Selmer groups for elliptic curves in l d -extensions of function fields of characteristic p
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2301-2327.

Let F be a function field of characteristic p>0, /F a l d -extension (for some prime lp) 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 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 /F.

Soit F un corps de fonctions de caractéristique p>0, /F une l d -extension (pour un nombre premier lp) et E/F une courbe elliptique non-isotrivale. Nous étudions le comportement des r-parties des groupes de Selmer pour les sous-extensions de 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 /F.

DOI: 10.5802/aif.2491
Classification: 11G05, 11R23
Keywords: Selmer groups, elliptic curves, function fields, Iwasawa theory
Mot clés : groupes de Selmer, courbes elliptiques, corps de fonctions, théorie d’Iwasawa
Bandini, Andrea 1; Longhi, Ignazio 2

1 Università della Calabria Dipartimento di Matematica via P. Bucci - Cubo 30B 87036 Arcavacata di Rende (CS) (Italy)
2 National Taiwan University Department of Mathematics N ∘  1 section 4 Roosevelt Road Taipei 106 (Taiwan)
@article{AIF_2009__59_6_2301_0,
     author = {Bandini, Andrea and Longhi, Ignazio},
     title = {Selmer groups for elliptic curves in $\mathbb{Z}_l^d$-extensions of function fields of characteristic $p$},
     journal = {Annales de l'Institut Fourier},
     pages = {2301--2327},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     doi = {10.5802/aif.2491},
     zbl = {1207.11061},
     mrnumber = {2640921},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2491/}
}
TY  - JOUR
AU  - Bandini, Andrea
AU  - Longhi, Ignazio
TI  - Selmer groups for elliptic curves in $\mathbb{Z}_l^d$-extensions of function fields of characteristic $p$
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 2301
EP  - 2327
VL  - 59
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2491/
DO  - 10.5802/aif.2491
LA  - en
ID  - AIF_2009__59_6_2301_0
ER  - 
%0 Journal Article
%A Bandini, Andrea
%A Longhi, Ignazio
%T Selmer groups for elliptic curves in $\mathbb{Z}_l^d$-extensions of function fields of characteristic $p$
%J Annales de l'Institut Fourier
%D 2009
%P 2301-2327
%V 59
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2491/
%R 10.5802/aif.2491
%G en
%F AIF_2009__59_6_2301_0
Bandini, Andrea; Longhi, Ignazio. Selmer groups for elliptic curves in $\mathbb{Z}_l^d$-extensions of function fields of characteristic $p$. Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2301-2327. doi : 10.5802/aif.2491. https://aif.centre-mersenne.org/articles/10.5802/aif.2491/

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

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

[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.)

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

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

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

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

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

[9] Igusa, Jun-ichi Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves, Amer. J. Math., Volume 81 (1959), pp. 453-476 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

[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), Volume 155 (2002) no. 1, pp. 295-315 | DOI | MR | Zbl

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

Cited by Sources: