no. 3
Sur une condition suffisante pour l’existence de mesures p-adiques admissibles
Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 805-829.

On donne une nouvelle condition suffisante pour l’existence des mesures p-adiques admissibles μ obtenues à partir de suites de distributions Φ j (j≥0) à valeurs dans les espaces de formes modulaires. On utilise la projection caractéristique sur le sous-espace primaire associé à une valeur propre non nulle α de l’opérateur U d’Atkin. Notre condition est exprimée en termes des congruences entre les coefficients de Fourier des formes modulaires Φ j . On montre comment vérifier ces congruences, et on traite plusieurs applications. On obtient donc une explication conceptuelle des formules de Yu.Manin pour les distributions attachées à la fonction L f (s,χ)=∑ n≥1 χ(n)a n n -s d’une forme parabolique primitive f=∑ n≥1 a n q n ∈S k (Γ 0 (N),ψ) de poids k≥2.

We give a new sufficient condition for the existence of admissible p-adic measures μ obtained from sequences of distributions Φ j (j≥0) with values in spaces of modular forms. We use the characteristic projection on the primary subspace associated to a non zero eigenvalue α of the Atkin operator U. Our condition is expressed in terms of congruences between the Fourier coefficients of the modular forms Φ j . We show how to verify these congruences and we give several applications. So we get a conceptual explanation for the Yu.Manin’s formulas for the distributions attached to the L-function, L f (s,χ)=∑ n≥1 χ(n)a n n -s , of a primitive cuspform f=∑ n≥1 a n q n ∈S k (Γ 0 (N),ψ) of weight k≥2.

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     title = {Sur une condition suffisante pour l{\textquoteright}existence de mesures $p$-adiques admissibles},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {805--829},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
     mrnumber = {2142237},
     zbl = {1078.11038},
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Panchishkin, Alexei. Sur une condition suffisante pour l’existence de mesures $p$-adiques admissibles. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 805-829. http://archive.numdam.org/item/JTNB_2003__15_3_805_0/

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