论文标题
在功能场的中央点上的dirichlet l功能消失
Vanishing of Dirichlet L-functions at the central point over function fields
论文作者
论文摘要
我们给出了与循环字符相关的dirichlet $ l $ functions $ \ mathbb {f} _q（t）$在中央点$ s = 1/2 $上消失的几何标准。这个想法是基于这样的观察结果，即可以将消失在中心点的消失被解释为从与角色相关的投影曲线到$ \ mathbb {f} _q $的某些Abelian品种的图。使用此几何标准，我们在$ \ mathbb {f} _q（t）$上获得了一个立方字符数量的下限，其$ l $ functions在中央点消失了，其中$ q = p = p = p^{4n} $，对于任何有理的prime prime $ prime $ p \ equiv 2 \ equiv 2 \ bmod 3 $。我们还使用有关存在超椭圆形曲线存在的最新结果，以推断出其他订单的Dirichlet字符的$ l $ functions的后果。
We give a geometric criterion for Dirichlet $L$-functions associated to cyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanish at the central point $s=1/2$. The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over $\mathbb{F}_q$. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over $\mathbb{F}_q(t)$ whose $L$-functions vanish at the central point where $q=p^{4n}$ for any rational prime $p \equiv 2 \bmod 3$. We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the $L$-functions of Dirichlet characters of other orders.
