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

Blomer, Valentin

doi:10.5802/aif.2035

Non-vanishing of class group

L

-functions at the central point
[Non-annulation des fonctions

L

de groupes de classes au point central]

Blomer, Valentin ¹

¹ University of Toronto, Department of Mathematics, 100 St. George Street, Toronto M5S 3G3, Ontario, (Canada)

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

Résumé
Abstract

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

Let $K = ℚ (\sqrt{- D})$ 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 $c \cdot h \prod_{p ∣ D} (1 - p^{- 1})$ of the $h$ distinct $L$ -functions $L_{K} (s, χ)$ do not vanish at the central point $s = 1 / 2$ .

MR Zbl

DOI : 10.5802/aif.2035

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

Affiliations des auteurs :

Blomer, Valentin ¹

¹ University of Toronto, Department of Mathematics, 100 St. George Street, Toronto M5S 3G3, Ontario, (Canada)

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     title = {Non-vanishing of class group $L$-functions at the central point},
     journal = {Annales de l'Institut Fourier},
     pages = {831--847},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {4},
     year = {2004},
     doi = {10.5802/aif.2035},
     mrnumber = {2111013},
     zbl = {1063.11040},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2035/}
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Blomer, Valentin. Non-vanishing of class group $L$-functions at the central point. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 831-847. doi : 10.5802/aif.2035. http://www.numdam.org/articles/10.5802/aif.2035/

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