Classification 11G40, 11G05.     

Springer Correspondence for Disconnected Exceptional Groups

The determination of the generalized Springer correspondencefor disconnected reductive groups is completed here by beingachieved in the case of disconnected groups of types D₄ in characteristic3 and E₆ in characteristic 2. It is expected that this correspondencewill provide an important ingredient for the computation ofcharacter values of finite groups of Lie type. Some evidenceis provided in support of the latter statement. 2000 MathematicsSubject Classification 20G05 (primary), 20C33 (secondary).     

Large Lattices Over Orders

Let K be the quotient field of a Dedekind domain R. We characterizethe R-orders in a separable K-algebra for which every R-projective-module decomposes into -lattices. Butler, Campbell and Kovácshave recently shown that the latter holds for the integral groupring of a cyclic group of prime order, as well as for lattice-finiteorders over a complete discrete valuation domain. 2000 MathematicsSubject Classification 16H05, 16G30 (primary), 05C65 (secondary).     

Vanishing Theorems for Quaternionic Complexes

1991 Mathematics Subject Classification 58G05, 53C07.     

A Hyperelliptic Smoothness Test, II

This series of papers presents and rigorously analyzes a probabilisticalgorithm for finding small prime factors of an integer. Thealgorithm uses the Jacobian varieties of curves of genus 2 inthe same way that the elliptic curve method uses elliptic curves.This second paper in the series is concerned with the orderof the group of rational points on the Jacobian of a curve ofgenus 2 defined over a finite field. We prove a result on thedistribution of these orders. 2000 Mathematical Subject Classification:11Y05, 11G10, 11M20, 11N25.     

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.     

On Suborbital Graphs for the Modular Group

This paper proves a conjecture of G. A. Jones, D. Singermanand K. Wicks, that a suborbital graph for the modular groupis a forest if and only if it contains no triangles. 2000 MathematicsSubject Classification 05C05, 05C20, 20H35.     

Some Characterizations of Commutative Subspace Lattices

Let H be a not necessarily separable Hilbert space, and letBH denote the space of all bounded linear operators on H. Itis proved that a commutative lattice D of self-adjoint projectionsin H that contains 0 and I is spatially complete if and onlyif it is a closed subset of BH in the strong operator topology.Some related results are obtained concerning commutative lattice-orderedcones of self-adjoint operators that contain D. 2000 MathematicsSubject Classification 47D03, 47L35, 47L07, 46L10, 54F05, 54G05,46E05.     

Toroidal and Annular Dehn Fillings

Suppose that M is a hyperbolic 3-manifold which admits two Dehnfillings M(r₁) and M(r₂) such that M(r₁) contains an essentialannulus, and M(r₂) contains an essential torus. It is knownthat = (r₁, r₂) 5. We will show that if = 5 then M is theWhitehead sister link exterior, and if = 4 then M is the exteriorof either the Whitehead link or the 2-bridge link associatedto the rational number . There are infinitely many examples with = 3. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.