$L$-polynomials of function fields and Fibonacci Numbers

Abstract

We show that for any natural number $m$, there is an integer $g_0\ge m$ such that for any $g\ge g_0$ there exists a function field $F/\mathbb {F}_q$ of genus $g$ whose $L$-polynomial satisfies

$$\begin{aligned} L(t)\equiv 1+F_1 t+\cdots +F_{m}t^{m}\pmod {t^{m+1}}, \end{aligned}$$

where $F_1,\ldots , F_{m}$ are the first $m$ Fibonacci numbers. This fact shows that the set $\{F_1,\ldots ,F_m\}$ is a non-trivial example of a set of integers satisfying the conditions of a recent theorem by Anbar and Stichtenoth [(Bull Braz Math Soc 44:173–193, 2013), Thm. 8.1] about the first coefficients of the $L$-polynomial of a function field.