Asymptotically good towers of function fields with small p-rank

Over any quadratic finite field we construct function fields of large genus that have simultaneously many rational places, small p-rank, and many automorphisms.     

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).     

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 natural graph of finite fields distinguishing between models

We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields. Certain basic notions in finite field theory have interpretations in terms of standard graph properties. We show that the graphs are connected and provide an estimate of their diameter. An accidental graph isomorphism is uncovered and proved. The smallest non-trivial Laplace eigenvalue is given some attention, in particular for a specific family of 8-regular graphs showing that it is not an expander. We introduce a regular covering graph and show that it is connected if and only if the root is primitive.     

A note on character sums in function fields

We give an improvement on a character sum estimate in ${\mathbb{F}}_q[t]$ and answer a question of Shparlinski.     

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.     

The ABC theorem for higher-dimensional function fields

We generalize the ABC theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non-singular in codimension one. We also obtain an upper bound for the minimal order sequence of Wronskians over such function fields of positive characteristic.

     

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.     

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.     

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).     

On rational embeddings of curves in the second Garcia–Stichtenoth tower

In 1995, Garcia and Stichtenoth explicitly constructed a tower of projective curves over a finite field with q² elements which reaches the Drinfeld–Vlăduţ bound. These curves are given recursively by covers of Artin–Schreier type where the curve on the nth level of the tower has a natural model in . In this paper, for q an even prime power, we use point projections in order to embed these curves into projective space of the lowest possible dimension.     

关于临界图的一个定理的推广

叶宏博证明了当Δ≥５时没有度序列是２ｒΔ２ｒ的Δ－临界图．Ｋａｙａｔｈｒｉ推广了上述结果,证明了当Δ≥５时,没有同时满足下列两个条件的Δ－临界图：（ａ）Ｇ有一个２度点ｘ;设ｙ,ｚ是ｘ的两个邻接点;（ｂ）有一主项点ｙ１∈ＮＧ（ｙ）（ｙ１≠ｙ）与－２度点邻接．我们对上述结果进一步推广,证明了条件（ｂ）不是必要的;只要ｙ１与一个度数小于Δ－１的点邻接即可（可以不是２度点）．     

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.     

Extensions of valuations to rational function fields over completions

Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of v to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$. The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.