Function field genus theory for non-Kummer extensions

In this paper we first obtain the genus field of a finite abelian non-Kummer l–extension of a global rational function field. Then, using 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 and our previous results, we deduce the general expression of the genus field of a finite abelian extension of a global rational function field.     

Corrigendum to “Genus fields of abelian extensions of rational congruence function fields” [Finite Fields Appl. 20 (2013) 40–54]

There is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statements for Theorems 4.2, 4.5 and 5.2.     

Quadratic extensions of totally real quintic fields

In this work, we establish lists for each signature of tenth degree number fields containing a totally real quintic subfield and of discriminant less than in absolute value. For each field in the list we give its discriminant, the discriminant of its subfield, a relative polynomial generating the field over one of its subfields, the corresponding polynomial over , and the Galois group of its Galois closure.

We have examined the existence of several non-isomorphic fields with the same discriminants, and also the existence of unramified extensions and cyclic extensions.

     

On the number of certain Galois extensions of local fields

In this paper, we will calculate the number of Galois extensions of local fields with Galois group or .

     

2-Descent on elliptic curves and rational points on certain Kummer surfaces

The first part of this paper further refines the methodology for 2-descents on elliptic curves with rational 2-division points which was introduced in [J.-L. Colliot-Thélène, A.N. Skorobogatov, Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579-650]. To describe the rest, let E⁽¹⁾ and E⁽²⁾ be elliptic curves, D⁽¹⁾ and D⁽²⁾ their respective 2-coverings, and X be the Kummer surface attached to D⁽¹⁾×D⁽²⁾. In the appendix we study the Brauer-Manin obstruction to the existence of rational points on X. In the second part of the paper, in which we further assume that the two elliptic curves have all their 2-division points rational, we obtain sufficient conditions for X to contain rational points; and we consider how these conditions are related to Brauer-Manin obstructions. This second part depends on the hypothesis that the relevent Tate-Shafarevich group is finite, but it does not require Schinzel's Hypothesis.     

On the number of solutions of some Kummer equations over finite fields

Let l be a prime number and let $k={\mathbb{F}}_q$ be a finite field of characteristic $p\ne l$ with $q=p^f$ elements. Let $n\ge 0$. We determine the number N of solutions $(x,y)$ in k of the Kummer equation$y^l=x(x^{l^n}-1),$ in terms of the trace of a certain Jacobi sum.     

Rational places in extensions and sequences of function fields of Kummer type

In this paper we prove results on the number of rational places in extensions of Kummer type over finite fields and give sufficient conditions for non-trivial lower bounds on the number of rational places at each step of sequences of function fields over a finite field, that we call (a, b)-sequences. In the case of a prime field, we apply these results to the study of rational places in certain sequences of function fields of Kummer type.     

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers Ok of k, which, together with the degree [K:k] of the extension determines the Ok-module structure of OK. We call Rt(k,G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A^′-groups inductively, starting with abelian groups and then considering semidirect products of A^′-groups with abelian groups of relatively prime order and direct products of two A^′-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A^′-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine Rt(k,Dn) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).     

Class numbers of some abelian extensions of rational function fields

Let be a monic irreducible polynomial. In this paper we generalize the determinant formula for of Bae and Kang and the formula for of Jung and Ahn to any subfields of the cyclotomic function field By using these formulas, we calculate the class numbers of all subfields of when and are small.

     

A mean value theorem for discriminants of abelian extensions of a number field

Let k be an algebraic number field and let N(k,C?;m) denote the number of abelian extensions K of k with G(K/k)≅C?, the cyclic group of prime order ?, and the relative discriminant D(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for _∑m?XN(k,C?;m) using the class field theory and a method, developed by Wright. We show that our result is identical to a result of Cohen, Diaz y Diaz and Olivier, obtained by methods of classical algebraic number theory, although our methods allow for a more elegant treatment and reduce a global calculation to a series of local calculations.     

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).     

The genus fields of Kummer function fields

In this paper, we give a definition of genus field of function field with one variable over finite fields. And we explicitly describe the genus fields of Kummer function fields. For quadratic function fields case, our results are analogous to the genus fields of quadratic number fields.     

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 cubic Galois field extensions

We study Morton's characterization of cubic Galois extensions F/K by a generic polynomial depending on a single parameter s∈K. We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F. For K=Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.     

A Class of Artin-Schreier Towers with Finite Genus

We study the asymptotic behaviour of the genus in some Artin-Schreier towers of function fields over a finite field, and we present a new class of Artin-Schreier towers having finite genus.     

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 $(\widehat{K}(X),v)$.     

Tenth degree number fields with quintic fields having one real place

In this paper, we enumerate all number fields of degree of discriminant smaller than in absolute value containing a quintic field having one real place. For each one of the (resp. found fields of signature (resp. the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative discriminant, the corresponding polynomial over , and the Galois group of the Galois closure are given.

In a supplementary section, we give the first coincidence of discriminant of (resp. nonisomorphic fields of signature (resp. .