Twists of the Albanese varieties of cyclic multiple planes with large ranks over higher dimension function fields
Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 861-876.

Dans [17], nous avons prouvé un théorème de structure pour les groupes de Mordell–Weil de variétés abéliennes définies sur des corps de fonctions, obtenues comme tordues de variétés abéliennes par des revêtements cycliques de variétés projectives, et ce en terme des variétés de Prym associées à ces revêtements. Dans ce nouvel article, nous donnons une méthode explicite pour construire des variétés abéliennes de grands rangs sur les corps de fonctions. Pour ce faire, nous appliquons le théorème mentionné ci-dessus aux twists des variétés d’Albanese des plans multiples cycliques.

In [17], we proved a structure theorem on the Mordell–Weil group of abelian varieties over function fields that arise as the twists of abelian varieties by the cyclic covers of projective varieties in terms of the Prym varieties associated with covers. In this paper, we provide an explicit way to construct the abelian varieties with large ranks over the higher dimension function fields. To do so, we apply the above-mentioned theorem to the twists of Albanese varieties of the cyclic multiple planes.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1144
Classification : 11G10, 14H40, 14H05
Mots clés : Mordell–Weil rank, Twists, Albanese and Prym varieties, Cyclic multiple planes, Higher dimension function fields
Salami, Sajad 1

1 Instítuto de Matemática e Estatística Universidade Estadual do Rio do Janeiro, Brazil
@article{JTNB_2020__32_3_861_0,
     author = {Salami, Sajad},
     title = {Twists of the {Albanese} varieties of cyclic multiple planes with large ranks over higher dimension function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {861--876},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {3},
     year = {2020},
     doi = {10.5802/jtnb.1144},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1144/}
}
TY  - JOUR
AU  - Salami, Sajad
TI  - Twists of the Albanese varieties of cyclic multiple planes with large ranks over higher dimension function fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
SP  - 861
EP  - 876
VL  - 32
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1144/
DO  - 10.5802/jtnb.1144
LA  - en
ID  - JTNB_2020__32_3_861_0
ER  - 
%0 Journal Article
%A Salami, Sajad
%T Twists of the Albanese varieties of cyclic multiple planes with large ranks over higher dimension function fields
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 861-876
%V 32
%N 3
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1144/
%R 10.5802/jtnb.1144
%G en
%F JTNB_2020__32_3_861_0
Salami, Sajad. Twists of the Albanese varieties of cyclic multiple planes with large ranks over higher dimension function fields. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 861-876. doi : 10.5802/jtnb.1144. http://www.numdam.org/articles/10.5802/jtnb.1144/

[1] Beauville, Arnaud Variétés de Prym et Jacobiennes intermédiares, Ann. Sci. Éc. Norm. Supér., Volume 10 (1977), pp. 309-391 | DOI | Numdam | Zbl

[2] Borel, Armand; Serre, Jean-Pierre Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv., Volume 39 (1964), pp. 111-164 | DOI | Zbl

[3] Catanese, Fabrizio; Ciliberto, Ciro On the irregularity of cyclic coverings of algebraic surfaces, Geometry of complex projective varieties (Cetraro, 1990) (Seminars and Conferences), Volume 9, Mediterranean Press, 1993, pp. 89-115 | MR | Zbl

[4] Comessatti, Annibale Sui piani tripli ciclici irregolari, Palermo Rend., Volume 31 (1911), pp. 369-386 | DOI | Zbl

[5] De Franchis, Michele I piani doppi dotati di due o piú differenziali totah di prima specie, Rom. Acc. L. Rend. (5), Volume 13 (1904) no. 1, pp. 688-695 | Zbl

[6] Hazama, Fumio Rational points on certain abelian varieties over function fields, J. Number Theory, Volume 50 (1995) no. 2, pp. 278-285 | DOI | MR | Zbl

[7] Hindry, Marc; Silverman, Joseph H. Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, 201, Springer, 2001 | Zbl

[8] Khashin, Sergej I. The irregularity of double surfaces, Math. Notes, Volume 33 (1983), pp. 233-235 | DOI | Zbl

[9] Kulikov, Viktor S. On plane algebraic curves of positive Albanese dimension, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 59 (1995) no. 6, pp. 75-94 translation in Izv. Math. 59 (1955), no. 6, p. 1173-1192 | MR | Zbl

[10] Lang, Serge Abelian Varieties, Springer, 1983 | Zbl

[11] Lang, Serge Fundamentals of Diophantine Geometry, Springer, 1983 | Zbl

[12] Lange, Herbert; Ortega, Angela Prym varieties of cyclic coverings, Geom. Dedicata, Volume 150 (2011), pp. 391-403 | DOI | MR | Zbl

[13] Lapin, Andreĭ I. Subfields of hyperelliptic fields. I, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 28 (1964), pp. 953-988 translation in Am. Math. Soc., Transl. 69 (1968), p. 204-240 | MR | Zbl

[14] Lapin, Andreĭ I. On the rational points of an elliptic curves, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 29 (1965), pp. 701-716 translation in Am. Math. Soc., Transl. 69 (1968), p. 231-245 | MR

[15] Libgober, Anatoly Factors of Jacobians and isotrivial elliptic surfaces, J. Singul., Volume 5 (2012), pp. 115-123 | MR | Zbl

[16] Libgober, Anatoly On Mordell–Weil groups of isotrivial abelian varieties over function fields, Math. Ann., Volume 357 (2013) no. 2, pp. 605-629 | DOI | MR | Zbl

[17] The LMFDB Collaboration The L-functions and Modular Forms Database, 2019 (http://www.lmfdb.org)

[18] Mérindol, Jean-Yves Variétés de Prym d’un revêtement galoisien, J. Reine Angew. Math. (1995), pp. 49-61 | Zbl

[19] Mumford, David Prym varieties (I), Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press Inc., 1974, pp. 325-350 | Zbl

[20] Sakai, Fumio On the irregularity of cyclic coverings of the projective plane, Classification of algebraic varieties (Contemporary Mathematics), Volume 162, American Mathematical Society, 1994, pp. 359-369 | DOI | MR | Zbl

[21] Salami, Sajad The rational points on certain abelian varieties over function fields, J. Number Theory, Volume 195 (2019), pp. 330-337 | DOI | MR | Zbl

[22] Serre, Jean-Pierre Morphismes universels et variétés d’Albanese, Séminaire Claude Chevalley: Variétés de Picard, Volume 4, 1960 | Zbl

[23] Shioda, Tetsuji Mordell–Weil Lattices for higher genus fibrations, Proc. Japan Acad., Ser. A, Volume 68 (1992) no. 9, pp. 247-250 | DOI | MR | Zbl

[24] Shioda, Tetsuji Some remarks on elliptic curves over function fields, Journées arithmétiques (Astérisque), Volume 209, Société Mathématique de France, 1992, pp. 99-114 | Numdam | Zbl

[25] Shioda, Tetsuji Constructing curves with high rank via symmetry, Am. J. Math., Volume 120 (1998) no. 3, pp. 551-566 | DOI | MR | Zbl

[26] Shioda, Tetsuji Mordell–Weil lattices for higher genus fibration over a curve, New trends in algebraic geometry (London Mathematical Society Lecture Note Series), Volume 264, Cambridge University Press, 1999, pp. 359-373 | DOI | MR | Zbl

[27] Silverman, Joseph H. Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math., Volume 342 (1983), pp. 197-211 | MR | Zbl

[28] Tate, John T.; Shafarevich, Igor R. The rank of elliptic curves, Dokl. Akad. Nauk SSSR, Volume 175 (1967), pp. 770-773 | MR | Zbl

[29] Tokunaga, Hiro-o Irreducible plane curves with the Albanese dimension 2, Proc. Am. Math. Soc., Volume 127 (1999) no. 7, pp. 1935-1940 | DOI | MR | Zbl

[30] Ulmer, Douglas Elliptic curves with large rank over function fields, Ann. Math., Volume 155 (2002), pp. 295-315 | DOI | MR | Zbl

[31] Ulmer, Douglas L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields, Invent. Math., Volume 167 (2007) no. 2, pp. 379-408 | DOI | MR | Zbl

[32] Ulmer, Douglas On Mordell–Weil groups of Jacobians over function fields, J. Inst. Math. Jussieu, Volume 12 (2013) no. 1, pp. 1-29 | DOI | MR | Zbl

[33] Ulmer, Douglas Rational curves on elliptic surfaces, J. Algebr. Geom., Volume 26 (2017), pp. 357-377 | DOI | MR | Zbl

[34] Wang, Wang B. On the twist of abelian varieties defined by the Galois extension of prime degree, J. Algebra, Volume 163 (1994) no. 3, pp. 813-818 | DOI | MR | Zbl

Cité par Sources :