Leading coefficient of the Goss Zeta value and p-ranks of Jacobians of Carlitz cyclotomic covers
Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 963-995.

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 de 𝔽 q (t) quand on fait varier l’idéal premier de 𝔽 q [t]. 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.

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 -th Carlitz cyclotomic extension of 𝔽 q (t), as we vary the prime of 𝔽 q [t]. 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.

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DOI : 10.5802/jtnb.1008
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
Gebhard Böckle 1 ; Dinesh S. Thakur 2

1 IWR, Universität Heidelberg 69120 Heidelberg, Germany
2 Department of Mathematics University of Rochester Rochester, NY 14627, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Gebhard Böckle; Dinesh S. Thakur. Leading coefficient of the Goss Zeta value and $p$-ranks of Jacobians of Carlitz cyclotomic covers. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 963-995. doi : 10.5802/jtnb.1008. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1008/

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