The size function for imaginary cyclic sextic fields
Ha Thanh Nguyen Tran1 ; Peng Tian2 ; Amy Feaver3
1 Department of Mathematical and Physical Sciences Concordia University of Edmonton 7128 Ada Blvd NW Edmonton, AB T5B 4E4, Canada
2 Department of Mathematics East China University of Science and Technology Meilong Road 130 200237, Shanghai, P. R. China
3 Department of Mathematics and Computer Science Gordon College 255 Grapevine Rd Wenham, MA 01984, United States
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 841-866.

Dans cet article, nous étudions la fonction de taille h 0 pour les corps de nombres. Cette fonction est analogue à la fonction donnant la dimension de l’espace de Riemann–Roch d’un diviseur sur une courbe algébrique. Van der Geer et Schoof ont conjecturé que h 0 atteint son maximum sur la classe triviale des diviseurs d’Arakelov. Cette conjecture a été prouvée pour tous les corps de nombres dont le groupe des unités est de rang 0 et 1, ainsi que pour les corps cubiques cycliques dont le groupe des unités est de rang 2. Nous prouvons que cette conjecture est également valable pour les corps sextiques cycliques totalement imaginaires, une autre classe de corps de nombres dont le groupe des unités est de rang 2.

In this paper, we investigate the size function h 0 for number fields. This size function is analogous to the dimension of the Riemann–Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that h 0 attains its maximum at the trivial class of Arakelov divisors. This conjecture was proved for all number fields with the unit group of rank 0 and 1, and also for cyclic cubic fields which have unit group of rank two. We prove the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1266
Classification : 11Y40, 11H06, 11R20
Mots clés : Arakelov divisor, size function, imaginary cyclic sextic fields, hexagonal lattice, unit lattice, cyclic cubic field
Ha Thanh Nguyen Tran 1 ; Peng Tian 2 ; Amy Feaver 3

1 Department of Mathematical and Physical Sciences Concordia University of Edmonton 7128 Ada Blvd NW Edmonton, AB T5B 4E4, Canada
2 Department of Mathematics East China University of Science and Technology Meilong Road 130 200237, Shanghai, P. R. China
3 Department of Mathematics and Computer Science Gordon College 255 Grapevine Rd Wenham, MA 01984, United States
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2023__35_3_841_0,
     author = {Ha Thanh Nguyen Tran and Peng Tian and Amy Feaver},
     title = {The size function for imaginary cyclic sextic fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {841--866},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {35},
     number = {3},
     year = {2023},
     doi = {10.5802/jtnb.1266},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1266/}
}
TY  - JOUR
AU  - Ha Thanh Nguyen Tran
AU  - Peng Tian
AU  - Amy Feaver
TI  - The size function for imaginary cyclic sextic fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2023
SP  - 841
EP  - 866
VL  - 35
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1266/
DO  - 10.5802/jtnb.1266
LA  - en
ID  - JTNB_2023__35_3_841_0
ER  - 
%0 Journal Article
%A Ha Thanh Nguyen Tran
%A Peng Tian
%A Amy Feaver
%T The size function for imaginary cyclic sextic fields
%J Journal de théorie des nombres de Bordeaux
%D 2023
%P 841-866
%V 35
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1266/
%R 10.5802/jtnb.1266
%G en
%F JTNB_2023__35_3_841_0
Ha Thanh Nguyen Tran; Peng Tian; Amy Feaver. The size function for imaginary cyclic sextic fields. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 841-866. doi : 10.5802/jtnb.1266. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1266/

[1] Emil Artin Die gruppentheoretische Struktur des Diskriminanten algebraischer Zahlkörper., J. Reine Angew. Math., Volume 164 (1931), pp. 1-11 | DOI | Zbl

[2] Ulrich Fincke; Michael Pohst Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comput., Volume 44 (1985) no. 170, pp. 463-471 | DOI | MR | Zbl

[3] Paolo Francini The size function h 0 for quadratic number fields, J. Théor. Nombres Bordeaux, Volume 13 (2001) no. 1, pp. 125-135 21st Journées Arithmétiques (Rome, 2001) | DOI | Numdam | MR | Zbl

[4] Paolo Francini The size function h ∘ for a pure cubic field, Acta Arith., Volume 111 (2004) no. 3, pp. 225-237 | DOI | MR | Zbl

[5] Gerard van der Geer; René Schoof Effectivity of Arakelov divisors and the theta divisor of a number field, Sel. Math., New Ser., Volume 6 (2000) no. 4, pp. 377-398 | DOI | MR | Zbl

[6] Richard P. Groenewegen The size function for number fields, Ph. D. Thesis, Universiteit van Amsterdam (1999)

[7] Richard P. Groenewegen An arithmetic analogue of Clifford’s theorem, J. Théor. Nombres Bordeaux, Volume 13 (2001) no. 1, pp. 143-156 21st Journées Arithmétiques (Rome, 2001) | DOI | Numdam | MR | Zbl

[8] Helmut Hasse Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage, Math. Z., Volume 31 (1930) no. 1, pp. 565-582 | DOI | MR | Zbl

[9] Helmut Hasse; Jacques Martinet Über die Klassenzahl abelscher Zahlkorper, 195, Citeseer, 1952 | DOI | Zbl

[10] Mikihito Hirabayashi; Ken-ichi Yoshino Remarks on unit indices of imaginary abelian number fields, Manuscr. Math., Volume 60 (1988) no. 4, pp. 423-436 | DOI | MR | Zbl

[11] Hendrik W. Lenstra Lattices, Algorithmic number theory: lattices, number fields, curves and cryptography (Mathematical Sciences Research Institute Publications), Volume 44, Cambridge University Press, 2008, pp. 127-181 | MR | Zbl

[12] The PARI Group PARI/GP version 2.13.4, 2022 (available from http://pari.math.u-bordeaux.fr/)

[13] René Schoof Computing Arakelov class groups, Algorithmic number theory: lattices, number fields, curves and cryptography (Mathematical Sciences Research Institute Publications), Volume 44, Cambridge University Press, 2008, pp. 447-495 | MR | Zbl

[14] Ha Thanh Nguyen Tran Computing dimensions of spaces of Arakelov divisors of number fields, Int. J. Number Theory, Volume 13 (2017) no. 2, pp. 487-512 | DOI | MR | Zbl

[15] Ha Thanh Nguyen Tran The size function for quadratic extensions of complex quadratic fields, J. Théor. Nombres Bordeaux, Volume 29 (2017) no. 1, pp. 243-259 | DOI | Numdam | MR | Zbl

[16] Ha Thanh Nguyen Tran; Peng Tian The size function for cyclic cubic fields, Int. J. Number Theory, Volume 14 (2018), pp. 399-415 | DOI | MR | Zbl

[17] Wolfram Research, Inc. Mathematica, Version 13.1 (Champaign, IL, 2022)

Cité par Sources :