Average value of the divisor class numbers of real cubic function fields

1 Introduction

There have been many developments in the study of moments of L -function families since they have many connections to the famous Lindelöf hypothesis for such L -functions [1]. In fact, Gauss [2] made two conjectures on the mean value of the class numbers of quadratic number fields: one is for the imaginary case and the other is for the real case. The conjecture on the imaginary case was proved by Lipschitz [3], Mertens [4], Siegel [5], and Vinogradov [6], and the conjecture on the real case was proved by Siegel [5].

In the function field context as well, there has been active research done on the study of moments of L -function families and class numbers of global function fields (e.g., refer to [7–16]). Let k ≔ F q ( T ) be the rational function field and A ≔ F q [ T ] , where F q is a finite field of order q . Let K be a global function field, which is an algebraic extension over k . We say that K is real if the infinite place ∞ of k splits completely in K ; otherwise, we call K imaginary. Inspired by Gauss’s conjectures, Hoffstein and Rosen [9] computed the mean value of the (divisor) class numbers h m of quadratic function fields k ( m ) , where m is a nonsquare polynomial in A . We point out that they computed both cases of imaginary fields and real fields. In detail, for a positive integer n , let A n + be the set of monic polynomials in A of degree n . They obtained the following results: if n is odd (in this case, k ( m ) is imaginary), then

and if n is even (in this case, k ( m ) is real), then

where ζ ( s ) is the zeta function of A . In fact, we note that h m = h ˜ m R m [17, Prop. 14.7], where h ˜ m is the ideal class number of k ( m ) (the order of the ideal class group of the maximal order O k ( m ) of k ( m ) ) and R m is the regulator of k ( m ) .

Now, we discuss the case of cubic fields. As a matter of fact, in the number field situation, there has been no work done on the mean value computation for cubic fields yet. On the other hand, in the function field context, Lee et al. [18] obtained an asymptotic formula for the mean value of the divisor class numbers of cubic function fields K m = k ( m 3 ) , where q ≡ 1 ( mod 3 ) , m ∈ F q [ T ] is a cube-free polynomial, and deg ( m ) ≡ 1 ( mod 3 ) ; in this case, K m is imaginary. Therefore, the goal of this article is to compute an asymptotic formula for the mean value of the divisor class numbers of real cubic function fields K m ; in this case, deg ( m ) is divisible by 3. We note that the infinite place ∞ of k splits completely in K m when the degree of m is divisible by 3. To achieve our goal, we compute the mean value of ∣ L ( s , χ ) ∣ 2 evaluated at s = 1 when χ average runs through the primitive cubic even Dirichlet characters of F q [ T ] as in Theorem 1.1, where L ( s , χ ) is the associated Dirichlet L -function.

We state the main results as follows.

Notation 1.

As a consequence, we find an asymptotic formula for the average value of the class numbers of cubic real function fields in Theorem 1.2.

We briefly mention the difference between our current work and the previous work [18] as follows. For computation of the mean value of ∣ L ( s , χ ) ∣ 2 evaluated at s = 1 , in [18], χ runs through the primitive cubic odd Dirichlet characters of F q [ T ] ; in this article, we deal with the case of the even Dirichlet characters of F q [ T ] . We emphasize that the computational complexity for ∣ L ( 1 , χ ) ∣ 2 with even characters χ increases significantly compared with the case of odd characters χ . In fact, the major difference between the even case and the odd case in terms of complexity comes from the difference between two functional equations of L ( s , χ ) for odd and even primitive characters as follows. Let χ be a primitive character of modulus R ≠ 1 . By [1, Theorem 3.9], if χ is odd, then the functional equation is

and if χ is even, then the functional equation satisfies the following:

with ∣ W ( χ ) ∣ = 1 .

For the case where χ is odd, taking the squared modulus of both sides of the functional equation and letting s = 1 , we obtain [18, Lemma 3.1] the following:

where L i ( χ ) ≔ ∑ deg a = i , a ∈ A + χ ( a ) .

Unlike the odd case, if χ is even, we need to take derivatives of both sides of the functional equation of L ( s , χ ) with respect to s twice. Letting s = 1 , we obtain

where M i ( χ ) ≔ q L i − 1 ( χ ) − L i ( χ ) , L − 1 ( χ ) ≔ 0 , and L g + 2 ( χ ) = 0 . Due to this difference, we point out that Lemma 3.2 plays a significant role for our main computation.

This article is organized as follows: in Section 2, we recall some basic definitions and necessary lemmas that are useful for our main results; in Section 3, we estimate the value ∑ χ ∣ L ( 1 , χ ) ∣ 2 (Lemmas 3.1–3.5), and for the computation of ∑ χ ∣ L ( 1 , χ ) ∣ 2 , we divide the formula of ∑ χ ∣ L ( 1 , χ ) ∣ 2 into three parts; and finally in Section 4, we give the proofs of our main results: Theorems 1.1 and 1.2.

2 Preliminaries

Let F q be a finite field of order q , where q is an odd prime power with q ≡ 1 ( mod 3 ) . Let k ≔ F q ( T ) be the rational function field and A ≔ F q [ T ] be a polynomial ring. For a nonzero polynomial f ∈ A , the norm of f is defined as ∣ f ∣ ≔ q deg ( f ) . We denote the set of monic polynomials of A by A + .

The trivial Dirichlet character of modulus h is defined by χ ( a ) = 1 if ( a , h ) = 1 , 0 if ( a , h ) ≠ 1 ; we denote this by χ 0 . The inverse of a Dirichlet character χ , denoted by χ ¯ , is defined by χ ¯ ( a ) = χ ( a ) ¯ for all a ∈ A , where χ ( a ) ¯ is a complex conjugate of χ ( a ) . We say that a character χ is even if χ ( c ) = 1 for all c ∈ F q × ; otherwise, it is called an odd character. A character χ such that χ 3 = χ 0 and χ ≠ χ 0 is called a cubic Dirichlet character.

A Dirichlet character of modulus h induces a homomorphism ( A ∕ h A ) × → C × . Conversely, given such a homomorphism, there is a uniquely corresponding Dirichlet character [17, p. 35]. Abusing the notation, let χ : ( A ∕ h A ) × → C × be a Dirichlet character of modulus h . For a Dirichlet character χ of modulus h , we say that we may define χ mod f for f ∣ h if there exists ξ : ( A ∕ f A ) × → C × such that ξ ∘ φ h , f = χ , where φ h , f is a canonical homomorphism from ( A ∕ h A ) × to ( A ∕ f A ) × . We note that given a Dirichlet character χ of modulus h , there exists a unique monic polynomial f ∈ A of minimal degree dividing h such that χ can be defined mod f [19, Theorem 12.6.3].

We now introduce the definition of the cubic character χ p defined by the cubic residue symbol, where p ∈ A + is an irreducible polynomial.

We now define an important set M g as follows. Let M g be the set of monic cube-free polynomials m in A ≔ F q [ T ] such that the degree of m is divisible by 3 and the genus of K m = k ( m 3 ) is g , where g is a positive integer. Since m is a monic cube-free polynomial, there are monic square-free polynomials m 1 and m 2 in A with ( m 1 , m 2 ) = 1 such that m = m 1 m 2 2 . By [13, Lemma 3.2], we obtain ( 3 − 1 ) ( deg ( m 1 ) + deg ( m 2 ) − 2 ) 2 = g , i.e., deg ( m 1 ) + deg ( m 2 ) = g + 2 . Therefore, the set M g can be written as follows:

Let χ m be the cubic character associated with K m , where m ∈ M g . From the condition that 3 divides the degree of m , the character χ m is even [8, p. 1273]. In addition, we note that the conductor of χ m is m 1 m 2 .

3 Necessary lemmas for main computations

In this section, we prove five important lemmas for finding the second moment of the class numbers of real cubic function fields with q ≡ 1 ( mod 3 ) .

We define a set S g to be

we note that S g is a set of primitive cubic even characters with conductor whose degree is g .

For our computation, we need the following lemma.

We recall that A + refers to the set of monic polynomials of A = F q [ T ] and A d is the set of monic square-free polynomials of A of degree d . From now on, we denote by A the set of monic square-free polynomials of A . In addition, for some f ∈ A + , we denote by a b 2 = □ if a b 2 can be written as f 3 . If not, we put a b 2 ≠ □ .

Notation 2.