$p$-adic $L$-functions and the geometry of Hida families

Kramer-Miller, Joe

Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 727-770.

Une question centrale dans la théorie de la déformations $p$-adique des formes automorphes est la suivante : quelle est la relation entre une $L$-fonction $p$-adic sur une variété propre et la géométrie de cette variété propre ? Dans l’article nous montrons certains résultats dans cet esprit pour la partie ordinaire de la courbe propre (i.e., les familles d’Hida). Plus précisément, nous montrons que l’intersection de deux familles d’Hida est déterminée par le comportement local des $L$-fonctions $p$-adic des familles. Nous prouvons aussi que le comportement local des $L$-fonctions $p$-adique nous dit quand une famille d’Hida est ramifiée sur l’espace de poids. La technique principale consiste à prouver une réciproque d’un théorème de Vatsal et puis à interpoler $p$-adiquement certaines congruences.

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this article we prove results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We show that the crossing of two Hida families is determined by the local behavior of $p$-adic $L$-functions on those Hida families. In addition, we prove that the local behavior of $p$-adic $L$-functions determines when a Hida family ramifies over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L$-values and then $p$-adically interpolating congruences.

Classification : 11F33, 11F67
Keywords: $p$-adic $L$-functions, Hida theory, $\Lambda $-adic modular forms
Mot clés : $L$-fonction $p$-adique, Théorie d’Hida

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