Real 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.

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.

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.

Explicit infrastructure for real quadratic function fields and real hyperelliptic curves

In 1989, Koblitz first proposed the Jacobian of a an imaginary hyperelliptic curve for use in public-key cryptographic proto- cols. This concept is a generalization of elliptic curve cryptography. It can be used with the same assumed key-per-bit strength for small genus. More recently, real hyperelliptic curves of small genus have been introduced as another source for cryptographic protocols. The arithmetic is more in- volved than its imaginary counterparts and it is based on the so-called infrastructure of the set of reduced principal ideals in the ring of regular functions of the curve. This infrastructure is an interesting phenomenon. The main purpose of this article is to explain the infrastructure in explicit terms and thus extend Shanks' infrastructure ideas in real quadratic num- ber fields to the case of real quadratic congruence function fields and their curves. Hereby, we first present an elementary introduction to the contin- ued fraction expansion of real quadratic irrationalities and then generalize important results for reduced ideals.

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.

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 .

Exponents of class groups of real quadratic function fields (II)

Let g g be an even positive integer. We show that there are ≫ q l / g / l 2 \gg q^{l/g}/l^2 polynomials D ∈ F q [ t ] D\in \mathbb F_q[t] with deg ⁡ ( D ) ≤ l \deg (D)\le l such that the ideal class group of the real quadratic extensions F q ( t , D ) \mathbb F_q(t,\sqrt D) have an element of order g g .

An Improved Real-Quadratic-Field-Based Key Exchange Procedure

To date, the only non-group structure that has been suitably employed as the key space for Diffie-Hellman-type cryptographic key exchange is the infrastructure of a real quadratic (number or function) field. We present an implementation of a Diffie-Hellman-type protocol based on real quadratic number field arithmetic that provides a significant improvement in performance over previous versions of this scheme. This dramatic speed-up is achieved by replacing the ordinary multiplication and reduction procedures for reduced ideals by a new version of the NUCOMP algorithm due to Shanks.

Exponents of class groups of real quadratic function fields

We show that there are ≫ q l / ( 2 g ) \gg q^{l/(2g)} polynomials D ∈ F q [ t ] D\in \mathbb {F}_q[t] with deg ⁡ ( D ) ≤ l \deg (D)\le l such that the ideal class group of the real quadratic extensions F q ( t , D ) \mathbb {F}_q(t,\sqrt D) has an element of order g g .

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields $$ K = k{\left( {{\sqrt D }} \right)} $$ over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.

The parallelized Pollard kangaroo method in real quadratic function fields

We show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields. Specifically, we show how to apply the kangaroo method to the infrastructure in these fields. We also show how to speed up the computation by using heuristics on the distribution of the divisor class number, and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field. Furthermore, we provide examples for regulators and class numbers of hyperelliptic function fields of genus 3 that are larger than those ever reported before.

Computing discrete logarithms in real quadratic congruence function fields of large genus

The discrete,logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields.

On real quadratic function fields of Chowla type with ideal class number one

Let F q \mathbb {F}_q be the finite field with q q elements, (2 ⧸ | q {\not |}q ), k = F q ( x ) k= \mathbb {F}_q(x) , K = k ( D ) K=k(\sqrt {D}) where D = D ( x ) = A ( x ) 2 + a D=D(x) =A(x)^2+a is a square-free polynomial in F q [ x ] \mathbb {F}_q[x] with deg ⁡ A ( x ) ≥ 1 \deg A(x)\geq 1 and a ∈ F q ∗ a\in \mathbb { F}_q^* . In this paper several equivalent conditions for the ideal class number h ( O K ) h(O_K) to be one are presented and all such quadratic function fields with h ( O K ) = 1 h(O_K)=1 are determined.

Some Methods for Evaluating the Regulator of a Real Quadratic Function Field

We develop methods for the rapid computation of the regulator of a real quadratic congruence function field K = k(x)(√D). By extending Shanks' infrastructure ideas in real quadratic number fields to real quadratic congruence function fields we obtain a baby step–giant step method for evaluating the regulator of K in O(|D|¼) polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in O(|D|⅕) polynomial operations. By implementing both methods on a computer, we found that the O(|D|⅕) algorithm tends to be far better than the baby step–giant step algorithm in those cases where the regulator exceeds 108.

Estimates for Coefficients ofL-Functions for Function Fields

We consider the Dirichlet characters for polynomial rings Fq[T] and the associatedL-functions. By Weil's result, the associatedL-functions are all polynomials. Applying Burgess' idea, we obtain an upper bound for the coefficients of theseL-functions. As an application, using our estimates, we obtain an upper bound for the degree of the fundamental unit in real quadratic function fields.