Comparaison des métriques d’Arakelov et de Poincaré sur X0(N)

On the Sup-Norm of Maass Cusp Forms of Large Level: II

“…The Laplacian eigenvalue of f j is denoted by 1 4 + r 2 j with r j ∈ R ∪ [− i 2 , i 2 ], and the Hecke eigenvalues of f j are denoted by λ j (n). Correspondingly, the Laplacian eigenvalue and Hecke eigenvalues of f are denoted by 1 4 + r 2 f and λ f (n). The basic idea of amplification in the current setting is the inequality…”

“…To be specific, we consider the noncompact arithmetic surface Γ 0 (N )\H equipped with its hyperbolic metric and associated measure dxdy/y 2 ; the total volume is then asymptotically equal to N 1+o (1) . We shall L 2 -normalize all Hecke-Maass cuspidal newforms with respect to that measure, namely (1) Γ0(N )\H |f (z)| 2 dxdy y 2 = 1.…”

“…We shall L 2 -normalize all Hecke-Maass cuspidal newforms with respect to that measure, namely (1) Γ0(N )\H |f (z)| 2 dxdy y 2 = 1.…”

“…In this paper we focus mainly on the N -aspect and, thanks to the new ideas explained below, we are able to provide a short but self-contained treatment. For several reasons, see [1,2,5], the "trivial" bound is f ∞ λ, N , while the most optimistic bound would be f ∞ λ, N −1/2+ . Here and later, the dependence on λ is understood to be continuous.…”

An omega result for supremum norms of Hecke-eigenforms in the level aspect

“…Let {f 1 , · · · , f g } be the set of all primitive forms of weight 2 for Γ. In [1] and [17], it was derived very good estimates for…”