Real Function Fields Research Articles

Indivisibility of divisor class numbers of Kummer extensions over the rational function field

We find a complete criterion for a Kummer extension K over the rational function field k=Fq(T) of degree ℓ to have indivisibility of its divisor class number hK by ℓ, where Fq is the finite field of order q and ℓ is a prime divisor of q−1. More importantly, when hK is not divisible by ℓ, we have hK≡1(modℓ). In fact, the indivisibility of hK by ℓ depends on the number of finite primes ramified in K/k and whether or not the infinite prime of k is unramified in K. Using this criterion, we explicitly construct an infinite family of the maximal real cyclotomic function fields whose divisor class numbers are divisible by ℓ.

Algorithms for quadratic forms over real function fields

Algorithms for quadratic forms over real function fields

Lower bounds for Pythagoras numbers of function fields

We show that the transcendence degree of a real function field over an arbitrary real base field is a strict lower bound for its Pythagoras number and a weak lower bound for all its higher Pythagoras numbers.

HILBERT 3-CLASS FIELD TOWERS OF REAL CUBIC FUNCTION FIELDS

In this paper we study the infiniteness of Hilbert 3-class field tower of real cubic function fields over , where mod 3. We also give various examples of real cubic function fields whose Hilbert 3-class field tower is infinite.

A DETERMINANT FORMULA FOR CONGRUENT ZETA FUNCTIONS OF REAL ABELIAN FUNCTION FIELDS

In this paper we give a determinant formula for congruent zeta functions of real Abelian function fields. We also give some example of congruent zeta functions when the conductor of real Abelian function field is monic irreducible.

IMAGINARY BICYCLIC FUNCTION FIELDS WITH THE REAL CYCLIC SUBFIELD OF CLASS NUMBER ONE

Let <TEX>$k={\mathbb{F}}_q(T)$</TEX> and <TEX>${\mathbb{A}}={\mathbb{F}}_q[T]$</TEX>. Fix a prime divisor <TEX>${\ell}$</TEX> q-1. In this paper, we consider a <TEX>${\ell}$</TEX>-cyclic real function field <TEX>$k(\sqrt[{\ell}]P)$</TEX> as a subfield of the imaginary bicyclic function field K = <TEX>$k(\sqrt[{\ell}]P,\;(\sqrt[{\ell}]{-Q})$</TEX>, which is a composite field of <TEX>$k(\sqrt[{\ell}]P)$</TEX> wit a <TEX>${\ell}$</TEX>-cyclic totally imaginary function field <TEX>$k(\sqrt[{\ell}]{-Q})$</TEX> of class number one. und give various conditions for the class number of <TEX>$k(\sqrt[{\ell}]{P})$</TEX> to be one by using invariants of the relatively cyclic unramified extensions <TEX>$K/F_i$</TEX> over <TEX>${\ell}$</TEX>-cyclic totally imaginary function field <TEX>$F_i=k(\sqrt[{\ell}]{-P^iQ})$</TEX> for <TEX>$1{\leq}i{\leq}{\ell}-1$</TEX>.

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field F q , there exist infinitely many real quadratic function fields over F q such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field F q , of infinitely many real function fields over F q with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over F q such that the numerator of their zeta function is an irreducible polynomial.

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.

The Scholz theorem in function fields

The Scholz theorem in function fields states that the l-rank difference between the class groups of an imaginary quadratic function field and its associated real quadratic function field is either 0 or 1 for some prime l. Furthermore, Leopoldt's Spiegelungssatz (= the Reflection theorem) in function fields yields a comparison between the m-rank of some subgroup of the class group of an imaginary cyclic function field L 1 and the m-rank of some subgroup of the class group of its associated real cyclic function field L 2 for some prime number m; then their m-ranks also equal or differ by 1. In this paper we find an explicit necessary condition for their m-ranks (respectively l-ranks) to be the same in the case of cyclic function fields (respectively quadratic function fields). In particular, in the case of quadratic function fields, if l does not divide the regulator of L 2 , then their l-ranks are the same, equivalently if their l-ranks differ by 1, then l divides the regulator of L 2 .

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.

Pythagoras numbers of function fields of hyperelliptic curves with good reduction

It is shown that the Pythagoras number of a real function field of a hyperelliptic curve C, with good reduction, defined over the real formal power series field is equal to 2. A main tool in the proof is the explicit description of the Brauer group of C. If the function field of such a curve is non-real then it is shown that its Pythagoras number is 3.

Abelian subgroups of any order in class groups of global function fields

Let F be a finite field with q elements, and T a transcendental element over F . In this paper, we construct infinitely many real function fields of any fixed degree over F(T) with ideal class numbers divisible by any given positive integer greater than 1. For imaginary function fields, we obtain a stronger result which shows that for any relatively prime integers m and n with m, n>1 and relatively prime to the characteristic of F , there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to ( Z/n Z) m−1 .

SUBGROUP THEOREMS FOR FREE PROFINITE PRODUCTS WITH AMALGAMATION

Binz, Neukirch and Wenzel proved a profinite version of the Kurosh subgroup theorem. In the paper a subgroup theorem for certain profinite fundamental groups of graphs of profinite groups is shown. In particular, the case of free products with amalgamation is discussed and illustrated in the context of real function fields in one variable.

Levels of Function Fields of Surfaces over Number Fields

We study the level of nonformally real function fields of surfaces over number fields and show that it is at most 4 for a large class of surfaces.

Sums of powers in function fields

When is a rational function on a smooth variety V whose values are sums of 2 nth powers, itself a sum of 2 nth powers in the field of rational functions? We investigate this property and some weaker form of it. The aim is to gain a better understanding of sums of powers in formally real function fields.

Null- and Positivstellensa¨tze for generalized real closed fields

Null- and Positivstellensa¨tze for all generalized real closed fields are proved. These fields play an important role for the study of sums of powers in fields. The results of this paper are especially related to the study of sums of powers in real function fields.

Some local-global principles in the arithmetic of algebraic groups over real function fields

Some local-global principles in the arithmetic of algebraic groups over real function fields

Some local-global principles in the arithmetic of algebraic groups over real function fields

Some local-global principles in the arithmetic of algebraic groups over real function fields

Corrigendum to the paper “Hermitian forms over division algebras over real function fields”

In the previous paper [T] we gave a classification of hermitian forms over the real function fieldk=R(t) and its completionsk v with respect to valuationsv trivial onR. Unfortunately in the local case the arguments given for cases A and D, in general, were not correct. Therefore the resulting local and local-global classifications obtained were incorrect. I would like also to thank Dr. D. Hoffmann for pointing out these mistakes and the referee for useful comments. Here we would like to make necessary corrections to [T]. We keep the same notation used there, except that in the first paragraph,J is not the standard involution of a quaternion division algebraD (with basis {1,i,j,ij}). All hermitian forms will be hermitian forms with respect toJ, with values inD.

On some new local-global principles over a real function field

We prove here the validity of the cohomological Hasse principle for almost simple simply connected groups of classical type defined over the rational real function field R(t) and the Hasse principle (analog of the Platonov conjecture) about the projective simplicity of the groups of rational points of such groups.