Selmer groups for elliptic curves in $\mathbb{Z}_l^d$-extensions of function fields of characteristic $p$

[Groupes de Selmer pour les courbes elliptiques en $\mathbb{Z}_l^d$-extensions de corps de fonctions de caractéristique $p$]

Bandini, Andrea ; Longhi, Ignazio

Annales de l'Institut Fourier, Tome 59 (2009), p. 2301-2327 / Harvested from Numdam

Résumé

Soit $F$ un corps de fonctions de caractéristique $p>0$, $\mathcal{F}/F$ une $\mathbb{Z}_l^d$-extension (pour un nombre premier $l\ne p$) et $E/F$ une courbe elliptique non-isotrivale. Nous étudions le comportement des $r$-parties des groupes de Selmer pour les sous-extensions de $\mathcal{F}$ par des variantes du Théorème de contrôle de Mazur. Conséquemment, nous démontrons que la limite des groupes de Selmer est un module finiment co-engendré (parfois de cotorsion) sur l’algèbre d’Iwasawa de $\mathcal{F}/F$.

Abstract

Let $F$ be a function field of characteristic $p>0$, $\mathcal{F}/F$ a $\mathbb{Z}_l^d$-extension (for some prime $l\ne p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of the $r$-parts of the Selmer groups ($r$ any prime) in the subextensions of $\mathcal{F}$ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of $\mathcal{F}/F$.

Détail

Publié le : 2009-01-01

MR 2640921 | Zbl pre05673897 | Zbl 1207.11061

DOI : https://doi.org/10.5802/aif.2491
Classification:  11G05,  11R23
Mots clés: groupes de Selmer, courbes elliptiques, corps de fonctions, théorie d’Iwasawa

Bibliographie

[1] Balister, P. N.; Howson, S. Note on Nakayama’s lemma for compact $\Lambda $-modules, Asian J. Math., Tome 1 (1997) no. 2, pp. 224-229 | MR 1491983 | Zbl 0904.16019

[2] Bandini, A.; Longhi, I. Control theorems for elliptic curves over function fields, Int. J. Number Theory, Tome 5 (2009) no. 2, pp. 229-256 | Article | MR 2502807 | Zbl pre05556399

[3] Bandini, A.; Longhi, I.; Vigni, S. Torsion points on elliptic curves over function fields and a theorem of Igusa (to appear on Expo. Math.) | Zbl pre05572047

[4] Ellenberg, Jordan S. Selmer groups and Mordell-Weil groups of elliptic curves over towers of function fields, Compos. Math., Tome 142 (2006) no. 5, pp. 1215-1230 | Article | MR 2264662 | Zbl 1106.11021

[5] Fastenberg, Lisa A. Mordell-Weil groups in procyclic extensions of a function field, Duke Math. J., Tome 89 (1997) no. 2, pp. 217-224 | Article | MR 1460621 | Zbl 0903.14006

[6] Greenberg, Ralph Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin (Lecture Notes in Math.) Tome 1716 (1999), pp. 51-144 | MR 1754686 | Zbl 0946.11027

[7] Greenberg, Ralph Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry (Park City, UT, 1999), Amer. Math. Soc., Providence, RI (IAS/Park City Math. Ser.) Tome 9 (2001), pp. 407-464 | MR 1860044 | Zbl 1002.11048

[8] Grothendieck, A. Revêtements étales et groupe fondamental (SGA 1), Société Mathématique de France, Paris, Documents Mathématiques (Paris), 3 (2003) | MR 2017446

[9] Igusa, Jun-Ichi Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves, Amer. J. Math., Tome 81 (1959), pp. 453-476 | Article | MR 104669 | Zbl 0115.38904

[10] Mazur, Barry Rational points of abelian varieties with values in towers of number fields, Invent. Math., Tome 18 (1972), pp. 183-266 | Article | MR 444670 | Zbl 0245.14015

[11] Milne, J. S. Étale cohomology, Princeton University Press, Princeton, N.J., Princeton Mathematical Series, Tome 33 (1980) | MR 559531 | Zbl 0433.14012

[12] Milne, J. S. Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York (1986), pp. 167-212 | MR 861976 | Zbl 0604.14018

[13] Neukirch, Jürgen Algebraic number theory, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome 322 (1999) (Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder) | MR 1697859 | Zbl 0956.11021

[14] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of number fields, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 323 (2000) | MR 1737196 | Zbl 0948.11001

[15] Ochiai, Tadashi; Trihan, Fabien On the Selmer groups of abelian varieties over function fields of characteristic $p>0$, Math. Proc. Cambridge Philos. Soc., Tome 146 (2009) no. 1, pp. 23-43 | Article | MR 2461865 | Zbl 1156.14037

[16] Pál, Ambrus Proof of an exceptional zero conjecture for elliptic curves over function fields, Math. Z., Tome 254 (2006) no. 3, pp. 461-483 | Article | MR 2244360 | Zbl pre05064684

[17] Serre, Jean-Pierre Local fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 67 (1979) (Translated from the French by Marvin Jay Greenberg) | MR 554237 | Zbl 0423.12016

[18] Shioda, Tetsuji An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math., Tome 108 (1986) no. 2, pp. 415-432 | Article | MR 833362 | Zbl 0602.14033

[19] Shioda, Tetsuji On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul., Tome 39 (1990) no. 2, pp. 211-240 | MR 1081832 | Zbl 0725.14017

[20] Silverman, Joseph H. The arithmetic of elliptic curves, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 106 (1986) | MR 817210 | Zbl 0585.14026

[21] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 151 (1994) | MR 1312368 | Zbl 0911.14015

[22] Silverman, Joseph H. The rank of elliptic surfaces in unramified abelian towers, J. Reine Angew. Math., Tome 577 (2004), pp. 153-169 | Article | MR 2108217 | Zbl 1105.11016

[23] Trihan, Fabien On the Iwasawa Main Conjecture of abelian varieties over function fields of characteristic $p>0$ (in progress)

[24] Ulmer, Douglas Elliptic curves with large rank over function fields, Ann. of Math. (2), Tome 155 (2002) no. 1, pp. 295-315 | Article | MR 1888802 | Zbl 1109.11314

[25] Ulmer, Douglas Jacobi sums, Fermat Jacobians, and ranks of abelian varieties over towers of function fields, Math. Res. Lett., Tome 14 (2007) no. 3, pp. 453-467 | MR 2318649 | Zbl 1127.14021