An ergodic approach towards an equidistribution result of Ferrero–Washington

Lee, Jungwon; Palvannan, Bharathwaj

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 805-833

An important ingredient in the Ferrero–Washington proof of the vanishing of cyclotomic $\mu$-invariant for Kubota–Leopoldt $p$-adic $L$-functions is an equidistribution result which they established using the Weyl criterion. The purpose of our manuscript is to provide an alternative proof by adopting a dynamical approach. A key ingredient to our methods is studying an ergodic skew-product map on ${\mathbb{Z}}_p\times [0,1]$, which is then suitably identified as a factor of the $2$-sided Bernoulli shift on the sample space ${\{0,1,2,\cdots ,p-1\}}^{\mathbb{Z}}$.

Un ingrédient important de la preuve de Ferrero et Washington de la nullité de l’invariant cyclotomique $\mu$ pour les fonctions $L$ $p$-adiques de Kubota–Leopoldt est un résultat d’équidistribution qu’ils ont établi en utilisant le critère de Weyl. Le but de notre manuscrit est de fournir une preuve alternative en adoptant une approche dynamique. L’une des idées clés de notre méthode est l’étude d’une application semi-directe ergodique sur ${\mathbb{Z}}_p\times [0,1]$, qui est ensuite identifiée de manière appropriée comme un facteur du décalage de Bernoulli bilatéral sur l’espace ${\{0,1,2,\cdots ,p-1\}}^{\mathbb{Z}}$.

Classification : 11R23, 37A05, 37A44, 11K06, 11K41
Keywords: Iwasawa theory, Ergodic theory, Bernoulli shifts, Equidistribution modulo $1$

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