On class numbers of division fields of abelian varieties

Garnek, Jędrzej

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

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 $n\gg 0$, 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 $p$.

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 $p$.

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

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

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

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

[4] Clark, Pete L.; Lacy, Allan 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] David, Chantal; Weston, Tom 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] Dieulefait, Luis V. Explicit determination of the images of the Galois representations attached to abelian surfaces with $\mathrm{End}\left(A\right)=\mathbb{Z}$, Exp. Math., Volume 11 (2003) no. 4, pp. 503-512 | DOI | MR | Zbl

[7] Fukuda, Takashi; Komatsu, Keiichi; Yamagata, Shuji Iwasawa $\lambda$-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] Garnek, Jędrzej On $p$-degree of elliptic curves, Int. J. Number Theory, Volume 14 (2018) no. 3, pp. 693-704 | DOI | MR | Zbl

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

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

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

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

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

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

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

[18] Ogus, Arthur 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] Arias-de-Reyna, Sara; Gajda, Wojciech; Petersen, Sebastian 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] Ribet, Kenneth A. Kummer theory on extensions of abelian varieties by tori, Duke Math. J., Volume 46 (1979) no. 4, pp. 745-761 | DOI | MR | Zbl

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

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

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

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

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

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

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

[28] Suprunenko, Irina D.; Zalesski, Alexandre E. Reduced symmetric powers of natural realizations of the groups $s{l}_m\left(p\right)$ and $s{p}_m\left(p\right)$ 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)