Algebraic Function Fields Research Articles

A note on defining transcendentals in function fields

AbstractThe work [11] deals with questions of first-order definability in algebraic function fields. In particular, it exhibits new cases in which the field of constant functions is definable, and it investigates the phenomenon of definable transcendental elements. We fix some of its proofs and make additional observations concerning definable closure in these fields.

The classical reciprocity law for power residues as an analog of the Abelian integral theorem

A formula for power residue symbols is deduced, which can be treated as an analog of the Abelian integral theorem for number fields. §0. Introduction It was Leopold Kronecker who first understood that there is a deep relationship between algebraic numbers and algebraic functions. He stated that the prime ideals in the fields of algebraic functions play the same role as the points of Riemann surfaces: the ramification points of a Riemann surface correspond to prime divisors of the discriminant of a number field, etc. The theory of Abelian integrals is the most studied part of the theory of algebraic functions. David Hilbert was apparently the first who initiated an analog of this theory in the fields of algebraic numbers. In particular, he noticed that his reciprocity law for the product of norm residue symbols (Hilbert symbols),

Rational separability of the integral closure

We investigate the following question. Let K be a global field, i.e. a number field or an algebraic function field of one variable over a finite field of constants. Let W K be a set of primes of K, possibly infinite, such that in some fixed finite separable extension L of K, all the primes of W K do not have factors of relative degree 1. Let M be a finite extension of K and let W M be the set of all the M-primes above the primes of W K . Then does W M have the same property? The answer is “always” for one variable algebraic function fields over finite fields of constants and “not always” for number fields. In this paper we give a complete description of the conditions under which W M inherits and does not inherit the above described property.

Every place admits local uniformization in a finite extension of the function field

We prove that every place P of an algebraic function field F | K of arbitrary characteristic admits local uniformization in a finite extension F of F. We show that F | F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension F P | F P of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no extension of F is needed. Our proofs are based solely on valuation theoretical theorems, which are of particular importance in positive characteristic. They are also applicable when working over a subring R ⊂ K and yield similar results if R is regular and of dimension smaller than 3.

Bounding the number of [formula omitted]-rational places in algebraic function fields using Weierstrass semigroups

We present a new bound on the number of F q -rational places in an algebraic function field. It uses information about the generators of the Weierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields.

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Let L/K be an l-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠ l. In this paper, the \({\mathbb{Z}_{\ell}[G]}\)-module structure of the l-torsion of the Jacobian associated to L is explicitly determined.

New Linear Codes and Algebraic Function Fields Over Finite Fields

In this correspondence, we present 129 new linear codes over F8 and F9 based on the construction by Xing and Niederreiter using algebraic function fields and places of small degrees. In addition, we construct some global function fields in which the number of rational places improves the lower bounds given by van der Geer and van der Vlugt.

A Note on Optimal Towers over Finite Fields

Recently, under the influence of Elkies' conjecture [2], the optimal recursive towers of algebraic function fields (of one variable) over the finite fields with square cardinality are studied [3, 5, 6]. In this paper, we define the limit $$ \lim_{i \to \infty} \text{(the number of places of degree $n$ in $F_{i}/\mathbf{F}_{q}$)/(genus of $F_{i}$)} $$ of a tower $F_{0} \subseteq F_{1} \subseteq F_{2} \subseteq \cdots$ over the finite field $\F_{q}$. Using this limit, we prove that all the proper constant field extensions of all the optimal towers over the finite fields with square cardinality are not optimal, and we show a simple criterion whether a tower is optimal or not. Moreover, we give many new recursive towers of finite ramification type.

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g − 1 in the algebraic function fields of genus g defined over F q . Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields F q by using Shimura and modular curves defined over F q . From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields F q , notably in the case of F 2 . These upper bounds attain the best asymptotic upper bounds of Shparlinski–Tsfasman–Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145–169].

On existence of tame Harrison map

Abstract We present here two new criteria for existence of a tame Harrison map of two formally real algebraic function fields over a fixed real closed field of constants. The first criterion (c.f. Theorem 2.5) shows that a square class group isomorphism is a tame Harrison map if it induces an isomorphism of the coproduct rings of residue Witt rings. The other result (c.f. Proposition 3.5) associates a tame Harrison map to an integral quaternion-symbol equivalence.

Tractability of algebraic function fields in one variable of genus one over global fields

A field k of characteristic unequal to 2 is called tractable if for every nonzero a i , b i ∈ k , i = 1 , 2 , 3 , whenever the quaternion algebra ( a i , b j / k ) is split for all i ≠ j and ( a 1 , b 1 / k ) ≅ ( a 2 , b 2 / k ) ≅ ( a 3 , b 3 / k ) , then ( a i , b i / k ) is split. In the present paper, we study tractability of algebraic function fields in one variable over global fields and give specific examples of tractable function fields and intractable function fields of genus one over Q , the rationals.

Miura conjecture on Affine curves

Shinji Miura gave certain multivariable polynomials that express an Affine curve for a given algebraic function field F and its degree one place O, if F contains such an O. Suppose the equations contain t ( 2) variables, and that the pole orders at O are a1, , at 1, where GCD a1, , at = 1. If ai di a1 di 1 N + + ai 1 di 1 N, di = GCD a 1, , ai

On automorphisms of generalized algebraic-geometry codes

We consider a class of generalized algebraic-geometry codes based on places of the same degree of a fixed algebraic function field over a finite field F / F q . We study automorphisms of such codes which are associated with automorphisms of F / F q .

On splitting the Knebusch–Milnor exact sequence

We show here that the Witt ring of the ring of regular functions is a direct summand of the Witt ring of a formally real algebraic function field over a real closed field. Next, we use this result to show that the Knebusch–Milnor exact sequence of these Witt rings fulfills a certain analogy of splitting.

Toroidalization of generating sequences in dimension two function fields

Let k be an algebraically closed field of characteristic 0, and let K ∗ / K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let ν ∗ be a k-valuation of K ∗ with valuation ring V ∗ , and let ν be the restriction of ν ∗ to K. Suppose that R → S is an extension of algebraic regular local rings with quotient fields K and K ∗ respectively, such that V ∗ dominates S and S dominates R. We prove that there exist sequences of quadratic transforms R → R ¯ and S → S ¯ along ν ∗ such that S ¯ dominates R ¯ and the map between generating sequences of ν and ν ∗ has a toroidal structure. Our result extends the Strong Monomialization theorem of Cutkosky and Piltant.

Asymptotically Good Codes from Generalized Algebraic-Geometry Codes

We consider generalized algebraic-geometry codes, based on places of the same degree of a fixed algebraic function field over a finite field. In this note, using a method similar to the Justesen's one, we construct a family of such codes which is asymptotically good.

Abhyankar places admit local uniformization in any characteristic

We prove that every place P of an algebraic function field F | K of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field FP over K is equal to the transcendence degree of F | K , and the extension F P | K is separable. We generalize this result to the case where P dominates a regular local Nagata ring R ⊆ K of Krull dimension dim R ⩽ 2 , assuming that the valued field ( K , v P ) is defectless, the factor group v P F / v P K is torsion-free and the extension of residue fields F P | K P is separable. The results also include a form of monomialization.

β-Expansions in algebraic function fields over finite fields

The present paper deals with an algebraic function field analogue of β-expansions of real numbers. It completely characterizes the sets with eventually periodic and finite expansions. These characterizations are unknown in the real case.

Towards an Algebraic Proof of Deligne’s Regularity Criterion. An Informal Survey of Open Problems

We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in [1, I.4]. Our main interest lies in the algebraic form of Deligne’s regularity criterion [2, II.4.4 (iii)], asserting that, on a normal compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book [1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements in our book [1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle. We report on ongoing correspondence with Y. Andre and N. Tsuzuki, leading to partial results and provide examples indicating the subtlety of the problem.

Resolution of curve and surface singularities in characteristic zero

Preface. Note to the Reader. Terminology. I: Valuation Theory. 1. Marot Rings. 2. Manis Valuation Rings. 3. Valuation Rings and Valuations. 4. The Approximate Theorem for Independent Valuations. 5. Extensions of Valuations. 6. Extending Valuations to Algebraic Overfields. 7. Extensions of Discrete Valuations. 8. Ramification Theory of Valuations. 9. Extending Valuations to Non-Algebraic Overfields. 10. Valuations of Algebraic Function Fields. 11. Valuations Dominating a Local Domain. II: One-Dimensional Semilocal Cohen-Macaulay Rings. 1. Transversal Elements. 2. Integral Closure of One-Dimensional Semilocal Cohen-Macaulay Rings. 3. One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings. 4. Blowing up Ideals. 5. Infinitely Near Rings. III: Differential Modules and Ramification. 1. Introduction. 2. Norms and Traces. 3. Formally Unramified and Ramified Extensions. 4. Unramified Extensions and Discriminants. 5. Ramification for Quasilocal Rings. 6. Integral Closure and Completion. IV: Formal and Convergent Power Series Rings. 1. Formal Power Series Rings. 2. Convergent Power Series Rings. 3. Weierstrass Preparation Theorem. 4. The Category of Formal and Analytic Algebras. 5. Extensions of Formal and Analytic Algebras. V: Quasiordinary Singularities. 1. Fractionary Power Series. 2. The Jung-Abhyankar Theorem: Formal Case. 3. The Jung-Abhyankar Theorem: Analytic Case. 4. Quasiordinary Power Series.5. A Generalized Newton Algorithm. 6. Strictly Generated Semigroups. VI: The Singularity Zq = XYp. 1. Hirzebruch-Jung Singularities. 2. Semigroups and Semigroup Rings. 3. Continued Factions. 4. Two-Dimensional Cones. 5. Resolution of Singularities. VII: Two-Dimensional Regular Local Rings. 1. Ideal Transform. 2. Quadratic Transforms and Ideal Transforms. 3. Complete Ideals. 4. Factorization of Complete Ideals. 5. The Predecessors of a Simple Ideal. 6. Uniformization. 7. Resolution of Surface Singularities II: Blowing up and Normalizing. Appendices. A: Results from Classical Algebraic Geometry. 1. Generalities. 2. Affine and Finite Morphisms. 3. Products. 4. Proper Morphisms. 5. Algebraic Cones and Projective Varieties. 6. Regular and Singular points. 7. Normalization of a Variety. 8. Desingularization of a Variety. 9. Dimension of Fibres. 10. Quasifinite Morphisms and Ramification. 11. Divisors. 12. Some Results on Projections. 13. Blowing up. 14. Blowing up: the Local Rings. B: Miscellaneous Results. 1. Ordered Abelian Groups. 2. Localization. 3. Integral Extensions. 4. Some Results on Graded Rings and Modules. 5. Properties of the Rees Ring. 6. Integral Closure of Ideals. 7. Decomposition Group and Inertia Group. 8. Decomposable Rings. 9.