On several families of elliptic curves with arbitrary large Selmer groups

In this paper,we calculate the ()-Selmer groups S()(E/Q) and S()(E/Q) of elliptic curves y2 = x(x + εpD)(x + εqD) via the descent method.In particular,we show that the Selmer groups of several families of such elliptic curves can be arbitrary large.     

Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli

We prove that any ordinary symplectic separable isogeny class in the moduli space of principally polarized abelian varieties over a field of positive characteristic is dense in the Zariski topology.Oblatum 24-X-1994This research is partially supported by grant DMS90-02574 from the National Science Foundation and by a grant from the National Science Council of Taiwan, R.O.C.     

Two-parameter families of predictor-corrector methods for the solution of ordinary differential equations

Two-parameter families of predictor-corrector methods based upon a combination of Adams- and Nyström formulae have been developed. The combinations use correctors of order one higher than that of the predictors. The methods are chosen to give optimal stability properties with respect to a requirement on the form and size of the regions of absolute stability. The optimal methods are listed and their regions of absolute stability are presented. The efficiency of the methods is compared to that of the corresponding Adams methods through numerical results from a variable order, variable stepsize program package.     

Best Chebyshev approximation from families of ordinary differential equations

Within the framework of the Meinardus and Schwedt theory ofnonlinear Chebyshev approximation we consider the approximationof functions and data by approximants generated from the solutionof parameter dependent initial value problems in ordinary differentialequations. New results are obtained for the uniqueness and characterizationof best approximations in terms of properties of the differentialequation. Some examples of new approximating families are givenalong with computed best approximations.     

The growth rate of symplectic Floer homology

The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence. This also establishes a connection between Floer homology and geometry of 3-manifolds.     

The algebraic functional equation of Selmer groups for CM fields

Because the analytic functional equation holds for Katz p-adic L-function for CM fields, the algebraic functional equation of the Selmer groups for CM fields is expected to hold. In this note we prove it following the specialization principle developed by T. Ochiai (2005) in [Och05].     

Factorization and symplectic uniton numbers for harmonic maps into symplectic groups

It is proved that any harmonic map ϕ : Ω →Sp(N) from a simply connected domain Ω ⊆R ²⋃ | ∞ | into the symplectic groupSp(N) ⊂U(2N) with finite uniton number can be factorized into a product of a finite number of symplectic unitons. Based on this factorization, it is proved that the minimal symplectic uniton number of ϕ is not larger thanN, and the minimal uniton number of ϕ is not larger than 2N - 1. The latter has been shown in literature in a quite different way.     

Galois Theory for the Selmer Group of an Abelian Variety

This paper concerns the Galois theoretic behavior of the p-primary subgroup Sel _A (F) _p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map s _K/F Sel _A (F) _p Sel _A (K) _p ^G(K/F) where F is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B.Mazurand M. Harris.     

Descent sets for symplectic groups

The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.     

Some families of complete caps of quadrics and symplectic polarities over GF(8)

A cap on a quadric is a set of its points whose pairwise joins are all chords. A cap is complete if it is not part of a larger one. The only field for which all complete quadric caps are known is GF(2). Those caps are small; the biggest for each quadric is of order the dimension of the ambient space. Apart from information about ovoids in dimensions at most 7, little else is known. Here, the evidence is increased by providing caps over GF(2), odd, which, if >1, have size of order the dimension cubed. In particular, complete caps are obtained for the quadrics Q _2m (8), Q ₊ ^8k+7 (8), Q _- ^8k+3 (8), Q ₊ ^8k+1 (8) and Q _- ^8k+5 (8). These caps on Q ₊ ^8k+7 (8) and Q _- ^8k+3 (8) are complete on any Q _n(8) of which their quadrics are sections; so is that that of Q ₄₊₂(8) for any Q _2n (8) of which Q ₄₊₂(8) is a section with the same kernel. From the correspondence with Q _2n (8) complete caps are obtained for symplectic polarities over GF(8).     

Donaldson's Q-operators for symplectic manifolds

We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold.This estimate is an ingredient in the recent result of Keller and Lejmi(2017) about a symplectic generalization of Donaldson's lower bound for the L~2-norm of the Hermitian scalar curvature.     

Isometric,holomorphic and symplectic reflections

As an extension of local geodesic symmetries we study here local reflections with respect to a topologically embedded submanifoldP in a Riemannian manifold (M, g). First we derive a criterion for isometric reflections. Then we study holomorphic and symplectic reflections on an almost Hermitian manifold. In particular we focus on the influence of these reflections on the intrinsic and extrinsic geometry of the submanifold. Finally we treat these three kinds of reflections and their relationship when the ambient manifold is a locally Hermitian symmetric space. The results are derived by the use of Jacobi vector fields.     

Contact Dehn surgery, symplectic fillings, and Property P for knots

These are notes of a talk given at the Mathematische Arbeitstagung 2005 in Bonn. Following ideas of Özbağcı–Stipsicz, a proof based on contact Dehn surgery is given of Eliashberg's concave filling theorem for contact 3-manifolds. The role of the theorem in the Kronheimer–Mrowka proof of Property P for nontrivial knots is sketched.     

Schur functors for the symplectic group

The purpose of this work is to describe some new connections between the characteristic-free representation theories of the symplectic group and the corresponding general linear group (Theorem 2.2 and Theorem 2.6).     

Derived equivalence for quantum symplectic resolutions

Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson–Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon–Stafford (Gordon and Stafford in Adv Math 198(1):222–274, 2005; Duke Math J 132(1):73–135, 2006) and Kashiwara–Rouquier (Kashiwara and Rouquier in Duke Math J 144(3):525–573, 2008) as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities.