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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.


[1] Gebhard Böckle, The distribution of the zeros of the Goss zeta-function for $A=\mathbb{F}_2[x,y]/(y^2+y+x^3+x+1)$, Math. Z. 275 (2013), p. 835-861 Article
[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 9, European Mathematical Society, 2009
[4] Wieb Bosma, John J. Cannon, C. Fieker & A. Steel (ed.), Handbook of Magma functions, 2010, Edition 2.16
[5] Leonard Carlitz, Sums of products of multinomial coefficients, Elem. Math. 18 (1963), p. 37-39
[6] Ching-Li Chai, Brian Conrad & Frans Oort, Complex multiplication and lifting problems, Mathematical Surveys and Monographs 195, American Mathematical Society, 2014
[7] Steven Galovich & Michael Rosen, The class number of cyclotomic function fields, J. Number Theory 13 (1981), p. 363-375 Article
[8] Ernst-Ulrich Gekeler, On power sums of polynomials over finite fields, J. Number Theory 30 (1988), p. 11-26 Article
[9] David Goss, Basic Structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 35, Springer, 1996
[10] David Goss & Warren Sinnott, Class groups of function fields, Duke Math. J. 52 (1985), p. 507-516 Article
[11] Herbert Lange & Sevin Recillas Pishmish, Abelian varieties with group action, J. Reine Angew. Math. 575 (2004), p. 135-155
[12] Yuri Ivanovich Manin, On the Hasse-Witt matrix of an algebraic curve, Izv. Akad. Nauk SSSR, Ser. Mat. 25 (1961), p. 1513-172
[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,
[15] Barry Mazur, How can we construct abelian Galois extensions of basic number fields?, Bull. Am. Math. Soc. 48 (2011), p. 155-209 Article
[16] Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics 210, Springer, 2002
[17] Daisuke Shiomi, The Hasse-Witt invariant of cyclotomic function fields, Acta Arith. 150 (2011), p. 227-240 Article
[18] Daisuke Shiomi, Ordinary cyclotomic function fields, J. Number Theory 133 (2013), p. 523-533 Article
[19] Henning Stichtenoth, Die Hasse-Witt Invariante eines Kongruenzfunktionenkörpers, Arch. Math. 33 (1980), p. 357-360 Article
[20] Lenny Taelman, A Herbrand-Ribet theorem for function fields, Invent. Math. 188 (2012), p. 253-275 Article
[21] Selmo Tauber, On multinomial coefficients, Am. Math. Mon. 70 (1963) Article
[22] Dinesh S. Thakur, Function Field Arithmetic, World Scientific, 2004
[23] Dinesh S. Thakur, Power sums with applications to multizeta and zeta zero distribution for $\mathbb{F}_q[t]$, Finite Fields Appl. 15 (2009), p. 534-552 Article
[24] Dinesh S. Thakur, Valuations of $\nu $-adic power sums and zero distribution for the Goss’ $v$-adic zeta function for $\mathbb{F}_q[t]$, J. Integer Seq. 16 (2013)
[25] The Sage Developers, “SageMath, the Sage Mathematics Software System (Version 6.2)” 2014,
[26] William C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér. 2 (1969), p. 521-560 Article