Quadratic Function Fields Research Articles

Diophantine approximation with prime denominator in real quadratic function fields

In the thirties of the last century, I.M. Vinogradov proved that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent 1/5 by 1/4 using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. Recently, in [3], the authors established an analogue of Matomäki's result on imaginary quadratic extensions of the function field k=Fq(T). In this paper, we consider the case of real quadratic extensions of k of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent θ=1/4). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper [3] on the imaginary quadratic case. We also simplify arguments in the paper [1] on the same problem for real quadratic number fields by D. Mazumder and the first-named author.

The moments and statistical distribution of class number of primes over function fields

We investigate the moment and the distribution of L(1,χP), where χP varies over quadratic characters associated to irreducible polynomials P of degree 2g+1 over Fq[T] as g→∞. In the first part of the paper, we compute the integral moments of the class number hP associated to quadratic function fields with prime discriminants P, and this is done by adapting to the function field setting some of the previous results carried out by Nagoshi in the number field setting. In the second part of the paper, we compute the complex moments of L(1,χP) in large uniform range and investigate the statistical distribution of the class numbers by introducing a certain random Euler product. The second part of the paper is based on recent results carried out by Lumley when dealing with square-free polynomials.

Quasilinear quadratic forms and function fields of quadrics

Let p and q be anisotropic quadratic forms of dimension $$\ge 2$$ over a field F. In a recent article, we formulated a conjecture describing a general constraint which the dimensions of p and q impose on the isotropy index of q after scalar extension to the function field of p. This can be viewed as a generalization of Hoffmann’s Separation Theorem which simultaneously incorporates and refines some well-known classical results on the Witt kernels of function fields of quadrics. Using algebro-geometric methods, it was shown that large parts of this conjecture hold in the case where the characteristic of F is not 2. In the present article, we prove similar (in fact, somewhat sharper) results in the case where F has characteristic 2 and q is a so-called quasilinear form. In contrast to the situation where $$\mathrm {char}(F) \ne 2$$, the methods used to treat this case are purely algebraic.

Non-abelian Cohen–Lenstra heuristics over function fields

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$extensions of imaginary quadratic number fields for$p$an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of$\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$extensions, as$q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

Class number parities of compositum of quadratic function fields

Abstract The parities of ideal class numbers of compositum of quadratic function fields are studied. Especially, the parities of ideal class numbers of F q ( t ) ( P , Q ) and F q ( t ) ( P , Q , R ) $\mathbb{F}_q(t)(\sqrt P,\sqrt Q) \text{ and } \mathbb{F}_q(t)(\sqrt P,\sqrt Q,\sqrt R)$ are completely determined in detail, where P,Q,R are monic irreducible polynomials of even degrees.

Imaginary quadratic function fields with ideal class group of prime exponent

For K, an imaginary quadratic extension of a rational function field over a finite field, in which the infinite place ramifies, we give necessary conditions (illustrating for exponent three) for the ideal class group to have odd prime exponent. For exponent two we classify all such extensions, taking this opportunity to complete the list that we previously had.

Average values of L-series for real characters in function fields

We establish asymptotic formulae for the first and second moments of quadratic Dirichlet L-functions, at the center of the critical strip, associated to the real quadratic function field \(k(\sqrt{P})\) and inert imaginary quadratic function field \(k(\sqrt{\gamma P})\) with P being a monic irreducible polynomial over a fixed finite field \(\mathbb {F}_{q}\) of odd cardinality q and \(\gamma \) a generator of \(\mathbb {F}_{q}^{\times }\). We also study mean values for the class number and for the cardinality of the second K-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over \(\mathbb {F}_{q}\). One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet L-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average L-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of L-functions over function fields.

Indivisibility of class numbers of real quadratic function fields

In this paper we work on indivisibility of the class numbers of real quadratic function fields. We find an explicit expression for a lower bound of the density of real quadratic function fields (with constant field F) whose class numbers are not divisible by a given prime ℓ. We point out that the explicit lower bound of such a density we found only depends on the prime ℓ, the degrees of the discriminants of real quadratic function fields, and the condition: either |F|≡1(modℓ) or not.

Construction of all cubic function fields of a given square-free discriminant

For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.

DENSITIES FOR 4-RANKS OF REAL QUADRATIC FUNCTION FIELDS

In this paper we study of densities of the 4-rank of narrow ideal class groups of real quadratic function fields over the rational function field Fq(T ) when q ≡ 3 mod 4.

HILBERT 2-CLASS FIELD TOWERS OF REAL QUADRATIC FUNCTION FIELDS

In this paper we study the infiniteness of Hilbert 2-class field towers of real quadratic function fields over Fq(T), where q is a power of an odd prime number.

REAL QUADRATIC FUNCTION FIELDS OF MINIMAL TYPE

In this paper, we will introduce the notion of the real quadratic function fields of minimal type, which is a function field analogue to Kawamoto and Tomita's notion of real quadratic fields of minimal type. As number field cases, we will show that there are exactly 6 real quadratic function fields of class number one that are not of minimal type.

HILBERT 2-CLASS FIELD TOWERS OF INERT IMAGINARY QUADRATIC FUNCTION FIELDS

In this paper we study the infniteness of Hilbert 2-class field towers of inert imaginary quadratic function fields over <TEX>$\mathbb{F}_q(T)$</TEX>, where <TEX>$q$</TEX> is a power of an odd prime number.

HILBERT 2-CLASS FIELD TOWERS OF IMAGINARY QUADRATIC FUNCTION FIELDS

In this paper we study the infiniteness of Hilbert 2-class field towers of imaginary quadratic function fields over , where is a power of an odd prime number.

Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field \(\mathbb{F}_{q}(x)\) whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥ 3, we show that for every ε > 0, there are \(\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon)}\) polynomials \(f \in \mathbb{F}_{q}[x]\) with \(\deg f=L\), for which the class group of the quadratic extension \(\mathbb{F}_{q}(x, \sqrt{f})\) has an element of order g. This sharpens the previous lower bound \(q^{L(\frac{1}{2}+\frac{1}{g})}\) of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.

ONO INVARIANTS OF IMAGINARY QUADRATIC FUNCTION FIELDS

In this paper, we define the Ono invariants of imaginary quadratic function fields and obtained several results concerning the relations between the Ono invariants and the class numbers.

4-RANKS OF CLASS GROUPS OF QUADRATIC EXTENSIONS OF CERTAIN QUADRATIC FUNCTION FIELDS

We obtain some density results for the 4-ranks of class groups of quadratic extensions of quadratic function fields analogous to those results of F. Gerth in the classical case.

On the 4-rank of ideal class groups of quadratic function fields

On the 4-rank of ideal class groups of quadratic function fields

Real quadratic function fields of Richaud-Degert type with ideal class number one

We determine all real quadratic function fields of Richaud-Degert type with ideal class number one.

Density of class groups of imaginary ℓ-cyclic function fields

In this paper we generalize a recent result of Wittmann on densities of the 4-rank of class groups of imaginary quadratic function fields to the ℓ-rank of class groups of imaginary ℓ-cyclic extensions of F q ( T ) where ℓ is any odd prime dividing q − 1 .