Geodesic
Theta functions, fourth moments of eigenforms and the sup-norm problem II
Forum of Mathematics, Pi, Tome 12 (2024)

Voir la notice de l'article provenant de la source Cambridge University Press

Let f be an $L^2$-normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$. Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate

$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$

with absolute implied constant. For a cuspidal Maaß newform $\varphi $ of eigenvalue $\lambda $ on such a surface, we prove that

$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$

We establish analogous estimates in the setting of definite quaternion algebras. 
@article{10_1017_fmp_2024_9,
     author = {Ilya Khayutin and Paul D. Nelson and Raphael S. Steiner},
     title = {Theta functions, fourth moments of eigenforms and the sup-norm problem {II}},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {12},
     year = {2024},
     doi = {10.1017/fmp.2024.9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2024.9/}
}
TY  - JOUR
AU  - Ilya Khayutin
AU  - Paul D. Nelson
AU  - Raphael S. Steiner
TI  - Theta functions, fourth moments of eigenforms and the sup-norm problem II
JO  - Forum of Mathematics, Pi
PY  - 2024
VL  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2024.9/
DO  - 10.1017/fmp.2024.9
LA  - en
ID  - 10_1017_fmp_2024_9
ER  - 
%0 Journal Article
%A Ilya Khayutin
%A Paul D. Nelson
%A Raphael S. Steiner
%T Theta functions, fourth moments of eigenforms and the sup-norm problem II
%J Forum of Mathematics, Pi
%D 2024
%V 12
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2024.9/
%R 10.1017/fmp.2024.9
%G en
%F 10_1017_fmp_2024_9
Ilya Khayutin; Paul D. Nelson; Raphael S. Steiner. Theta functions, fourth moments of eigenforms and the sup-norm problem II. Forum of Mathematics, Pi, Tome 12 (2024). doi: 10.1017/fmp.2024.9

[AL70] Atkin, A. O. L. and Lehner, J., ‘Hecke operators on ’, Math. Ann. 185 (1970), 134–160.Google Scholar | DOI

[Ass21] Assing, E., ‘The sup-norm problem beyond the newform’, Preprint, 2021, .Google Scholar | arXiv

[Ass24] Assing, E., ‘On sup-norm bounds part I: Ramified Maaß newforms over number fields’, J. Eur. Math. Soc. (JEMS) 2024. Published online first.Google Scholar | DOI

[Ber31] Bernstein, S., ‘Sur les polynomes orthogonaux relatifs à un segment fini (seconde partie)’, Journal de Mathématiques Pures et Appliquées 10 (1931), 219–286.Google Scholar

[Bér77] Bérard, P. H., ‘On the wave equation on a compact Riemannian manifold without conjugate points’, Math. Z. 155(3) (1977), 249–276.Google Scholar

[BH10] Blomer, V. and Holowinsky, R., ‘Bounding sup-norms of cusp forms of large level’, Invent. Math. 179(3) (2010), 645–681.Google Scholar | DOI

[BHM16] Blomer, V., Harcos, G. and Milićević, D., ‘Bounds for eigenforms on arithmetic hyperbolic -manifolds’, Duke Math. J. 165(4) (2016), 625–659.Google Scholar | DOI

[BHMM20] Blomer, V., Harcos, G., Maga, P. and Milićević, D., ‘The sup-norm problem for over number fields’, J. Eur. Math. Soc. (JEMS) 22(1) (2020), 1–53.Google Scholar | DOI

[BHMM22] Blomer, V., Harcos, G., Maga, P. and Milićević, D., ‘Beyond the spherical sup-norm problem’, J. Math. Pures Appl. (9) 168 (2022), 1–64.Google Scholar | DOI

[BHW93] Betke, U., Henk, M. and Wills, J. M., ‘Successive-minima-type inequalities’, Discrete Comput. Geom. 9(2) (1993), 165–175.Google Scholar | DOI

[BK15] Buttcane, Jack and Khan, Rizwanur. -norms of Hecke newforms of large level. Math. Ann. 362(3-4):699–715, 2015.Google Scholar | DOI

[Blo13] Blomer, V., ‘On the 4-norm of an automorphic form’, J. Eur. Math. Soc. (JEMS) 15(5) (2013), 1825–1852.Google Scholar | DOI

[BM11] Blomer, V. and Michel, P., ‘Sup-norms of eigenfunctions on arithmetic ellipsoids’, Int. Math. Res. Not. IMRN (21) (2011), 4934–4966.Google Scholar

[BM13] Blomer, V. and Michel, P., ‘Hybrid bounds for automorphic forms on ellipsoids over number fields’, J. Inst. Math. Jussieu 12(4) (2013), 727–758.Google Scholar | DOI

[Bor63] Borel, A., ‘Some finiteness properties of adele groups over number fields’, Inst. Hautes Études Sci. Publ. Math. (16) (1963), 5–30.Google Scholar

[Bum97] Bump, D.. Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, vol. 55 (Cambridge University Press, Cambridge, 1997).Google Scholar | DOI

[Cas73] Casselman, W., ‘On some results of Atkin and Lehner’, Math. Ann. 201 (1973), 301–314.Google Scholar | DOI

[Com21] Comtat, F., ‘Optimal sup norm bounds for newforms on with maximally ramified central character’, Forum Math. 33(1) (2021), 1–16.Google Scholar

[Del71] Deligne, P., ‘Formes modulaires et représentations -adiques’, Séminaire Bourbaki , Vol. 1968/1969, Exp. 347-363 179 (1971), 139–172.Google Scholar

[Del74] Deligne, P., ‘La conjecture de Weil. I’, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.Google Scholar | DOI

[GL87] Gruber, P. M. and Lekkerkerker, C. G., Geometry of Numbers, second edn., North-Holland Mathematical Library, vol. 37 (North-Holland Publishing Co., Amsterdam, 1987).Google Scholar | DOI

[Har03] Harcos, G., ‘An additive problem in the Fourier coefficients of cusp forms’, Math. Ann. 326(2) (2003), 347–365.Google Scholar | DOI

[HC19] Hou, F. and Chen, B., ‘Level aspect subconvexity for twisted -functions’, J. Number Theory 203 (2019), 12–31.Google Scholar | DOI

[HL94] Hoffstein, J. and Lockhart, P., ‘Coefficients of Maass forms and the Siegel zero’, Ann. of Math. (2) 140(1) (1994), 161–181. With an appendix by Dorian Goldfeld, Hoffstein and D. Lieman.Google Scholar

[HM06] Harcos, G. and Michel, P., ‘The subconvexity problem for Rankin-Selberg -functions and equidistribution of Heegner points. II’, Invent. Math. 163(3) (2006), 581–655.Google Scholar | DOI

[HN18] Hu, Y. and Nelson, P. D., ‘New test vector for Waldspurger’s period integral, relative trace formula, and hybrid subconvexity bounds’, Preprint, 2018, .Google Scholar | arXiv

[HNS19] Hu, Y., Nelson, P. D. and Saha, A., ‘Some analytic aspects of automorphic forms on of minimal type’, Comment. Math. Helv. 94(4) (2019), 767–801.Google Scholar

[HS20] Hu, Y. and Saha, A., ‘Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: Newforms and subconvexity’, Compos. Math. 156(11) (2020), 2368–2398.Google Scholar | DOI

[HT12] Harcos, G. and Templier, N., ‘On the sup-norm of Maass cusp forms of large level: II’, Int. Math. Res. Not. IMRN (20) (2012), 4764–4774.Google Scholar | DOI

[HT13] Harcos, G. and Templier, N., ‘On the sup-norm of Maass cusp forms of large level. III’, Math. Ann. 356(1) (2013), 209–216.Google Scholar | DOI

[IS95] Iwaniec, H. and Sarnak, P., ‘ norms of eigenfunctions of arithmetic surfaces’, Ann. of Math. (2) 141(2) (1995), 301–320.Google Scholar | DOI

[Iwa90] Iwaniec, H., ‘Small eigenvalues of Laplacian for ’, Acta Arith. 56(1) (1990), 65–82.Google Scholar

[Iwa97] Iwaniec, H., Topics in Classical Automorphic, Forms Graduate Studies in Mathematics, vol. 17 (American Mathematical Society, Providence, RI, 1997).Google Scholar

[JL70] Jacquet, H. and Langlands, R. P., Automorphic forms on , Lecture Notes in Mathematics, vol. 114 (Springer-Verlag, Berlin, 1970).Google Scholar

[KR94] Kudla, S. S. and Rallis, S., ‘A regularized Siegel–Weil formula: The first term identity’, Ann. of Math. (2) 140(1) (1994), 1–80.Google Scholar | DOI

[KS20] Khayutin, I. and Steiner, R. S., ‘Theta functions, fourth moments of eigenforms, and the sup-norm problem I’, Preprint, 2020, .Google Scholar | arXiv

[LPS87] Lubotzky, A., Phillips, R. and Sarnak, P., ‘Hecke operators and distributing points on ’, II. Comm. Pure Appl. Math. 40(4) (1987), 401–420.Google Scholar | DOI

[Mar16] Marshall, S., ‘Local bounds for norms of Maass forms in the level aspect’, Algebra Number Theory 10(4) (2016), 803–812.Google Scholar | DOI

[Mœg97] Mœglin, C., ‘Quelques propriétés de base des séries théta’, J. Lie Theory 7(2) (1997), 231–238.Google Scholar

[Nel15] Nelson, P. D., ‘Evaluating modular forms on Shimura curves’, Math. Comp. 84(295) (2015), 2471–2503.Google Scholar | DOI

[Nel16] Nelson, P. D., ‘Quantum variance on quaternion algebras, I’, Preprint, 2016, .Google Scholar | arXiv

[Nel17] Nelson, P. D., ‘Quantum variance on quaternion algebras, II’, Preprint, 2017, .Google Scholar | arXiv

[Nel19] Nelson, P. D., ‘Quantum variance on quaternion algebras, III’, Preprint, 2019, .Google Scholar | arXiv

[Nel20] Nelson, P. D., ‘Bounds for twisted symmetric square -functions via half-integral weight periods’, Forum Math. Sigma 8 (2020), Paper No. e44, 21.Google Scholar | DOI

[Nel21] Nelson, P. D., ‘The spectral decomposition of ’, Math. Z. 298(3-4) (2021), 1425–1447.Google Scholar | DOI

[Nor21] Nordentoft, A. C., ‘Hybrid subconvexity for class group -functions and uniform sup norm bounds of Eisenstein series’, Forum Math. 33(1) (2021), 39–57.Google Scholar | DOI

[PY19] Petrow, I. and Young, M. P., ‘The fourth moment of Dirichlet -functions along a coset and the Weyl bound’, arXiv e-prints, 2019, .Google Scholar | arXiv

[Ral84] Rallis, S., ‘On the Howe duality conjecture’, Compositio Math. 51(3) (1984), 333–399.Google Scholar

[RS75] Rallis, S. and Schiffmann, G., ‘Distributions invariantes par le groupe orthogonal’, in Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973–75), Lecture Notes in Math., vol. 497 (1975), 494–642.Google Scholar | DOI

[Sah17] Saha, A., ‘On sup-norms of cusp forms of powerful level’, J. Eur. Math. Soc. (JEMS) 19(11) (2017), 3549–3573.Google Scholar | DOI

[Sah20] Saha, A., ‘Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces’, Math. Ann. 376(1-2) (2020), 609–644.Google Scholar | DOI

[Saw21] Sawin, W., ‘A geometric approach to the sup-norm problem for automorphic forms: the case of newforms on with squarefree level’, Proc. Lond. Math. Soc. (3) 123(1) (2021), 1–56.Google Scholar

[Shi72] Shimizu, H., ‘Theta series and automorphic forms on ’, J. Math. Soc. Japan 24 (1972), 638–683.Google Scholar | DOI

[Ste20] Steiner, R. S., ‘Sup-norm of Hecke-Laplace eigenforms on ’, Math. Ann. 377(1-2) (2020), 543–553.Google Scholar

[Ste23] Steiner, R. S., ‘Small diameters and generators for arithmetic lattices in and certain Ramanujan graphs’, Ramanujan J. 62(4) (2023), 953–966.Google Scholar | DOI

[Tem10] Templier, N., ‘On the sup-norm of Maass cusp forms of large level’, Selecta Math. (N.S.) 16(3) (2010), 501–531.Google Scholar | DOI

[Tem15] Templier, N., ‘Hybrid sup-norm bounds for Hecke-Maass cusp forms’, J. Eur. Math. Soc. (JEMS) 17(8) (2015), 2069–2082.Google Scholar | DOI

[Tom23] Toma, R., ‘Hybrid bounds for the sup-norm of automorphic forms in higher rank’, Trans. Amer. Math. Soc. 376(8) (2023), 5573–5600.Google Scholar | DOI

[vdC36] Van Der Corput, J. G., ‘Verallgemeinerung einer Mordellschen Beweismethode in der Geometrie der Zahlen, Zweite Mitteilung’, Acta Arithmetica 2(1) (1936), 145–146.Google Scholar

[Vig77] Vignéras, M.-F., ‘Séries thêta des formes quadratiques indéfinies’, in Séminaire Delange-Pisot-Poitou, 17e année (1975/76), Théorie des nombres: Fasc. 1, Exp. No. 20 (1977), 3.Google Scholar

[Voi18] Voight, J., ‘Quaternion algebras’, 2018, https://math.dartmouth.edu/jvoight/quat.html.Google Scholar

[Wal85] Waldspurger, J.-L., ‘Sur les valeurs de certaines fonctions automorphes en leur centre de symétrie’, Compositio Math. 54(2) (1985), 173–242.Google Scholar

[Wat08] Watson, T. C., ‘Rankin triple products and quantum chaos’, Preprint, 2008, .Google Scholar | arXiv

[Wei64] Weil, A., ‘Sur certains groupes d’opérateurs unitaires’, Acta Math. 111 (1964), 143–211.Google Scholar

[Xia07] Xia, H., ‘On norms of holomorphic cusp forms’, J. Number Theory 124(2) (2007), 325–327.Google Scholar | DOI

[You17] Young, M. P., ‘Weyl-type hybrid subconvexity bounds for twisted -functions and Heegner points on shrinking sets’, J. Eur. Math. Soc. (JEMS) 19(5) (2017), 1545–1576.Google Scholar | DOI

Cité par Sources :