On class numbers of division fields of abelian varieties
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 227-242.

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 n0, 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 : 2018-05-28
Revised : 2018-09-24
Accepted : 2018-11-16
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1077
Classification:  11R29,  11G10
Keywords: division fields, class number, abelian varieties
     author = {J\k edrzej Garnek},
     title = {On class numbers of division fields of abelian varieties},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {227-242},
     doi = {10.5802/jtnb.1077},
     language = {en},
Garnek, Jędrzej. 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/item/JTNB_2019__31_1_227_0/

[1] Grzegorz Banaszak; Wojciech Gajda; Piotr Krasoń Detecting linear dependence by reduction maps, J. Number Theory, Tome 115 (2005) no. 2, pp. 322-342 | Zbl 1089.11030

[2] M. I. Bashmakov The cohomology of abelian varieties over a number field, Russ. Math. Surv., Tome 27 (1972) no. 6, pp. 25-70 | Zbl 0271.14010

[3] Siegfried Bosch; Werner Lütkebohmert; Michel Raynaud Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 21, Springer, 1990 | Zbl 0705.14001

[4] Pete L. Clark; Allan Lacy There are genus one curves of every index over every infinite, finitely generated field, J. Reine Angew. Math., Tome 794 (2019), pp. 65-86 | Zbl 07050841

[5] Chantal David; Tom Weston Local torsion on elliptic curves and the deformation theory of Galois representations, Math. Res. Lett., Tome 15 (2008) no. 2-3, pp. 599-611 | Zbl 1222.14070

[6] Luis V. Dieulefait Explicit determination of the images of the Galois representations attached to abelian surfaces with End (A)=, Exp. Math., Tome 11 (2003) no. 4, pp. 503-512 | Zbl 1162.11347

[7] Takashi Fukuda; Keiichi Komatsu; Shuji Yamagata Iwasawa λ-invariants and Mordell–Weil ranks of abelian varieties with complex multiplication, Acta Arith., Tome 127 (2007) no. 4, pp. 305-307 | Zbl 1188.11055

[8] Jędrzej Garnek On p-degree of elliptic curves, Int. J. Number Theory, Tome 14 (2018) no. 3, pp. 693-704 | Zbl 06855926

[9] Ralph Greenberg Iwasawa theory—past and present, Class field theory—its centenary and prospect (Tokyo, 1998) (Advanced Studies in Pure Mathematics) Tome 30, Mathematical Society of Japan, 2001, pp. 335-385 | Zbl 0998.11054

[10] Benedict H. Gross; Joe Harris Real algebraic curves, Ann. Sci. Éc. Norm. Supér., Tome 14 (1981) no. 2, pp. 157-182 | Zbl 0533.14011

[11] Chris Hall An open-image theorem for a general class of abelian varieties, Bull. Lond. Math. Soc., Tome 43 (2011) no. 4, pp. 703-711 | Zbl 1225.11083

[12] Robin Hartshorne Algebraic geometry, Graduate Texts in Mathematics, Tome 52, Springer, 1977 | Zbl 0367.14001

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

[14] Michael Larsen Maximality of Galois actions for compatible systems, Duke Math. J., Tome 80 (1995) no. 3, pp. 601-630 | Zbl 0912.11026

[15] Gunter Malle; Donna Testerman Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, Tome 133, Cambridge University Press, 2011 | Zbl 1256.20045

[16] Arthur Mattuck Abelian varieties over p-adic ground fields, Ann. Math., Tome 62 (1955), pp. 92-119 | Zbl 0066.02802

[17] David Mumford Abelian varieties, Tata Institute of Fundamental Research, 2008 (corrected reprint of the second (1974) edition) | Zbl 1177.14001

[18] Arthur Ogus Hodge Cycles and Crystalline Cohomology, Hodge cycles, motives, and Shimura varieties (Lecture Notes in Mathematics) Tome 900, Springer, 1981, pp. 357-414 | Zbl 0538.14010

[19] Sara Arias-de-Reyna; Wojciech Gajda; Sebastian Petersen Big monodromy theorem for abelian varieties over finitely generated fields, J. Pure Appl. Algebra, Tome 217 (2013) no. 2, pp. 218-229 | Zbl 1294.11095

[20] Kenneth A. Ribet Kummer theory on extensions of abelian varieties by tori, Duke Math. J., Tome 46 (1979) no. 4, pp. 745-761 | Zbl 0428.14018

[21] Fumio Sairaiji; Takuya Yamauchi On the class numbers of the fields of the p n -torsion points of certain elliptic curves over , J. Number Theory, Tome 156 (2015), pp. 277-289 | Zbl 1328.11064

[22] Fumio Sairaiji; Takuya Yamauchi On the class numbers of the fields of the p n -torsion points of elliptic curves over (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 0709.14002

[24] Jean-Pierre Serre Lie algebras and Lie groups. 1964 lecture,s given at Harvard University, Lecture Notes in Mathematics, Tome 1500, Springer, 1992 | Zbl 0742.17008

[25] Jean-Pierre Serre Oeuvres/Collected papers. IV. 1985–1998, Springer Collected Works in Mathematics, Springer, 2013 | Zbl 1278.01034

[26] Joseph H. Silverman Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, Tome 151, Springer, 1994 | Zbl 0911.14015

[27] Yu. V. Sosnovskij Commutator structure of symplectic groups, Mat. Zametki, Tome 24 (1978) no. 5, pp. 641-648 | Zbl 0409.20037

[28] Irina D. Suprunenko; Alexandre E. Zalesski Reduced symmetric powers of natural realizations of the groups sl m (p) and sp m (p) and their restrictions to subgroups, Sib. Math. J., Tome 31 (1990) no. 4, pp. 33-46 | Zbl 0793.20042

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