Decision Problems in Quadratic Function Fields of High Genus

This paper provides verification procedures for a number of decision problems in quadratic function fields of odd characteristic, thereby establishing membership of these problems in both NP and co-NP. The problems include determining the ideal and divisor class numbers of the field, the regulator of the field (in the real case), a generating system of the ideal class group, a basis of the ideal class group, the pricipality of an ideal, the equivalence of two ideals, the discrete logarithm of an ideal class with respect to another ideal class, and the order of a class in the ideal class group. While several of these problems belong to the aforementioned complexity classes unconditionally, others require a certain assumption to ensure that the verification procedures can be done in polynomial time; so far, this assumption has only been verified for fields of high genus.     

Symmetric algebras of modules arising from a fixed submatrix of a generic matrix

We analyze symmetric algebras which arise from rather ‘bad’ ideals and modules. For example, the ideals are mixed, and every value ≠ 0 occurs as the projective dimension of one of the modules. We are interested in the Cohen-Macaulay property, the canonical module, normality, and the divisor class group. The symmetric algebras under consideration can be defined as residue class rings modulo determinantal ideals covered by the theory of Hochster-Eagon. Part of the results can be regarded as an extension of work of Andrade and Simis.     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

Class fields,Dirichlet characters, and extended genus fields of global function fields

We study the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses the class field theory. First, we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next, we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields. In the absolute abelian case, we compare this approach with the one given by Anglès and Jaulent.     

Arithmetic on superelliptic curves

This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form which has only one point at infinity. Divisors are represented as ideals, and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time. An algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field.

     

Hyperelliptic curves and homomorphisms to ideal class groups of quadratic number fields

For the function field K of hyperelliptic curves over Q we define a subgroup of the ideal class group called the group of Z-primitive ideals. We then show that there are homomorphisms from this subgroup to ideal class groups of certain quadratic number fields.     

Ideals Generated by Diagonal 2-Minors

With a simple graph G on [n], we associate a binomial ideal P_G generated by diagonal minors of an n × n matrix X = (x_ij) of variables. We show that for any graph G, P_G is a prime complete intersection ideal and determine the divisor class group of K[X]/P_G. By using these ideals, one may find a normal domain with free divisor class group of any given rank.     

A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field

Let k be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic forms over k. We have two main results. The first result is on the principal part of the global zeta function associated with the prehomogeneous vector space. The second result is on a mean value theorem for degree zero divisor class groups of quadratic extensions over k, which is a consequence of the first one.     

Generalized Calogero–Moser systems from rational Cherednik algebras

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals that are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero–Moser type which we explicitly specify. In the case of classical Coxeter groups, we also obtain generalized Calogero–Moser systems with added quadratic potential.     

Sums of squares in function fields of hyperelliptic curves

We study sums of squares, quadratic forms, and related field invariants in a quadratic extension of the rational function field in one variable over a hereditarily pythagorean base field.     

函数域常值域扩张的类数整除性

设K/F_q是亏格大于0的整体函数域,K_n:=KF_(q~n)是K上的n次常值域扩张.利用整体函数域zeta函数的整系数多项式的有理表达式,结合函数域常值域扩张的基本性质,对于满足特定条件的素数l,本文讨论了使得除子类群Pic~0(K_n)的Sylow-l子群为非平凡群的常值域扩张K_n的存在性.     

On positive definite quadratic forms in correlatedt variables

Summary In this paper we extend Ruben's [4] result for quadratic forms in normal variables. He represented the distribution function of the quadratic form in normal variables as an infinite mixture of chi-square distribution functions. In the central case, we show that the distribution function of a quadratic form int-variables can be represented as a mixture of beta distribution functions. In the noncentral case, the distribution function presented is an infinite series in beta distribution functions. An application to quadratic discrimination is given.     

The parallelized Pollard kangaroo method in real quadratic function fields

We show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields. Specifically, we show how to apply the kangaroo method to the infrastructure in these fields. We also show how to speed up the computation by using heuristics on the distribution of the divisor class number, and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field. Furthermore, we provide examples for regulators and class numbers of hyperelliptic function fields of genus that are larger than those ever reported before.

     

Application of Greenberg's conjecture in the capitulation problem

Let r be a positive integer. Assume Greenberg's conjecture for some totally real number fields, we show that there exists an infinite family of imaginary cyclic number fields F over the field of rational number field , with an elementary 2‐class group of rank equal to r that capitulates in an unramified quadratic extension over F. Also, we give necessary and sufficient conditions for the Galois group of the unramified maximal 2‐extension over F to be abelian.     

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers hq,0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak-Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.     

Divisor class group and canonical class of rings defined by ideals of pfaffians

In this paper we study the rings defined by ideals of pfaffians of a skew symmetric matrix of indeterminates. We analyze the case in which the pfaffians are not necessarily of fixed size. We prove that such rings are Cohen-Macaulay normal domains and we compute the divisor class group and the canonical class. It allows us to determine which of our rings are Gorenstein.