A bound for the average rank of a family of abelian varieties
Wazir, Rania
Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004), p. 241-252 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

In this note, we consider a one-parameter family of Abelian varieties A/Q⁢T, and find an upper bound for the average rank in terms of the generic rank. This bound is based on Michel's estimates for the average rank in a one-parameter family of Abelian varieties, and extends previous work of Silverman for elliptic surfaces.

Si considera una famiglia di varietà abeliane A/Q⁢T e si determina un estremo superiore per il rango di Mordell-Weil medio, in termini del rango di Mordell- Weil della fibra generica. Questo risultato è basato su stime di Michel per il rango medio di una famiglia di varietà abeliane, ed estende un lavoro precedente di Silverman sulle superficie ellittiche.

Publié le : 2004-02-01
@article{BUMI_2004_8_7B_1_241_0,
     author = {Rania Wazir},
     title = {A bound for the average rank of a family of abelian varieties},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {7-A},
     year = {2004},
     pages = {241-252},
     zbl = {1118.11030},
     mrnumber = {2044269},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2004_8_7B_1_241_0}
}

Wazir, Rania. A bound for the average rank of a family of abelian varieties. Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004) pp. 241-252. http://gdmltest.u-ga.fr/item/BUMI_2004_8_7B_1_241_0/

[Apo76] Apostol, T., Introduction to analytic number theory, Undergrad. Texts Math., Springer-Verlag, Berlin, 1976. | MR 434929 | Zbl 0335.10001

[BK94] Brumer, A.-Kramer, K., The conductor of an abelian variety, Comp. Math, 92 (1994), 227-248. | MR 1283229 | Zbl 0818.14016

[BLR90] Bosch, S.-Lutkebohmert, W.-Raynaud, M., Neron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 21, Springer Verlag, Berlin, 1990. | MR 1045822 | Zbl 0705.14001

[Del81] Deligne, P., La conjecture de Weil II, Publ. Math. IHES, 52 (1981), 313-428. | MR 601520 | Zbl 0456.14014

[FP93] Fouvry, E.-Pomykala, J., Rang des courbes elliptiques et sommes d'exponentielles, Monatsh. Math., 116 (1993), 115-125. | MR 1245858 | Zbl 0796.14022

[Kat74] Katz, N. M., Etude cohomologique des pinceaux de Lefschetz, SGA 7 II, LNM340, Springer Verlag, 1974, pp. 254-327. | Zbl 0284.14007

[Mic95] Michel, P., Rang moyen de familles de courbes elliptiques et lois de Sato-Tate, Monatsh. Math., 120 (1995), 127-136. | MR 1348365 | Zbl 0869.11052

[Mic97] Michel, P., Le rang de familles de variétés abéliennes, J. Alg. Geom., 6 (1997), 201-234. | MR 1489113 | Zbl 0882.11033

[Mil86] Milne, J. S., Jacobian varieties, Arithmetic Geometry, Springer-Verlag, Berlin (1986), 167-212. | MR 861976 | Zbl 0604.14018

[Ram89] Ramakrishnan, D., Regulators, algebraic cycles, and values of L-functions, Algebraic K-theory and Algebraic Number Theory (M. R. Stein and R. K. Dennis, eds.), Contemp. Math., vol. 83, AMS, Providence, 1989, pp. 183-307. | MR 991982 | Zbl 0694.14002

[RS98] Rosen, M.-Silverman, J., On the rank of an elliptic surface, Invent. Math., 133 (1998), 43-67. | MR 1626465 | Zbl 0905.14019

[Ser65] Serre, J.-P., Zeta and L functions, Harper and Row, New York, 1965, pp. 82-92. | MR 194396 | Zbl 0171.19602

[Shi72] Shioda, T., On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20-59. | MR 429918 | Zbl 0226.14013

[Shi82] Shioda, T., On the Picard number of a Fermat surface, J. Fac. Sci. Univ. Tokyo, Sec. IA, 28 (1982), 725-734. | MR 656049 | Zbl 0567.14021

[Shi92] Shioda, T., Some remarks on elliptic curves over function fields, Astérisque, 209 (1992), 99-114. | MR 1211006 | Zbl 0820.14016

[Shi99] Shioda, T., Mordell-Weil lattices for higher genus fibration over a curve, New trends in algebraic geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999, pp. 359-373. | MR 1714831 | Zbl 0947.14012

[Sil94] Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer Verlag, 1994. | MR 1312368 | Zbl 0911.14015

[Sil98] Silverman, J. H., The average rank of elliptic curves, J. reine angew. Math., 504 (1998), 227-236. | MR 1656771 | Zbl 0923.11087

[Tat65] Tate, J., Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, pp. 93-110. | MR 225778 | Zbl 0213.22804

[Waz03] Wazir, R., A local-global summation formula for Abelian varieties, Preprint, available on http://www.arxiv.org/abs/math.NT/0302266 (Feb. 2003).

[Won02] Wong, S., On the Néron-Severi groups of fibered varieties, To appear, J. reine Angew. Math. (2002)