Tamagawa defect of Euler systems

As remarked by Mazur and Rubin [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)] one does not expect the Kolyvagin system obtained from an Euler system for a p-adic Galois representation T to be primitive (in the sense of the above mentioned reference) if p divides a Tamagawa number at a prime ?≠p; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map in terms of the local Tamagawa numbers of T, refining a result of [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)]. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system.     

The loop orbifold of the symmetric product

Using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas-Sullivan orbifold product structure in the homology of the free loop space of the Borel construction of the symmetric product induces a ring structure in the homology of the inertia orbifold of the symmetric product. For a general almost complex orbifold, we define a new ring structure on the cohomology of its inertia orbifold which we call the virtual intersection ring. Finally we show that under Poincaré duality in the case of the symmetric product orbifold, both ring structures are isomorphic.     

Big Heegner point Kolyvagin system for a family of modular forms

The principal goal of this paper is to develop Kolyvagin’s descent to apply with the big Heegner point Euler system constructed by Howard  for the big Galois representation \(\mathbb T \) attached to a Hida family \(\mathbb F \) of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family \(\mathbb F \) at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.     

Cohomologies of Leibniz Algebras with Higher Derivations北大核心CSCD

本文研究具有高阶导子的莱布尼兹代数.我们称之为LeibHDer对.首先给出LeibHDer对的表示并构造半直积.最后,定义LeibHDer对的上同调并研究其中心扩张和形变理论.     

Heegner Point Kolyvagin System and Iwasawa Main Conjecture

We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis, at an ordinary prime p. It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch–Kato Selmer group(see Theorem 1.3 for precise statement). As a byproduct we also prove the equality in the Greenberg–Iwasawa main conjecture for certain Rankin–Selberg product(Theorem 1.7) under some local conditions, and an improvement of Skinner's result on a converse of Gross–Zagier and Kolyvagin theorem(Corollary 1.11).     

The Structure and Enumeration of Link Projections

We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well-known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition.

     

Betti Numbers of Hypergraphs

In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.     

The Universal Kolyvagin Recursion Implies the Kolyvagin Recursion

Let Hz. be the universal norm distribution and M a fixed power of prime p. By using the double complex method employed by Anderson, we study the universal Kolyvagin recursion occurring in the canonical basis in the cohomology group H^0（Gz,Hz/MHz）. We furthermore show that the universal Kolyvagin recursion implies the Kolyvagin recursion in the theory of Euler systems. One certainly hopes this could lead to a new way to find new Euler systems.     

Structures preserved by the QR-algorithm

In this paper we investigate some classes of structures that are preserved by applying a (shifted) QR-step on a matrix A. We will handle two classes of such structures: the first we call polynomial structures, for example a matrix being Hermitian or Hermitian up to a rank one correction, and the second we call rank structures, which are encountered for example in all kinds of what we could call Hessenberg-like and lower semiseparable-like matrices. An advantage of our approach is that we define a structure by decomposing it as a collection of ‘building stones’ which we call structure blocks. This allows us to state the results in their natural, most general context.     

Rings and groups of matrices with a nonstandard product

We define a new operation of multiplication on the set of square matrices. We determine when this multiplication is associative and when the set of matrices with this multiplication and the ordinary addition of matrices constitutes a ring. Furthermore, we determine when the nonstandard product admits the identity element and which elements are invertible. We study the relation between the nonstandard product and the affine transformations of a vector space. Using these results, we prove that the Mikha?lichenko group, which is a group of matrices with the nonstandard product, is isomorphic to a subgroup of matrices of a greater size with the ordinary product.     

On the notion of canonical dimension for algebraic groups

We define and study a numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P^n-1.     

Polynomial Ensembles and Pólya Frequency Functions

Journal of Theoretical Probability - We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of...     

A recurrence formula with respect to the Cameron–Storvick type theorem of the 𝒯-transform

ABSTRACT

In this paper, we define a transform which has the kernel in its definition and a concept of derivative for functionals on Wiener space. We then establish some results and formulas for the transforms of functionals on Wiener space. We also establish the Cameron–Storvick type theorem for the transform. Finally, we obtain the recurrence formula for the transforms to evaluate formulas involving the multi-dimensional derivative.     

Value of games with two-layered hypergraphs

This paper studies cooperative games with restricted cooperation among players. We define situations in which a priori unions and hypergraphs coexist simultaneously and mutually depend on each other. We call such structures two-layered hypergraphs. Using a two-step approach, we define a value of the games with two-layered hypergraphs. The value is characterized by Owen’s coalitional value of hypergraph-restricted games and in terms of weighted Myerson value. Further, our value is axiomatically characterized by component efficiency and a coalition size normalized balanced contributions property.     

The Tamari block lattice: An order on saturated chains in the Tamari lattice

In this article we define a partially-ordered set on equivalence classes of maximal chains of an interval of the Tamari lattice, which we call the Tamari Block Poset. We prove this is a graded lattice. Along the way we discuss some useful characterizations of its elements. In conclusion we define explicit maps between our work and the Higher Stasheff-Tamari orders in dimensions two and three.     

Quasi-Shuffle Products

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, *, ); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (U, III, ) onto (U, *, ) the set L of Lyndon words on A and their images { exp(w) w L} freely generate the algebra (U, *). We also consider the graded dual of (U, *, ). We define a deformation *_q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.     

Double Catalan monoids

In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.     

Commutator Calculus for Wreath Product Groups

We introduce a class of function spaces consisting of integer-valued functions of several integers which coordinatize the elements of certain subgroups of some finite regular wreath product groups. On each function space, we define operators which correspond to forming certain commutators relevent to computing the upper central series. We define an automorphism of the function space which enables us to define a class of subgroups that is useful for describing the upper central series of certain finite regular wreath product p-groups. Our results describe fundamental, interesting, and useful relationships between the automorphism and the operators. We describe some applications.     

Gears,pregears and related domains

We study conformal mappings from the unit disc to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or “gearlike”) domains fit into a more general category of domains we call “pregears” (images of gears under Möbius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.