论文标题
关于椭圆曲线扭曲的$ l $ functions在理性功能字段上的消失
On the vanishing of twisted $L$-functions of elliptic curves over rational function fields
论文作者
论文摘要
我们在本文中调查了$ s = 1 $的消失，椭圆曲线的扭曲$ l $ functions $ e $定义在有理函数字段$ \ mathbb {f} _q（t）$（$ \ mathbb {f} _q $ y Mathbb {f} _q $是$ q $ emements and tarnecormist $ ememitive $ e e e e e e geq 5 $ geq 5 $）的有限领域， 3 $，从理论和数值的角度来看。在数字字段的情况下，可以预测这种消失是一个非常罕见的事件，我们的数值数据似乎表明，对于非恒定曲线的功能字段，情况也是如此。 For constant curves, we adapt the techniques of Li and Donepudi--Li who proved vanishing at $s=1/2$ for infinitely many Dirichlet $L$-functions over $\mathbb{F}_q(t)$ based on the existence of one, and we can prove that if there is one $χ_0$ such that $L(E, χ_0, 1)=0$, then there are infinitely many.最后，我们提供了一些示例，这些示例表明，在$ \ mathbb {f} _q（t）$上的扭曲$ l $ functions的行为与一般的曲线不同。
We investigate in this paper the vanishing at $s=1$ of the twisted $L$-functions of elliptic curves $E$ defined over the rational function field $\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and characteristic $\geq 5$) for twists by Dirichlet characters of prime order $\ell \geq 3$, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of Li and Donepudi--Li who proved vanishing at $s=1/2$ for infinitely many Dirichlet $L$-functions over $\mathbb{F}_q(t)$ based on the existence of one, and we can prove that if there is one $χ_0$ such that $L(E, χ_0, 1)=0$, then there are infinitely many. Finally, we provide some examples which show that twisted $L$-functions of constant elliptic curves over $\mathbb{F}_q(t)$ behave differently than the general ones.
