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

Abstract

We compute an asymptotic formula for the divisor class numbers of real cubic function fields Km=k(m3){K}_{m}=k\left(\sqrt[3]{m}), where Fq{{\mathbb{F}}}_{q} is a finite field with qq elements, q≡1(mod3)q\equiv 1\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3), k≔Fq(T)k:= {{\mathbb{F}}}_{q}\left(T) is the rational function field, and m∈Fq[T]m\in {{\mathbb{F}}}_{q}\left[T] is a cube-free polynomial; in this case, the degree of mm is divisible by 3. For computation of our asymptotic formula, we find the average value of ∣L(s,χ)∣2{| L\left(s,\chi )| }^{2} evaluated at s=1s=1 when χ\chi goes through the primitive cubic even Dirichlet characters of Fq[T]{{\mathbb{F}}}_{q}\left[T], where L(s,χ)L\left(s,\chi ) is the associated Dirichlet LL-function.

Keywords

• l-function
• average value of class number
• cubic function field
• moment over function field
• 11m38
• 11r29
• 11m06
• 11r16
• 11r58