$p$-adic analytic twists  and strong subconvexity

Blomer, Valentin; Milićević, Djordje

doi:10.24033/asens.2252

p

-adic analytic twists and strong subconvexity
[Twists

p

-adiques analytiques et sous-convexité forte]

Blomer, Valentin ; Milićević, Djordje

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 561-605.

Résumé
Abstract

Soit $f$ une forme primitive nouvelle (holomorphe ou de Maass). Soient $p$ un nombre premier, $n \geq 1$ un entier, et $t$ un nombre réel. Nous démontrons une borne sous-convexe de type Weyl pour la fonction $L$ de $f$ , tordue par un caractère de Dirichlet $χ$ de conducteur $q = p^{n}$ . Plus précisément, on démontre $L (f \otimes χ, 1 / 2 + i t) ≪_{p, t} q^{1 / 3 + ε}$ , avec une dépendance polynomiale et explicite en $p$ et $t$ . La preuve repose sur la compensation entre les valeurs propres de Hecke de $f$ et les valeurs de $χ$ , dont l'oscillation est gouvernée par une phase $p$ -adique analytique. Au cours de la démonstration, on développe quelques outils $p$ -adiques, analogues de méthodes classiques ou archimédiennes, telles que la dissection de Farey et la méthode de van der Corput pour les sommes d'exponentielles.

Let $f$ be a fixed cuspidal (holomorphic or Maaß) newform. We prove a Weyl-exponent subconvexity bound $L (f \otimes χ, 1 / 2 + i t) ≪_{p, t} q^{1 / 3 + ε}$ for the twisted $L$ -function of $f$ with a Dirichlet character $χ$ of prime power conductor $q = p^{n}$ (with an explicit polynomial dependence on $p$ and $t$ ). We obtain our result by exhibiting strong cancellation between the Hecke eigenvalues of $f$ and the values of $χ$ , which act as twists by exponentials with a $p$ -adically analytic phase. Among the tools, we develop a general result on $p$ -adic approximation by rationals (a $p$ -adic counterpart to Farey dissection) and a $p$ -adic version of van der Corput's method for exponential sums.

Publié le : 2015-09-24

MR Zbl

DOI : 10.24033/asens.2252

Classification : 11F66, 11L40, 11L07
Keywords: Subconvexity, $L$-functions, character twists, depth aspect, character sums, exponential sums, method of stationary phase, $p$-adic analysis.
Mot clés : Sous-convexité, fonctions $L$, torsion par des caractères, sommes de caractères, sommes exponentielles, phase stationnaire, analyse $p$-adique.

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     author = {Blomer, Valentin and Mili\'cevi\'c, Djordje},
     title = {$p$-adic analytic twists  and strong subconvexity},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {561--605},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {3},
     year = {2015},
     doi = {10.24033/asens.2252},
     mrnumber = {3377053},
     zbl = {1401.11095},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2252/}
}

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Blomer, Valentin; Milićević, Djordje. $p$-adic analytic twists  and strong subconvexity. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 561-605. doi : 10.24033/asens.2252. http://www.numdam.org/articles/10.24033/asens.2252/

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