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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, 29 no. 3 (2017), p. 963-995, doi: 10.5802/jtnb.1008
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Class. Math.: 11M38, 14H05, 11R60, 14H40
Keywords: Bernoulli number, Artin–Schreier polynomial, Herbrand–Ribet theorem, Carlitz cyclotomic field, Hasse–Witt invariant, Goss $\zeta$-function, power sum, ordinariness

Résumé - Abstract

Let $\mathbb{F}_q$ 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 $\mathbb{F}_q(t)$, as we vary the prime $\wp$ of $\mathbb{F}_q[t]$. 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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