Bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes

We construct explicit bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes with good parameters. This correspondence is a generalization of a work of Maharaj, Matthews, and Pirsic.     

On some generalizations of the classical Riemann problem

The special class of the homogeneous vector boundary Riemann problems on the finite sequence of algebraic surfaces is investigated completely. Its coefficients are the noncommutative permutative matrices of the arbitrary but not prime order, and boundary conditions are given on the system of open contours. The constructive solution procedure and definite structure of the canonical solution matrix are obtained and present some generalizations of the classical Riemann problem. Simultaneously the corresponding class of algebraic equations for the appropriate covering surfaces is formed explicitly too. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Onk-stage Euclidean algorithms for Galois extensions of ℚ

LetR be an integral domain whose quotient field is an algebraic number field. Cooke and Weinberger [4] showed that, assuming the Generalized Riemann Hypothesis, ifR is a principal ideal domain and has infinite unit group, thenR is 4-stage Euclidean with the absolute value of the norm as algorithm. We remove the assumption of the Generalized Riemann Hypothesis from this result for totally real Galois extensions of ℚ of degree greater than or equal to three, replacing it with the requirement of finding sufficiently many prime elements ofR, ℚ such that the unit group ofR maps onto (R/((π₁⋯π _r )²))* via the reduction map. A similar result holds for real quadratic fields.     

On Extensions of Semilocal Prüfer Domains

One of the most important results of Chevalley's extension theorem states that every valuation domain has at least one extension to every extension field of its quotient field. We state a generalization of this result for Prüfer domains with any finite number of maximal ideals. Then we investigate extensions of semilocal Prüfer domains in algebraic field extensions. In particular, we find an upper bound for the cardinality of extensions of a semilocal Prüfer domain. Moreover, we show that any two extensions of a semilocal Prüfer domain are incomparable (by inclusion) in an algebraic extension of fields.     

Non-galois cubic fields which are euclidean but not norm-euclidean

Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm--Euclidean.

     

A foliated analogue of one- and two-dimensional Arakelov theory

The analogy between number fields and Riemann surfaces was an important source of motivation for mathematicians in the last century. We improve and extend this analogy by substituting Riemann surfaces with certain foliations by Riemann surfaces. In particular we show that coverings of these foliations lead to formulas having the same structure as formulas describing number field extensions. We also study higher dimensional foliations which have properties analogous to arithmetic surfaces. This provides more evidence for a conjecture of Deninger.     

Extensions of nilpotent blocks over arbitrary fields

In this paper, with a suitable condition, we describe the algebraic structure of block extensions of nilpotent blocks over arbitrary fields, thus generalize the main result of B. Külshammer and L. Puig on block extensions of nilpotent blocks over algebraically closed fields. Supported by NSFC (Grant No.: 10501016).     

Newton’s method as applied to the Riemann problem for media with general equations of state

An approach based on Newton’s method is proposed for solving the Riemann problem for media with normal equations of state. The Riemann integrals are evaluated using a cubic approximation of an isentropic curve that is superior to the Simpson method in terms of accuracy, convergence rate, and efficiency. The potentials of the approach are demonstrated by solving problems for media obeying the Mie-Grüneisen equation of state. The algebraic equation of the isentropic curve and some exact solutions for configurations with rarefaction waves are explicitly given.     

Invariante Zwischenkörper endlicher Körpererweiterungen

Let KM be a finite field extension. An intermediate field L is called invariant if there is an affine algebraic K-group acting on M with L as its invariant field. The question, which intermediate fields are invariant, was studied by Bégueri [1] for purely inseparable extensions and by Sweedler [6] for arbitrary extensions, but only for a restricted class of groups. In this paper Bégueri's result is generalized to arbitrary field extensions. Additionally it is shown that one can check whether a given intermediate field is invariant or not by computing the rank of certain matrices. As an application we get a class of invariant intermediate fields.     

Higher genus minimal surfaces by growing handles out of a catenoid

Starting with the catenoid we derive global Weierstraß representations for minimal surfaces obtained by pushing and pulling handles and by adding ends. The Weierstraß data are obtained from Riemann surfaces given by algebraic functions. Modifying the equations for these curves gives the new surfaces. New and well known examples are treated.     

A quadratic field which is Euclidean but not norm-Euclidean

The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integersR is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal ideal domains which are not norm-Euclidean, there should exist examples of rings of algebraic integers which are Euclidean but not norm-Euclidean. In this paper, we give the first example for quadratic fields, the ring of integers of .     

On a question of Abraham Robinson

In this note we give a negative answer to Abraham Robinson’s question of whether a finitely generated extension of an undecidable field is always undecidable. We construct ‘natural’ undecidable fields of transcendence degree 1 over Q all of whose proper finite extensions are decidable. We also construct undecidable algebraic extensions of Q that allow decidable finite extensions.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.     

Computation of Weng's rank 2 zeta function over an algebraic number field

In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng's rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable OF-lattices with rank r. For r=2, in the case of F=Q, Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng's rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions.     

Differentially algebraic gaps

H-fields are ordered differential fields that capture some basic properties of Hardy fields and fields of transseries. Each H-field is equipped with a convex valuation, and solving first-order linear differential equations in H-field extensions is strongly affected by the presence of a “gap” in the value group. We construct a real closed H-field that solves every first-order linear differential equation, and that has a differentially algebraic H-field extension with a gap. This answers a question raised in [1]. The key is a combinatorial fact about the support of transseries obtained from iterated logarithms by algebraic operations, integration, and exponentiation.     

Closure planes

We introduce a simple algebraic method for constructing infinite affine (and projective) planes from an infinite set of finite planes of prime power order stemming from a “root” plane. The construction uses finite fields and infinite extensions of finite fields in a critical way. We obtain a classical-looking result which states that if the construction succeeds over the algebraic closure of a finite field, then both the infinite plane and the original root plane must be Desarguesian. The Lenz–Barlotti types for these planes are then linked to the Lenz–Barlotti type of the root plane. Examples are then given. These show that under suitable conditions, the method can yield infinitely many non-isomorphic infinite planes. These examples are of Lenz–Barlotti types II.1 and V.1.     

On cyclic algebraic-geometry codes

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1. Received 21 July 1997; in revised form 5 February 1998     

Isomonodromy Approach to Boundary Value Problems for the Ernst Equation

We use the isomonodromy properties of theta-functional solutions of the Ernst equation and an asymptotic expansion in the spectral parameter to establish algebraic relations, enforced by the underlying Riemann surface, between the metric functions and their derivatives. These relations determine which classes of boundary value problems can be solved on a given surface. The situation on lower-genus Riemann surfaces is studied in detail.