A bound for the average rank of a family of abelian varieties

Rania Wazir

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 1, page 241-252
  • ISSN: 0392-4041

Abstract

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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.

How to cite

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Wazir, Rania. "A bound for the average rank of a family of abelian varieties." Bollettino dell'Unione Matematica Italiana 7-B.1 (2004): 241-252. <http://eudml.org/doc/195136>.

@article{Wazir2004,
abstract = {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.},
author = {Wazir, Rania},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {241-252},
publisher = {Unione Matematica Italiana},
title = {A bound for the average rank of a family of abelian varieties},
url = {http://eudml.org/doc/195136},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Wazir, Rania
TI - A bound for the average rank of a family of abelian varieties
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/2//
PB - Unione Matematica Italiana
VL - 7-B
IS - 1
SP - 241
EP - 252
AB - 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.
LA - eng
UR - http://eudml.org/doc/195136
ER -

References

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  1. APOSTOL, T., Introduction to analytic number theory, Undergrad. Texts Math., Springer-Verlag, Berlin, 1976. Zbl0335.10001MR434929
  2. BRUMER, A.- KRAMER, K., The conductor of an abelian variety, Comp. Math, 92 (1994), 227-248. Zbl0818.14016MR1283229
  3. BOSCH, S.- LUTKEBOHMERT, W.- RAYNAUD, M., Neron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 21, Springer Verlag, Berlin, 1990. Zbl0705.14001MR1045822
  4. DELIGNE, P., La conjecture de Weil II, Publ. Math. IHES, 52 (1981), 313-428. Zbl0456.14014MR601520
  5. FOUVRY, E.- POMYKALA, J., Rang des courbes elliptiques et sommes d'exponentielles, Monatsh. Math., 116 (1993), 115-125. Zbl0796.14022MR1245858
  6. KATZ, N. M., Etude cohomologique des pinceaux de Lefschetz, SGA 7 II, LNM340, Springer Verlag, 1974, pp. 254-327. Zbl0284.14007
  7. MICHEL, P., Rang moyen de familles de courbes elliptiques et lois de Sato-Tate, Monatsh. Math., 120 (1995), 127-136. Zbl0869.11052MR1348365
  8. MICHEL, P., Le rang de familles de variétés abéliennes, J. Alg. Geom., 6 (1997), 201-234. Zbl0882.11033MR1489113
  9. MILNE, J. S., Jacobian varieties, Arithmetic Geometry, Springer-Verlag, Berlin (1986), 167-212. Zbl0604.14018MR861976
  10. 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. Zbl0694.14002MR991982
  11. ROSEN, M.- SILVERMAN, J., On the rank of an elliptic surface, Invent. Math., 133 (1998), 43-67. Zbl0905.14019MR1626465
  12. SERRE, J.-P., Zeta and L functions, Harper and Row, New York, 1965, pp. 82-92. Zbl0171.19602MR194396
  13. SHIODA, T., On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20-59. Zbl0226.14013MR429918
  14. SHIODA, T., On the Picard number of a Fermat surface, J. Fac. Sci. Univ. Tokyo, Sec. IA, 28 (1982), 725-734. Zbl0567.14021MR656049
  15. SHIODA, T., Some remarks on elliptic curves over function fields, Astérisque, 209 (1992), 99-114. Zbl0820.14016MR1211006
  16. 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. Zbl0947.14012MR1714831
  17. SILVERMAN, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer Verlag, 1994. Zbl0911.14015MR1312368
  18. SILVERMAN, J. H., The average rank of elliptic curves, J. reine angew. Math., 504 (1998), 227-236. Zbl0923.11087MR1656771
  19. TATE, J., Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, pp. 93-110. Zbl0213.22804MR225778
  20. WAZIR, R., A local-global summation formula for Abelian varieties, Preprint, available on http://www.arxiv.org/abs/math.NT/0302266 (Feb. 2003). 
  21. WONG, S., On the Néron-Severi groups of fibered varieties, To appear, J. reine Angew. Math. (2002) 

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