Leading coefficient of the Goss Zeta value and $p$-ranks of Jacobians of Carlitz cyclotomic covers

Böckle, Gebhard; Thakur, Dinesh S.

Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 963-995.

Let ${𝔽}_q$ be a finite field of characteristic $p$. We study variations in slope zero multiplicities of the components of the Dieudonné module (or equivalently the $p$-divisible group) of the Jacobian of the $\wp$-th Carlitz cyclotomic extension of ${𝔽}_q\left(t\right)$, as we vary the prime $\wp$ of ${𝔽}_q\left[t\right]$. We also give some applications to the question of ordinariness and of $p$-ranks of the factors of these Jacobians. We do this, guided by numerical experiments, by proving and guessing some interesting structural patterns in prime factorizations of power sums representing the leading terms of the Goss zeta function at negative integers.

Soit ${𝔽}_q$ un corps fini de caractéristique $p$. Nous étudions la variation de la multiplicité de la pente nulle dans les composantes du module de Dieudonné (c’est-à-dire, du groupe $p$-divisible) associé à la Jacobienne de l’extension cyclotomique de Carlitz d’ordre $\wp$ de ${𝔽}_q\left(t\right)$ quand on fait varier l’idéal premier $\wp$ de ${𝔽}_q\left[t\right]$. Nous donnons quelques applications aux questions d’ordinarité et de calcul du $p$-rang des facteurs de ces Jacobiennes. Guidé par des expériences numériques, nous arrivons à nos résultats en démontrant et en conjecturant des propriétes structurales de la décomposition en facteurs premiers des sommes de puissances donnant les coefficients directeurs des valeurs de la fonction zêta de Goss aux entiers négatifs.

Classification: 11M38, 14H05, 11R60, 14H40
Mots-clés : Bernoulli number, Artin–Schreier polynomial, Herbrand–Ribet theorem, Carlitz cyclotomic field, Hasse–Witt invariant, Goss $\zeta $-function, power sum, ordinariness

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