$p$-adic analytic twists  and strong subconvexity

Blomer, Valentin; Milićević, Djordje

doi:10.24033/asens.2252

Geodesic

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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.

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Abstract (VO)
Résumé

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.

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.

Publié le : 2015-05-27

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.
Mots-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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/articles/10.24033/asens.2252/

Bibliographie
Cité par

Breuillard, E.; Gelander, T. A topological Tits alternative, Ann. of Math., Volume 166 (2007), pp. 427-474 (ISSN: 0003-486X) | MR | Zbl | DOI

Blomer, V.; Harcos, G. Hybrid bounds for twisted $L$ -functions, J. reine angew. Math., Volume 621 (2008), pp. 53-79 addendum: J. reine angew. Math. 694 (2014), 241–244 (ISSN: 0075-4102) | MR | Zbl | DOI

Blomer, V.; Khan, R.; Young, M. Distribution of mass of holomorphic cusp forms, Duke Math. J., Volume 162 (2013), pp. 2609-2644 (ISSN: 0012-7094) | MR | Zbl | DOI

Barban, M. B.; Linnik, Y. V.; Tshudakov, N. G. On prime numbers in an arithmetic progression with a prime-power difference, Acta Arith., Volume 9 (1964), pp. 375-390 (ISSN: 0065-1036) | MR | Zbl | DOI

Blomer, V.; Michel, P. Hybrid bounds for automorphic forms on ellipsoids over number fields, J. Inst. Math. Jussieu, Volume 12 (2013), pp. 727-758 (ISSN: 1474-7480) | MR | Zbl | DOI

Blomer, V.; Milićević, D. Kloosterman sums in residue classes, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 51-69 (ISSN: 1435-9855) | MR | Zbl | DOI

Blomer, V.; Milićević, D. The Second Moment of Twisted Modular $L$ -Functions, Geom. Funct. Anal., Volume 25 (2015), pp. 453-516 (ISSN: 1016-443X) | MR | Zbl | DOI

Burgess, D. A. On character sums and $L$ -series. II, Proc. London Math. Soc., Volume 13 (1963), pp. 524-536 (ISSN: 0024-6115) | MR | Zbl | DOI

Bykovskiĭ, V. A. A trace formula for the scalar product of Hecke series and its applications, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 226 (1996), pp. 14-36 ; English translation: J. Math. Sci. (New York) 89 (1998), 915–932 (ISSN: 0373-2703) | MR | Zbl | DOI

Conrey, J. B.; Iwaniec, H. The cubic moment of central values of automorphic $L$ -functions, Ann. of Math., Volume 151 (2000), pp. 1175-1216 (ISSN: 0003-486X) | MR | Zbl | DOI

Chambert-Loir, A.; Tschinkel, Y. Integral points of bounded height on partial equivariant compactifications of vector groups, Duke Math. J., Volume 161 (2012), pp. 2799-2836 (ISSN: 0012-7094) | MR | Zbl | DOI

Cluckers, R. Analytic van der Corput lemma for $p$ -adic and $𝐅_{q} ((t))$ oscillatory integrals, singular Fourier transforms, and restriction theorems, Expo. Math., Volume 29 (2011), pp. 371-386 (ISSN: 0723-0869) | MR | Zbl | DOI

Fouvry, É.; Kowalski, E.; Michel, P. Algebraic trace functions over the primes, Duke Math. J., Volume 163 (2014), pp. 1683-1736 (ISSN: 0012-7094) | MR | Zbl | DOI

Fouvry, É.; Kowalski, E.; Michel, P. Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal., Volume 25 (2015), pp. 580-657 (ISSN: 1016-443X) | MR | Zbl | DOI

Graham, S. W.; Kolesnik, G., London Mathematical Society Lecture Note Series, 126, Cambridge Univ. Press, Cambridge, 1991, 120 pages (ISBN: 0-521-33927-8) | MR | Zbl | DOI

Good, A. The square mean of Dirichlet series associated with cusp forms, Mathematika, Volume 29 (1982), pp. 278-295 (ISSN: 0025-5793) | MR | Zbl | DOI

Gradshteyn, I. S.; Ryzhik, I. M., Academic Press, Inc., San Diego, CA, 2000, 1163 pages (ISBN: 0-12-294757-6) | MR | Zbl

Heath-Brown, D. R. Hybrid bounds for Dirichlet $L$ -functions, Invent. Math., Volume 47 (1978), pp. 149-170 (ISSN: 0020-9910) | MR | Zbl | DOI

Harcos, G.; Michel, P. The subconvexity problem for Rankin-Selberg $L$ -functions and equidistribution of Heegner points. II, Invent. Math., Volume 163 (2006), pp. 581-655 (ISSN: 0020-9910) | MR | Zbl | DOI

Harcos, G.; Templier, N. On the sup-norm of Maass cusp forms of large level. III, Math. Ann., Volume 356 (2013), pp. 209-216 (ISSN: 0025-5831) | MR | Zbl | DOI

Iwaniec, H.; Kowalski, E., American Mathematical Society Colloquium Publications, 53, Amer. Math. Soc., Providence, RI, 2004, 615 pages (ISBN: 0-8218-3633-1) | MR | Zbl

Iwaniec, H., Graduate Studies in Math., 17, Amer. Math. Soc., Providence, RI, 1997, 259 pages (ISBN: 0-8218-0777-3) | MR | Zbl

Jutila, M.; Motohashi, Y. Uniform bound for Hecke $L$ -functions, Acta Math., Volume 195 (2005), pp. 61-115 (ISSN: 0001-5962) | MR | Zbl | DOI

Jutila, M., Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 80, Tata Institute of Fundamental Research, Bombay, 1987, 134 pages (ISBN: 3-540-18366-3) | MR | Zbl

Kowalski, E.; Michel, P.; VanderKam, J. Rankin-Selberg $L$ -functions in the level aspect, Duke Math. J., Volume 114 (2002), pp. 123-191 (ISSN: 0012-7094) | MR | Zbl | DOI

Landau, E. Über die $ζ$ -Funktion und die $L$ -Funktionen, Math. Z., Volume 20 (1924), pp. 105-125 (ISSN: 0025-5874) | MR | JFM | DOI

Littlewood, J. E. Researches in the theory of the Riemann $ζ$ -function, Proc. Lond. Math. Soc., Volume 20 (1922), pp. xxiv | JFM

Meurman, T., Number theory, Vol. I (Budapest, 1987) (Colloq. Math. Soc. János Bolyai), Volume 51, North-Holland, Amsterdam, 1990, pp. 325-354 | MR | Zbl

Milićević, D. Sub-Weyl subconvexity for Dirichlet $L$ -functions to prime power moduli (preprint arXiv:1407.4100, to appear in Compositio Math ) | MR

Munshi, R. Bounds for twisted symmetric square $L$ -functions—III, Adv. Math., Volume 235 (2013), pp. 74-91 (ISSN: 0001-8708) | MR | Zbl | DOI

Venkatesh, A. Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math., Volume 172 (2010), pp. 989-1094 (ISSN: 0003-486X) | MR | Zbl | DOI

Vishe, P. A fast algorithm to compute $L (1 / 2, f \times χ_{q})$ , J. Number Theory, Volume 133 (2013), pp. 1502-1524 (ISSN: 0022-314X) | MR | Zbl | DOI

Weyl, H. Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., Volume 77 (1916), pp. 313-352 (ISSN: 0025-5831) | MR | JFM | DOI

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