Non-vanishing of $L$-functions for cyclotomic characters in function fields

Abstract:

In the number field case, it is conjectured that the central values $L(\frac {1}{2}, \chi )$ of $L$-functions are nonzero, where $\chi :(\mathbb {Z}/m\mathbb {Z})^*\rightarrow \mathbb {C}^*$ is a primitive Dirichlet character with conductor $m$. We resolve this conjecture in the function field case by proving that there are infinitely many cyclotomic characters for which the central values of $L$-functions are nonzero. In detail, for a given positive integer $n$, we compute the mean value of $L(\frac {1}{2},\eta \chi _n)$ and that of $L(\frac {1}{2}, \chi _n)$ for $\chi _n \in O_n$, where $f$ is a monic irreducible polynomial in $A=\mathbb {F}_q[t]$, $\mathbb {F}_q$ is the finite field of characteristic $p$, $\chi _n : (A/f^{n} A)^* \rightarrow \mathbb {C}^*$ is a character with some $p$-power order, $O_n$ is the set of all the primitive cyclotomic characters $\chi _n$ modulo $f^n$ with $p$-power order, $g$ is a monic polynomial in $A$ that is relatively prime to $f$, and $\eta : (A/g A)^* \rightarrow \mathbb {C}^*$ is a primitive even character.