Truncations of principal geometries

We investigate the class of principal pregeometries (free simplicial geometries with spanning simplex) which form an important subclass of the class of transversal pregeometries (free simplicial geometries). We give a coordinate-free method for imbedding a transversal pregeometry on a simplex as a free simplicial pregeometry which makes use only of the set-theoretic properties of a presentation of the transversal pregeometry. We introduce the notion of an (r, k)-principal set as a generalization of principal basis and prove the collection of (r, k)-principal sets of a rank k pregeometry, if non-empty, are the bases of another pregeometry whose structure is determined. An algorithm for constructing principal sets is given. We then characterize truncations of principal geometries in terms of the existence of a principal set. We do this by erecting a given pregeometry to a free simplicial pregeometry with spanning simplex. The erection is the freest of all erections of the given pregeometry.     

Heegner Divisors and Nonholomorphic Modular Forms

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.     

The M3[D] construction and n-modularity

In 1968, Schmidt introduced the M ₃[D] construction, an extension of the five-element modular nondistributive lattice M ₃ by a bounded distributive lattice D, defined as the lattice of all triples satisfying . The lattice M ₃[D] is a modular congruence-preserving extension of D.? In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity such that is modularity and is properly weaker than . Let M _n denote the variety defined by , the variety of n-modular lattices. If L is n-modular, then M ₃[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M ₃[L] M ₃[Id L]. ? We provide an example of a lattice L such that M ₃[L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product of two lattices A and B with zero always a lattice. We complement this result by generalizing the M ₃[L] construction to an M ₄[L] construction. This yields, in particular, a bounded modular lattice L such that M ₄ L is not a lattice, thus providing a negative solution to Quackenbush’s problem in the variety M of modular lattices.? Finally, we sharpen a result of Dilworth: Every finite distributive lattice can be represented as the congruence lattice of a finite 3-modular lattice. We do this by verifying that a construction of Gr?tzer, Lakser, and Schmidt yields a 3-modular lattice. Received May 26, 1998; accepted in final form October 7, 1998.     

An Affine Representation for Transversal Geometries

Pregeometries (matroids) whose independent sets are the partial matchings of a relation (transversal pregeometries) can be canonically imbedded in a free-simplicial pregeometry (one whose points lie freely on flats spanned by a simplex). Conversely, all subgeometries of such free-simplicial pregeometries are transversal. Free-simplicial pregeometries are counted and their duals are naturally constructed and shown to be free-simplicial (showing self-dual free-simplexes corrspond to quasisymmetric relations). For more general transversal pregeometries, modular flats are characterized and transversal contractions are exemplified. Binary transversal pregeometries and their contractions (the class of binary gammoids) are shown to be the class of series-parallel networks, providing insight for further characterizations of (coordinatized) gammoids by excluded minors. Theorem. All principal transversal pregeometries and their truncations have critical exponent at most 2.     

Lattices of Intermediate Subfactors

LetNbe an irreducible subfactor of a typeII₁factorM. If the Jones index [M:N] is finite, then the set at(NM) of the intermediate subfactors for the inclusionNMforms afinitelattice. The commuting and co-commuting square conditions for intermediate subfactors are related to the modular identity in the lattice at(NN). In particular, simplicity of a finite groupGis characterized in terms of commuting square conditions of intermediate subfactors forNM=NG. We investigate the question of which finite lattices can be realized as intermediate subfactor lattices.     

Efficient calculation of Stark-Heegner points via overconvergent modular symbols

This note presents a qualitative improvement to the algorithm presented in [DG] for computing Stark-Heegner points attached to an elliptic curve and a real quadratic field. This algorithm computes the Stark-Heegner point with ap-adic accuracy ofM significant digits in time which is polynomial inM, the primep being treated as a constant, rather than theO(p ^M ) operations required for the more naive approach taken in [DG]. The key to this improvement lies in the theory of overconvergent modular symbols developed in [PS1] and [PS2].     

格的TL-模糊理想

本文研究了格的TL-模糊理想. 利用生成TL-模糊理想, 证明了一个模格的全体T_M-模糊理想形成一个完备的模格. 此外, 利用L-模糊集的投影和截影, 获得了将直积格的TL-模糊理想表示成分量格的TL-模糊理想的L-直积的一个充分必要条件. 所得结果进一步推广和发展了格的模糊理想的理论.     

Can Montgomery parasites be avoided? A design methodology based on key and cryptosystem modifications

Montgomery's algorithm [8], hereafter denotedM _n(...,...), is a process for computingM _n (A, B)=ABN modn whereN is a constant factor depending only onn.Usually,A B modn is obtained byM _n (M _n (A, B),N ^–2 modn) but in this article, we introduce an alternative approach consisting in pre-integratingN into cryptographic keys so that a singleM _n(...,...) will replace directly each modular multiplication.Except the advantage of halving the number of Montgomery multiplications, our strategy skips the precalculation (and the storage) of the constantN ^–2 modn and turns to be particularly efficient when a hardware device implementingM _n(...,...) is the basic computational tool at one's command.     

Dimension and automorphism groups of lattices

Every group is the automorphism group of a lattice of order dimension at most 4. We conjecture that the automorphism groups of finite modular lattices of bounded dimension do not represent every finite group. It is shown that ifp is a large prime dividing the order of the automorphism group of a finite modular latticeL then eitherL has high order dimension orM _p, the lattice of height 2 and orderp+2, has a cover-preserving embedding inL. We mention a number of open problems. Presented by C. R. Platt.     

The Schur property for subgroup lattices of groups

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups with modular subgroup lattice is described. Received: 16 October 2007, Final version received: 22 February 2008     

Bielliptic modular curves X 1(M,N)

In this work, we determine all modular curves X₁(M,N) which are bielliptic, and then we also discuss the problem of finding the curves X₁(M,N) which admit infinitely many quartic points over .     

Finitely generated lattices with <Emphasis Type="Italic">M</Emphasis>-standard elements among generators

We prove that a lattice is modular if it is generated by three elements, two of which are M-standard. We also show that a lattice generated by n, n > 3, M-standard elements should not necessarily be modular.     

Finite modular congruence lattices

We consider the variety of modular lattices generated by all finite lattices obtained by gluing together some M₃’s. We prove that every finite lattice in this variety is the congruence lattice of a suitable finite algebra (in fact, of an operator group). Received February 26, 2004; accepted in final form December 16, 2004.     

The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms. Partially supported by NSF Grant # 28976 X. Partially supported by NSF Grant # GP-28737 This revised version was published online in November 2006 with corrections to the Cover Date.     

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X ₀(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X ₀(N), if N=pM with p prime, p ∤ M and the genus of X ₀(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.      

Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.? It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A _i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal. Received July 19, 2001; accepted in final form July 11, 2002.     

Diophantine approximation for negatively curved manifolds

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant in its moduli space. Received: 24 October 2000; in final form: 10 November 2001 / Published online: 17 June 2002