On class numbers of division fields of abelian varieties

Jędrzej Garnek

doi:10.5802/jtnb.1077

Jędrzej Garnek ¹

¹ Graduate 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
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 $p$ .

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

MR: 3994728

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

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     title = {On class numbers of division fields of abelian varieties},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1077/

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 | DOI | MR | 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 | DOI | MR | 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 | DOI | MR | 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 | DOI | MR | 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 | DOI | MR | 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 | DOI | MR | Zbl

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

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

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