On class numbers of division fields of abelian varieties

Jędrzej Garnek

doi:10.5802/jtnb.1077

¹Graduate School, Adam Mickiewicz University Faculty of Mathematics and Computer Science Umultowska 87, 61-614 Poznan, Poland

Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 227-242

Abstract (Original language)
Résumé

Let $A$ be an abelian variety defined over a number field $K$ . Fix a prime $p$ and a natural number $n$ and consider the field $K_{n}$ , obtained by adjoining to $K$ all the coordinates of the $p^{n}$ -torsion points of $A$ . We give a lower bound on the $p$ -part of the class group of $K_{n}$ for large $n$ , by finding a large unramified extension of $K_{n}$ . This lower bound depends on the Mordell–Weil rank of $A$ and the reduction of $p$ -torsion points modulo primes above $p$ .

Soit $A$ une variété abélienne définie sur un corps de nombres $K$ . On fixe un nombre premier $p$ et pour tout nombre naturel $n,$ on note $K_{n}$ le corps engendré sur $K$ par les coordonnées des points de $p^{n}$ -torsion de $A$ . Nous donnons une minoration de l’ordre de la $p$ -partie du groupe de classes de $K_{n}$ pour $n ≫ 0$ , en construisant une extension non ramifiée suffisamment grande de $K_{n} .$ Cette minoration dépend du rang du groupe de Mordell–Weil de $A$ et de la réduction des points de $p$ -torsion en nombres premiers au-dessus de $p$ .

Article information
Cite this article

Received: 2018-05-27
Revised: 2018-09-23
Accepted: 2018-11-15
Published online: 2019-07-28

MR

DOI: 10.5802/jtnb.1077

Classification: 11R29, 11G10
Keywords: division fields, class number, abelian varieties

Author's affiliations:

Jędrzej Garnek ¹

¹ Graduate School, Adam Mickiewicz University Faculty of Mathematics and Computer Science Umultowska 87, 61-614 Poznan, Poland

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Jędrzej Garnek. On class numbers of division fields of abelian varieties. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 227-242. doi: 10.5802/jtnb.1077

@article{JTNB_2019__31_1_227_0,
     author = {J\k{e}drzej Garnek},
     title = {On class numbers of division fields of abelian varieties},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {227--242},
     year = {2019},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     doi = {10.5802/jtnb.1077},
     mrnumber = {3994728},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1077/}
}

TY  - JOUR
AU  - Jędrzej Garnek
TI  - On class numbers of division fields of abelian varieties
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
SP  - 227
EP  - 242
VL  - 31
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1077/
DO  - 10.5802/jtnb.1077
LA  - en
ID  - JTNB_2019__31_1_227_0
ER  -

%0 Journal Article
%A Jędrzej Garnek
%T On class numbers of division fields of abelian varieties
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 227-242
%V 31
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1077/
%R 10.5802/jtnb.1077
%G en
%F JTNB_2019__31_1_227_0

References
Cited by

[1] Grzegorz Banaszak; Wojciech Gajda; Piotr Krasoń Detecting linear dependence by reduction maps, J. Number Theory, Volume 115 (2005) no. 2, pp. 322-342 | MR | DOI | Zbl

[2] M. I. Bashmakov The cohomology of abelian varieties over a number field, Russ. Math. Surv., Volume 27 (1972) no. 6, pp. 25-70 | MR | DOI | Zbl

[3] Siegfried Bosch; Werner Lütkebohmert; Michel Raynaud Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 21, Springer, 1990 | Zbl

[4] Pete L. Clark; Allan Lacy There are genus one curves of every index over every infinite, finitely generated field, J. Reine Angew. Math., Volume 794 (2019), pp. 65-86 | MR | DOI | Zbl

[5] Chantal David; Tom Weston Local torsion on elliptic curves and the deformation theory of Galois representations, Math. Res. Lett., Volume 15 (2008) no. 2-3, pp. 599-611 | MR | DOI | Zbl

[6] Luis V. Dieulefait Explicit determination of the images of the Galois representations attached to abelian surfaces with $End (A) = ℤ$ , Exp. Math., Volume 11 (2003) no. 4, pp. 503-512 | MR | DOI | Zbl

[7] Takashi Fukuda; Keiichi Komatsu; Shuji Yamagata Iwasawa $λ$ -invariants and Mordell–Weil ranks of abelian varieties with complex multiplication, Acta Arith., Volume 127 (2007) no. 4, pp. 305-307 | MR | DOI | Zbl

[8] Jędrzej Garnek On $p$ -degree of elliptic curves, Int. J. Number Theory, Volume 14 (2018) no. 3, pp. 693-704 | MR | DOI | Zbl

[9] Ralph Greenberg Iwasawa theory—past and present, Class field theory—its centenary and prospect (Tokyo, 1998) (Advanced Studies in Pure Mathematics), Volume 30, Mathematical Society of Japan, 2001, pp. 335-385 | MR | DOI | Zbl

[10] Benedict H. Gross; Joe Harris Real algebraic curves, Ann. Sci. Éc. Norm. Supér., Volume 14 (1981) no. 2, pp. 157-182 | MR | Numdam | DOI | Zbl

[11] Chris Hall An open-image theorem for a general class of abelian varieties, Bull. Lond. Math. Soc., Volume 43 (2011) no. 4, pp. 703-711 | MR | DOI | Zbl

[12] Robin Hartshorne Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | Zbl

[13] Toshiro Hiranouchi Local torsion primes and the class numbers associated to an elliptic curve over $ℚ$ (2017) (https://arxiv.org/abs/1703.08275)

[14] Michael Larsen Maximality of Galois actions for compatible systems, Duke Math. J., Volume 80 (1995) no. 3, pp. 601-630 | MR | DOI | Zbl

[15] Gunter Malle; Donna Testerman Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, 133, Cambridge University Press, 2011 | MR | Zbl

[16] Arthur Mattuck Abelian varieties over $p$ -adic ground fields, Ann. Math., Volume 62 (1955), pp. 92-119 | MR | DOI | Zbl

[17] David Mumford Abelian varieties, Tata Institute of Fundamental Research, 2008 corrected reprint of the second (1974) edition | Zbl

[18] Arthur Ogus Hodge Cycles and Crystalline Cohomology, Hodge cycles, motives, and Shimura varieties (Lecture Notes in Mathematics), Volume 900, Springer, 1981, pp. 357-414 | DOI | Zbl

[19] Sara Arias-de-Reyna; Wojciech Gajda; Sebastian Petersen Big monodromy theorem for abelian varieties over finitely generated fields, J. Pure Appl. Algebra, Volume 217 (2013) no. 2, pp. 218-229 | MR | DOI | Zbl

[20] Kenneth A. Ribet Kummer theory on extensions of abelian varieties by tori, Duke Math. J., Volume 46 (1979) no. 4, pp. 745-761 | MR | DOI | Zbl

[21] Fumio Sairaiji; Takuya Yamauchi On the class numbers of the fields of the $p^{n}$ -torsion points of certain elliptic curves over $ℚ$ , J. Number Theory, Volume 156 (2015), pp. 277-289 | MR | DOI | Zbl

[22] Fumio Sairaiji; Takuya Yamauchi On the class numbers of the fields of the $p^{n}$ -torsion points of elliptic curves over $ℚ$ (2016) (https://arxiv.org/abs/1603.01296)

[23] Jean-Pierre Serre Abelian $l$ -adic representations and elliptic curves, Advanced Book Classics, Addison-Wesley Publishing Company, 1989 | Zbl

[24] Jean-Pierre Serre Lie algebras and Lie groups. 1964 lecture,s given at Harvard University, Lecture Notes in Mathematics, 1500, Springer, 1992 | Zbl

[25] Jean-Pierre Serre Oeuvres/Collected papers. IV. 1985–1998, Springer Collected Works in Mathematics, Springer, 2013 | Zbl

[26] Joseph H. Silverman Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994 | MR | Zbl

[27] Yu. V. Sosnovskij Commutator structure of symplectic groups, Mat. Zametki, Volume 24 (1978) no. 5, pp. 641-648 | MR | Zbl

[28] Irina D. Suprunenko; Alexandre E. Zalesski Reduced symmetric powers of natural realizations of the groups $s l_{m} (p)$ and $s p_{m} (p)$ and their restrictions to subgroups, Sib. Math. J., Volume 31 (1990) no. 4, pp. 33-46 | Zbl

[29] The LMFDB Collaboration The $L$ -functions and modular forms database, 2017 (http://www.lmfdb.org)

Cited by Sources: