Geodesic
A bound for the average rank of a family of abelian varieties
Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 1, pp. 241-252.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

In this note, we consider a one-parameter family of Abelian varieties $A/ \mathbb{Q}(T)$, 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 $A/ \mathbb{Q}(T)$ 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. 
@article{BUMI_2004_8_7B_1_a10,
     author = {Wazir, Rania},
     title = {A bound for the average rank of a family of abelian varieties},
     journal = {Bollettino della Unione matematica italiana},
     pages = {241--252},
     publisher = {mathdoc},
     volume = {Ser. 8, 7B},
     number = {1},
     year = {2004},
     zbl = {1118.11030},
     mrnumber = {434929},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_1_a10/}
}
TY  - JOUR
AU  - Wazir, Rania
TI  - A bound for the average rank of a family of abelian varieties
JO  - Bollettino della Unione matematica italiana
PY  - 2004
SP  - 241
EP  - 252
VL  - 7B
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_1_a10/
LA  - en
ID  - BUMI_2004_8_7B_1_a10
ER  - 
%0 Journal Article
%A Wazir, Rania
%T A bound for the average rank of a family of abelian varieties
%J Bollettino della Unione matematica italiana
%D 2004
%P 241-252
%V 7B
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_1_a10/
%G en
%F BUMI_2004_8_7B_1_a10
Wazir, Rania. A bound for the average rank of a family of abelian varieties. Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 1, pp. 241-252. http://geodesic.mathdoc.fr/item/BUMI_2004_8_7B_1_a10/

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

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

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

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

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

[Kat74] N. M. Katz, Etude cohomologique des pinceaux de Lefschetz, SGA 7 II, LNM 340, Springer Verlag, 1974, pp. 254-327. | Zbl

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

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

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

[Ram89] D. Ramakrishnan, Regulators, algebraic cycles, and values of $L$-functions, Algebraic $K$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 | Zbl

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

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

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

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

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

[Shi99] T. Shioda, 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 | Zbl

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

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

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

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

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