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.

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DOI : https://doi.org/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
@article{JTNB_2020__32_3_861_0,
     author = {Sajad Salami},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1144/}
}
Sajad Salami. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1144/

[1] Arnaud Beauville Variétés de Prym et Jacobiennes intermédiares, Ann. Sci. Éc. Norm. Supér., Volume 10 (1977), pp. 309-391 | Article | Numdam | Zbl 0368.14018

[2] Armand Borel; Jean-Pierre Serre Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv., Volume 39 (1964), pp. 111-164 | Article | Zbl 0143.05901

[3] Fabrizio Catanese; Ciro Ciliberto 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 1225590 | Zbl 0937.14007

[4] Annibale Comessatti Sui piani tripli ciclici irregolari, Palermo Rend., Volume 31 (1911), pp. 369-386 | Article | Zbl 42.0448.01

[5] Michele De Franchis 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 35.0423.01

[6] Fumio Hazama Rational points on certain abelian varieties over function fields, J. Number Theory, Volume 50 (1995) no. 2, pp. 278-285 | Article | MR 1316822 | Zbl 0838.14024

[7] Marc Hindry; Joseph H. Silverman Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, 201, Springer, 2001 | Zbl 0948.11023

[8] Sergej I. Khashin The irregularity of double surfaces, Math. Notes, Volume 33 (1983), pp. 233-235 | Article | Zbl 0564.14006

[9] Viktor S. Kulikov 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 1481615 | Zbl 0939.14013

[10] Serge Lang Abelian Varieties, Springer, 1983 | Zbl 0516.14031

[11] Serge Lang Fundamentals of Diophantine Geometry, Springer, 1983 | Zbl 0528.14013

[12] Herbert Lange; Angela Ortega Prym varieties of cyclic coverings, Geom. Dedicata, Volume 150 (2011), pp. 391-403 | Article | MR 2753712 | Zbl 1235.14028

[13] Andreĭ I. Lapin 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 199188 | Zbl 0192.26902

[14] Andreĭ I. Lapin 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 200778

[15] Anatoly Libgober Factors of Jacobians and isotrivial elliptic surfaces, J. Singul., Volume 5 (2012), pp. 115-123 | MR 2928938 | Zbl 1292.14026

[16] Anatoly Libgober On Mordell–Weil groups of isotrivial abelian varieties over function fields, Math. Ann., Volume 357 (2013) no. 2, pp. 605-629 | Article | MR 3096519 | Zbl 1283.14017

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

[18] Jean-Yves Mérindol Variétés de Prym d’un revêtement galoisien, J. Reine Angew. Math. (1995), pp. 49-61 | Zbl 0814.14043

[19] David Mumford Prym varieties (I), Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press Inc., 1974, pp. 325-350 | Zbl 0299.14018

[20] Fumio Sakai 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 | Article | MR 1272708 | Zbl 0830.14005

[21] Sajad Salami The rational points on certain abelian varieties over function fields, J. Number Theory, Volume 195 (2019), pp. 330-337 | Article | MR 3867445 | Zbl 06963711

[22] Jean-Pierre Serre Morphismes universels et variétés d’Albanese, Séminaire Claude Chevalley: Variétés de Picard, Volume 4, 1960 | Zbl 0123.13903

[23] Tetsuji Shioda Mordell–Weil Lattices for higher genus fibrations, Proc. Japan Acad., Ser. A, Volume 68 (1992) no. 9, pp. 247-250 | Article | MR 1202625 | Zbl 0788.14021

[24] Tetsuji Shioda Some remarks on elliptic curves over function fields, Journées arithmétiques (Astérisque), Volume 209, Société Mathématique de France, 1992, pp. 99-114 | Numdam | Zbl 0820.14016

[25] Tetsuji Shioda Constructing curves with high rank via symmetry, Am. J. Math., Volume 120 (1998) no. 3, pp. 551-566 | Article | MR 1623420 | Zbl 0919.14015

[26] Tetsuji Shioda 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 | Article | MR 1714831 | Zbl 0947.14012

[27] Joseph H. Silverman Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math., Volume 342 (1983), pp. 197-211 | MR 703488 | Zbl 0505.14035

[28] John T. Tate; Igor R. Shafarevich The rank of elliptic curves, Dokl. Akad. Nauk SSSR, Volume 175 (1967), pp. 770-773 | MR 237508 | Zbl 0168.42201

[29] Hiro-o Tokunaga Irreducible plane curves with the Albanese dimension 2, Proc. Am. Math. Soc., Volume 127 (1999) no. 7, pp. 1935-1940 | Article | MR 1637444 | Zbl 0917.14015

[30] Douglas Ulmer Elliptic curves with large rank over function fields, Ann. Math., Volume 155 (2002), pp. 295-315 | Article | MR 1888802 | Zbl 1109.11314

[31] Douglas Ulmer 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 | Article | MR 2270458 | Zbl 1110.11019

[32] Douglas Ulmer On Mordell–Weil groups of Jacobians over function fields, J. Inst. Math. Jussieu, Volume 12 (2013) no. 1, pp. 1-29 | Article | MR 3001733 | Zbl 1318.14025

[33] Douglas Ulmer Rational curves on elliptic surfaces, J. Algebr. Geom., Volume 26 (2017), pp. 357-377 | Article | MR 3606999 | Zbl 1357.14051

[34] Wang B. Wang On the twist of abelian varieties defined by the Galois extension of prime degree, J. Algebra, Volume 163 (1994) no. 3, pp. 813-818 | Article | MR 1265866 | Zbl 0811.14038