Representability of Hilbert Schemes and Hilbert Stacks of Points

We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary.     

ON MINIMAL SURFACES WHOSE GAUSS MAP IS INVARIANT BY A HOLOMORPHIC FOLIATION

We study real minimal surfaces : M^{2 4} under the hypothesisthat the holomorphic Gauss map of the immersion is invariantby a holomorphic foliation with singularities. We give a sortof Huber–Osserman theorem regarding the algebraicity ofthe holomorphic Gauss map.     

New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings

We construct the so-called right adjoint sequence of an matrix over an arbitrary ring. For an integer the right -adjoint and the right -determinant of a matrix is defined by the use of this sequence. Over -Lie nilpotent rings a considerable part of the classical determinant theory, including the Cayley-Hamilton theorem, can be reformulated for our right adjoints and determinants. The new theory is then applied to derive the PI of algebraicity for matrices over the Grassmann algebra.

     

q-Skew Derivations and Polynomial Identities

Let R be a prime ring and its left Martindale quotient ring. Assume that a q-skew -derivation of R satisfies the identity relationfor all x R, where the subring of constants of on R. It is proved that if R⁽⁾ satisfies nontrivial polynomial identities, then so does R. This answers affirmatively a problem raised in Bergen and Grzeszczuk [2] by removing the assumption on the algebraicity of .Mathematics Subject Classification (2000): 16W20, 16W25, 16W55Members of Mathematics Division, National Center for Theoretical Sciences at Taipei.Acknowledgement The authors are thankful to the referee for her/his useful suggestions and comments. This research was supported by the National Science Council of Taiwan.     

<Emphasis Type="Italic">p</Emphasis>-adic properties of division polynomials and elliptic divisibility sequences

For a fixed rational point P E (K) on an elliptic curve, we consider the sequence of values (F_n (P))_n1 of the division polynomials of E at P. For a finite field we prove that the sequence is periodic. For a local field we prove (under certain hypotheses) that there is a power q=p^e so that for all m1, the limit of exists in K and is algebraic over We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.Mathematics Subject Classification (1991): 11G07, 11D61, 14G20, 14H52The authors research supported by NSA grant H98230-04-1-0064.     

Explicit formulas for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift and their applications

We give an explicit formula for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift. As an application we prove a certain algebraicity result for the values of twisted Rankin–Selberg series at integers of half-integral weight modular forms, which was not treated by Shimura (J Math Soc Japan 33:649–672, 1981).     

Garrett's pullback formula for vector valued Siegel modular forms

Let be the Siegel Eisenstein series of degree n and weight k. Garrett showed a formula of on Hp×Hq, where Hn is the Siegel upper half space of degree n. This formula was useful for investigating the Fourier coefficients of the Klingen Eisenstein series and the algebraicity of the space of Siegel modular forms and of special values of the standard L-functions. We would like to generalize this formula in the case of vector valued Siegel modular forms. In this paper, using a differential operator D by Ibukiyama which sends a scalar valued Siegel modular form to the tensor product of two vector valued Siegel modular forms, under a certain condition, we give a formula of and investigate the Fourier coefficients of the Klingen Eisenstein series.     

Expanded Degenerations and Pairs

We provide a universal approach to the moduli of Jun Li's expanded pairs and expanded degenerations [20 Li , J. ( 2001 ). Stable morphisms to singular schemes and relative stable morphisms . J. Differential Geom. 57 : 509 – 578 .[Web of Science ®] , [Google Scholar]]. This enables us to prove algebraicity results, compare with Li's approach and with the approach of Graber and Vakil [16 Graber , T. , Vakil , R. ( 2005 ). Relative virtual localization and vanishing of tautological classes on moduli spaces of curves . Duke Math. J. 130 : 1 – 37 .[Crossref], [Web of Science ®] , [Google Scholar]], and generalize to the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneration formulas.     

Resultants and the algebraicity of the join pairing on Chow varieties

The Chow/Van der Waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension. The algebraicity of the join pairing on Chow varieties then follows. The approach implies a more algebraic proof of Lawson's complex suspension theorem in characteristic 0. The continuity of the action of the linear isometries operad on the group completion of the stable Chow variety is a consequence.

     

Hypersurfaces whose tangent geodesics do not cover the ambient space

Let be an immersion of an -dimensional connected manifold in an -dimensional connected complete Riemannian manifold without conjugate points. Assume that the union of geodesics tangent to does not cover . Under these hypotheses we have two results. The first one states that is simply connected provided that the universal covering of is compact. The second result says that if is a proper embedding and is simply connected, then is a normal graph over an open subset of a geodesic sphere. Furthermore, there exists an open star-shaped set such that is a manifold with the boundary .

     

Groups which do not admit ghosts

A ghost in the stable module category of a group is a map between representations of that is invisible to Tate cohomology. We show that the only non-trivial finite -groups whose stable module categories have no non-trivial ghosts are the cyclic groups and . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups the second power of the Jacobson radical of is stably isomorphic to a suspension of .

     

On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .

We prove that the following are equivalent:

1. is Lipschitz homeomorphic to .

2. is Lipschitz homeomorphic to .

3. is Lipschitz homeomorphic to .

This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.

     

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let be an imaginary quadratic field with ring of integers , where is a square free integer such that , and let is a linear code defined over . The level theta function of is defined on the lattice , where is the natural projection. In this paper, we prove that:

i) for any such that , and have the same coefficients up to ,

ii) for , determines the code uniquely,

iii) for , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to .

     

Higher order PDE's and iterated processes

We introduce a class of stochastic processes based on symmetric -stable processes, for . These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric -stable process. We call them -time processes. They generalize Brownian time processes studied in Allouba and Zheng (2001), Allouba (2002), (2003), and they introduce new interesting examples. We establish the connection of -time processes to some higher order PDE's for rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.

     

Linear difference equations with transition points

Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation

where and have power series expansions of the form

with . Our results hold uniformly for in an infinite interval containing the transition point given by . As an illustration, we present an asymptotic expansion for the monic polynomials which are orthogonal with respect to the modified Jacobi weight , , where , -1$"> and is real analytic and strictly positive on .