Non-vanishing of class group $L$-functions at the central point

Blomer, Valentin

Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 831-847.

Étant donné un corps quadratique imaginaire $K=\mathbb{Q}\left(\sqrt{-D}\right)$, notons $h$ son nombre de classes. Nous montrons qu’il existe une constante $c$ telle que pour $D$ assez grand, au moins $c·h{\prod }_{p\mid D}(1-p^{-1})$ des $h$ fonctions $L$ distinctes $L_K(s,\chi )$ ne s’annulent pas au point central $s=1/2$.

Let $K=\mathbb{Q}\left(\sqrt{-D}\right)$ be an imaginary quadratic field, and denote by $h$ its class number. It is shown that there is an absolute constant $c>0$ such that for sufficiently large $D$ at least $c·h{\prod }_{p\mid D}(1-p^{-1})$ of the $h$ distinct $L$-functions $L_K(s,\chi )$ do not vanish at the central point $s=1/2$.

Classification : 11R42, 11M41, 11F67
Keywords: non-vanishing results, $L$-functions, imaginary quadratic fields, mollifier
Mot clés : théorèmes de non-annulation, fonctions $L$, corps quadratique imaginaire, fonction de mollification

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