A tower with non-Galois steps which attains the Drinfeld-Vladut bound

We introduce a new tower of function fields over a finite field of square cardinality, which attains the Drinfeld-Vladut bound. One new feature of this new tower is that it is constructed with non-Galois steps; i.e., with non-Galois function field extensions. The exact value of the genus g(F_n) is also given (see Lemma 4).     

Graphs and recursively defined towers of function fields

In this paper we state and explore a connection between graph theory and the theory of recursively defined towers. This leads, among other things, to a generalization of Lenstra's identity (Finite Fields Appl. 8 (2001) 166) and the solution of an open problem concerning the Deuring polynomial posed in (J. Reine Angew. Math. 557 (2003) 53). Further we investigate the effect extension of the constant field has on the limit of certain towers.     

On towers of function fields of Artin-Schreier type

In this article we derive strong conditions on the defining equations of asymptotically good Artin-Schreier towers. We will show that at most three kinds of defining equations can give rise to a recursively defined good tower, if we restrict ourselves to prime degrees. ¹A. Garcia and H. Stichtenoth did part of thiswork during their stay at Sabanci University, Istanbul, Turkey (Sept. 2002). ²A. Garcia was partially supported by PRONEX # 662408/1996-3 (CNPq-Brazil).     

A simplified proof for the limit of a tower over a cubic finite field

Recently Bezerra, Garcia and Stichtenoth constructed an explicit tower F=(Fn)_n?0 of function fields over a finite field Fq³, whose limit λ(F)=limn→∞N(Fn)/g(Fn) attains the Zink bound λ(F)?2(q²−1)/(q+2). Their proof is rather long and very technical. In this paper we replace the complex calculations in their work by structural arguments, thus giving a much simpler and shorter proof for the limit of the Bezerra, Garcia and Stichtenoth tower.     

A graph aided strategy to produce good recursive towers over finite fields

We propose a systematic method to produce potentially good recursive towers over finite fields. The graph point of view, so as some magma and sage computations are used in this process. We also establish some theoretical functional criterion ensuring the existence of many rational points on a recursive tower. Both points are illustrated by an example, from the production process, to the theoretical study.     

A ruled residue theorem for function fields of conics

The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from 2 this theorem is extended here to function fields of conics. The main result is that there is at most one extension of a valuation on the base field to the function field of a conic for which the residue field extension is transcendental but not ruled. Furthermore the situation when this valuation is present is characterised.     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.     

The two-variable zeta function and the Riemann hypothesis for function fields

We present Bombieri's proof of the Riemann hypothesis for the zeta function of a curve over a finite field. We first briefly describe this zeta function and discuss the two-variable zeta function of Pellikaan. Then we give Naumann's proof that the numerator of this function is irreducible.     

An elementary proof of the law of quadratic reciprocity over function fields

Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in :

where is the Legendre symbol.

     

Integral closures and weight functions over finite fields

Curves and surfaces of type I are generalized to integral towers of rank r. Weight functions with values in N^r and the corresponding weighted total-degree monomial orderings lift naturally from one domain R_j−1 in the tower to the next, R_j, the integral closure of R_j−1[x_j]/φ(x_j). The qth power algorithm is reworked in this more general setting to produce this integral closure over finite fields, though the application is primarily that of calculating the normalizations of curves related to one-point AG codes arising from towers of function fields. Every attempt has been made to couch all the theory in terms of multivariate polynomial rings and ideals instead of the terminology from algebraic geometry or function field theory, and to avoid the use of any type of series expansion.     

A ruled residue theorem for function fields of elliptic curves

It is shown that a valuation of residue characteristic different from 2 and 3 on a field E has at most one extension to the function field of an elliptic curve over E, for which the residue field extension is transcendental but not ruled. The cases where such an extension is present are characterised.     

Witt equivalence of function fields of curves over local fields

Two fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over Archimedean local fields under Witt equivalence is well understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13 G?adki, P., Marshall, M. Witt equivalence of function fields over global fields. Trans. Am. Math. Soc., electronically published on April 11, 2017, doi: https://doi.org/10.1090/tran/6898 (to appear in print).[Crossref] , [Google Scholar]] by the authors and applied to study Witt equivalence of function fields of curves over global fields. In this paper, we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.     

On subfields of the second generalization of the GK maximal function field

The second generalized GK function fields $K_n$ are a recently found family of maximal function fields over the finite field with $q^{2n}$ elements, where q is a prime power and $n\ge 1$ an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of $K_n$. In case $\gcd (q+1,n)=1$ and either q is even or $q\equiv 1(\mathrm{mod}4)$, we find a complete list of Galois subfields of $K_n$. Our construction adds several previously unknown genera to the genus spectrum of maximal curves.     

Galois cohomology in degree 3 of function fields of curves over number fields

Let k be a field of characteristic not equal to 2. For n≥1, let denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements , there exist such that α_i=(a)∪(b_i), where for any λ∈k∗, (λ) denotes the image of k∗ in . In this paper we prove a higher dimensional analogue of the Tate's lemma.     

The Galois closure of Drinfeld modular towers

In this article we study Drinfeld modular curves X₀(pn) associated to congruence subgroups Γ₀(pn) of GL(2,Fq[T]) where p is a prime of Fq[T]. For n>r>0 we compute the extension degrees and investigate the structure of the Galois closures of the covers X₀(pn)→X₀(pr) and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies.