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 : https://doi.org/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 ζ-function, power sum, ordinariness
@article{JTNB_2017__29_3_963_0,
     author = {Gebhard B\"ockle and Dinesh S. Thakur},
     title = {Leading coefficient of the {Goss} {Zeta} value and $p$-ranks of Jacobians of Carlitz cyclotomic covers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {963--995},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {3},
     year = {2017},
     doi = {10.5802/jtnb.1008},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1008/}
}
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/

[1] Gebhard Böckle The distribution of the zeros of the Goss zeta-function for A=𝔽 2 [x,y]/(y 2 +y+x 3 +x+1), Math. Z., Volume 275 (2013) no. 3-4, pp. 835-861 | Article | Zbl 1282.11115

[2] Gebhard Böckle Cohomological theory of crystals over function fields and applications, Arithmetic Geometry over Global Function Fields (CRM Barcelona 2010) (Advanced Courses in Mathematics), Birkhäuser, 2014

[3] Gebhard Böckle; Richard Pink Cohomological theory of crystals over function fields, EMS Tracts in Mathematics, Volume 9, European Mathematical Society, 2009, viii+187 pages | Zbl 1186.14002

[4] Handbook of Magma functions (Wieb Bosma; John J. Cannon; C. Fieker; A. Steel, eds.), 2010, 5017 pages (Edition 2.16)

[5] Leonard Carlitz Sums of products of multinomial coefficients, Elem. Math., Volume 18 (1963), pp. 37-39 | Zbl 0116.25101

[6] Ching-Li Chai; Brian Conrad; Frans Oort Complex multiplication and lifting problems, Mathematical Surveys and Monographs, Volume 195, American Mathematical Society, 2014, ix+387 pages | Zbl 1298.14001

[7] Steven Galovich; Michael Rosen The class number of cyclotomic function fields, J. Number Theory, Volume 13 (1981), pp. 363-375 | Article | Zbl 0473.12014

[8] Ernst-Ulrich Gekeler On power sums of polynomials over finite fields, J. Number Theory, Volume 30 (1988) no. 1, pp. 11-26 | Article | Zbl 0656.12007

[9] David Goss Basic Structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, Volume 35, Springer, 1996, xiii+422 pages | Zbl 0874.11004

[10] David Goss; Warren Sinnott Class groups of function fields, Duke Math. J., Volume 52 (1985), pp. 507-516 | Article | Zbl 0571.12006

[11] Herbert Lange; Sevin Recillas Pishmish Abelian varieties with group action, J. Reine Angew. Math., Volume 575 (2004), pp. 135-155 | Zbl 1072.14053

[12] Yuri Ivanovich Manin On the Hasse-Witt matrix of an algebraic curve, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 25 (1961), p. 1513-172 (English translation in Amer. Math. Soc. Transl. 45 (1965), p. 245-264) | Zbl 0102.27802

[13] Maplesoft Maple mathematics software (a division of Waterloo Maple Inc., Waterloo, Ontario.)

[14] Maxima Maxima, a Computer Algebra System. Version 5.34.1, 2014 (http://maxima.sourceforge.net/)

[15] Barry Mazur How can we construct abelian Galois extensions of basic number fields?, Bull. Am. Math. Soc., Volume 48 (2011) no. 2, pp. 155-209 | Article | Zbl 1228.11163

[16] Michael Rosen Number theory in function fields, Graduate Texts in Mathematics, Volume 210, Springer, 2002, xii+358 pages | Zbl 1043.11079

[17] Daisuke Shiomi The Hasse-Witt invariant of cyclotomic function fields, Acta Arith., Volume 150 (2011) no. 3, pp. 227-240 | Article | Zbl 1244.11095

[18] Daisuke Shiomi Ordinary cyclotomic function fields, J. Number Theory, Volume 133 (2013) no. 2, pp. 523-533 | Article | Zbl 1286.11198

[19] Henning Stichtenoth Die Hasse-Witt Invariante eines Kongruenzfunktionenkörpers, Arch. Math., Volume 33 (1980), pp. 357-360 | Article | Zbl 0416.12010

[20] Lenny Taelman A Herbrand-Ribet theorem for function fields, Invent. Math., Volume 188 (2012) no. 2, pp. 253-275 | Article | Zbl 1278.11102

[21] Selmo Tauber On multinomial coefficients, Am. Math. Mon., Volume 70 (1963), 1058.1063 pages | Article | Zbl 0117.25902

[22] Dinesh S. Thakur Function Field Arithmetic, World Scientific, 2004, xv+388 pages | Zbl 1061.11001

[23] Dinesh S. Thakur Power sums with applications to multizeta and zeta zero distribution for 𝔽 q [t], Finite Fields Appl., Volume 15 (2009) no. 4, pp. 534-552 | Article | Zbl 1228.11139

[24] Dinesh S. Thakur Valuations of ν-adic power sums and zero distribution for the Goss’ v-adic zeta function for 𝔽 q [t], J. Integer Seq., Volume 16 (2013) no. 2 (Article 13.2.13, 18 p.) | Zbl 1288.11086

[25] The Sage Developers SageMath, the Sage Mathematics Software System (Version 6.2), 2014 (http://www.sagemath.org/)

[26] William C. Waterhouse Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér., Volume 2 (1969), pp. 521-560 | Article | Zbl 0188.53001