Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.     

On division algebras having the same maximal subfields

We show that if a field K of characteristic ≠ 2 satisfies the following property (*) for any two central quaternion division algebras D ₁ and D ₂ over K, the fact that D ₁ and D ₂ have the same maximal subfields implies that D ₁ ? D ₂ over K, then the field of rational functions K(x) also satisfies (*). This, in particular, provides an alternative proof for the result of S. Garibaldi and D. Saltman that the fields of rational functions k(x ₁, . . . , x _r ), where k is a number field, satisfy (*). We also show that K = k(x ₁, . . . , x _r ), where k is either a totally complex number field with a single dyadic place (e.g. ${k = \mathbb{Q}(\sqrt{-1})}$ ) or a finite field of characteristic ≠ 2, satisfies the analog of (*) for all central division algebras having exponent two in the Brauer group Br(K).     

Non-hyperelliptic curves of genus three over finite fields of characteristic two

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non-hyperelliptic locus of the moduli space M₃ of curves of genus 3. By adding to these computations the results of Oort [Moduli of abelian varieties and Newton polygons, C.R. Acad. Sci. Paris 312 (1991) 385-389] and Nart and Sadornil [Hyperelliptic curves of genus three over finite fields of characteristic two, Finite Fields Appl. 10 (2004) 198-200] on the hyperelliptic locus we obtain a complete picture of these strata for M₃.     

Capitulation problem for global function fields

Let q be a power of an odd prime number p, k=\mathbbF_q(t){p, k=\mathbb{F}_{q}(t)} be the rational function field over the finite field \mathbbF_q.{\mathbb{F}_{q}.} In this paper, we construct infinitely many real (resp. imaginary) quadratic extensions K over k whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. The proof of the infinity of such fields K relies on an estimation of certain character sum over finite fields.     

Calculating formula and divisibility for relative class numbers of abelian function fields

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.     

Cyclicity of the unramified Iwasawa module

For a number field k and a prime number p, let k _?? be the cyclotomic Z _p -extension of k with finite layers k _n . We study the finiteness of the Galois group X _?? over k _?? of the maximal abelian unramified p-extension of k _?? when it is assumed to be cyclic. We then focus our attention to the case where p?=?2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of k _n . As a consequence, we determine the complete list of real quadratic number fields for which X _?? is cyclic non trivial. We then apply these results to the study of Greenberg??s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya.     

Note on the number of integral ideals in Galois extensions

Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.     

Valuation rings and simple transcendental field extensions

If a valuation ring V on a simple transcendental field extension K₀(X) is such that the residue field k of V is not algebraic over the residue field k₀ of V₀=V∩K₀, then for k₀ a perfect field it is shown that k is obtained from k₀ by a finite algebraic followed by a simple transcendental field extension.     

On the birational anabelian program initiated by Bogomolov I

Recall that a program initiated by Bogomolov in 1990 aims at reconstructing function fields K|k with td(K|k)>1 and k algebraically closed from the maximal pro-ℓ abelian-by-central Galois group P^c  _K\Pi ^{\hbox to1pt{}{\rm c}_{\phantom {g}}}_{K} of K, where ℓ is any prime number ≠char(k). In this paper we complete that program in the case k is an algebraic closure of a finite field.     

On a conjecture of Vorst

Let k be an infinite perfect field of positive characteristic such that strong resolution of singularities holds over k. We prove that a localization of a d-dimensional commutative k-algebra R of finite type is K _d+1-regular if and only if it is regular. This partially affirms a conjecture of Vorst.     

A remark on the capitulation problem

We determine explicitly an infinite family of imaginary cyclic number fields k, such that the 2-class group of k is elementary with arbitrary large 2-rank and capitulates in an unramified quadratic extension K. The infinitely many number fields k and K have the same Hilbert 2-class field and an infinite Hilbert 2-class field tower.     

Quadratic function fields with invariant class group

Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case.     

On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt

Let K be a complete discrete valued field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G such that the induced extension of residue fields kL/kK is separable. Hesselholt (2004) [2] conjectured that the pro-abelian group {H¹(G,Wn(OL))}_n∈N is zero, where OL is the ring of integers of L and W(OL) is the ring of Witt vectors in OL w.r.t. the prime p. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt?s conjecture for all Galois extensions.     

Relative Brauer Groups of Function Fields of Curves of Genus One

Abstract

Let Kbe an algebraic function field in one variable over a constant field k. In this paper, we investigate the relative Brauer groups Br(K/k) of Kover kin various cases. When kis a global field, we focus on function fields K = k(C) of genus 1 where Cis the curve of the form y ² = at ⁴ + bwith a, b ∈ k ? {0}, and we describe the Brauer classes in Br(K/k). More precisely, we show that each algebra in Br(K/k) is a quaternion algebra which can be obtained by taking one of a finite number of the x-coordinates of k-rational points on the Jacobian of the curve C. In particular, for the field ? of rational numbers, we determine Br(K/?) precisely in numerous cases and give examples.     

The iwasawa invariant μ in the composite of two Zl-extensions

Let k₀ be a finite extension field of the rational numbers, and assume k₀ has at least two Z_l-extensions. Assume that at least one Z_l-extension $\text{K}\text{k}{}_0$ has Iwasawa invariant μ = 0, and let L be the composite of K and some other Z_l-extension of k₀. In this paper we find an upper bound for the number of Z_l-extensions of k₀ contained in L with nonzero μ.     

Weak approximation over function fields of curves over large or finite fields

Let K = k(C) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally connected K-variety with ${X(K)\neq\emptyset}Let K = k(C) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally connected K-variety with X(K) 1 ?{X(K)\neq\emptyset}. Under the assumption that X admits a smooth projective model p: X? C{\pi: \mathcal{X}\to C}, we prove the following weak approximation results: (1) if k is a large field, then X(K) is Zariski dense; (2) if k is an infinite algebraic extension of a finite field, then X satisfies weak approximation at places of good reduction; (3) if k is a nonarchimedean local field and R-equivalence is trivial on one of the fibers X_p{\mathcal{X}_p} over points of good reduction, then there is a Zariski dense subset W í C(k){W\subseteq C(k)} such that X satisfies weak approximation at places in W. As applications of the methods, we also obtain the following results over a finite field k: (4) if |k| > 10, then for a smooth cubic hypersurface X/K, the specialization map X(K)? ?_{p ? P}X_p(k(p)){X(K)\longrightarrow \prod_{p\in P}\mathcal{X}_p(\kappa(p))} at finitely many points of good reduction is surjective; (5) if char k 1 2, 3{\mathrm{char}\,k\neq 2,\,3} and |k| > 47, then a smooth cubic surface X over K satisfies weak approximation at any given place of good reduction.     

Equality of norm groups of subextensions of Sn(n?5) extensions of algebraic number fields

Let E/k be a Galois extension of algebraic number fields with the Galois group isomorphic to the symmetric group S_n on n?5 letters. For any field extensions k⊆K, L⊆E a necessary and a sufficient condition is given for the equality to hold, where is the group of norms from K to k of the elements of the multiplicative group K∗ of K.     

On the Norm Equation Over Function Fields

If K is an algebraic function field of one variable over analgebraically closed field k and F is a finite extension ofK, then any element a of K can be written as a norm of someb in F by Tsen's theorem. All zeros and poles of a lead to zerosand poles of b, but in general additional zeros and poles occur.The paper shows how this number of additional zeros and polesof b can be restricted in terms of the genus of K, respectivelyF. If k is the field of all complex numbers, then we use Abel'stheorem concerning the existence of meromorphic functions ona compact Riemann surface. From this, the general case of characteristic0 can be derived by means of principles from model theory, sincethe theory of algebraically closed fields is model-complete.Some of these results also carry over to the case of characteristicp>0 using standard arguments from valuation theory.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.