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Mosc. Math. J., 2002, Volume 2, Number 2, Pages 281–311 (Mi mmj56)  

This article is cited in 11 scientific papers (total in 11 papers)

Counting elliptic surfaces over finite fields

A. J. de Jong

Massachusetts Institute of Technology

Abstract: We count the number of isomorphism classes of elliptic curves of given height $d$ over the field of rational functions in one variable over the finite field of $q$ elements. We also estimate the number of isomorphism classes of elliptic surfaces over the projective line, which have a polarization of relative degree 3. This leads to an upper bound for the average 3-Selmer rank of the aforementionned curves. Finally, we deduce a new upper bound for the average rank of elliptic curves in the large $d$ limit, namely the average rank is asymptotically bounded by $1.5+O(1/q)$.

Key words and phrases: Elliptic curves, elliptic surfaces, rank, average rank, Selmer group.

DOI: https://doi.org/10.17323/1609-4514-2002-2-2-281-311

Full text: http://www.ams.org/.../abst2-2-2002.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 14G, 11G, 14H25, 1452
Received: December 13, 2001
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Citation: A. J. de Jong, “Counting elliptic surfaces over finite fields”, Mosc. Math. J., 2:2 (2002), 281–311

Citation in format AMSBIB
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\by A.~J.~de Jong
\paper Counting elliptic surfaces over finite fields
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 2
\pages 281--311
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1944508}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Poonen B., “Squarefree values of multivariable polynomials”, Duke Math. J., 118:2 (2003), 353–373  crossref  mathscinet  zmath  isi
    2. de Jong A.J., Freedman R., “On the Geometry of Principal Homogeneous Spaces”, Amer J Math, 133:3 (2011), 753–796  crossref  mathscinet  zmath  isi
    3. Poonen B., Rains E., “Random Maximal Isotropic Subspaces and Selmer Groups”, J Amer Math Soc, 25:1 (2012), 245–269  crossref  mathscinet  zmath  isi
    4. Kloosterman R., “The Average Rank of Elliptic N-Folds”, Indiana Univ. Math. J., 61:1 (2012), 131–146  crossref  mathscinet  zmath  isi
    5. Silverberg A., “Ranks “Cheat Sheet””, Women in Numbers 2: Research Directions in Number Theory, Contemporary Mathematics, 606, eds. David C., Lalin M., Manes M., Amer Mathematical Soc, 2013, 101–110  crossref  mathscinet  zmath  isi
    6. Poonen B., “Average Rank of Elliptic Curves [After Manjul Bhargava and Arul Shankar]”, Asterisque, 2013, no. 352, 187–204  mathscinet  zmath  isi
    7. Ho W., “How Many Rational Points Does a Random Curve Have?”, Bull. Amer. Math. Soc., 51:1 (2014), 27–52  mathscinet  zmath  isi
    8. Poonen B., Stoll M., “Most Odd Degree Hyperelliptic Curves Have Only One Rational Point”, Ann. Math., 180:3 (2014), 1137–1166  crossref  mathscinet  zmath  isi
    9. Ho Q.P., Le Hung V.B., Ngo B.C., “Average Size of 2-Selmer Groups of Elliptic Curves Over Function Fields”, Math. Res. Lett., 21:6 (2014), 1305–1339  crossref  mathscinet  zmath  isi
    10. Bhargava M., Shankar A., “Ternary Cubic Forms Having Bounded Invariants, and the Existence of a Positive Proportion of Elliptic Curves Having Rank 0”, Ann. Math., 181:2 (2015), 587–621  crossref  mathscinet  zmath  isi
    11. Bhargava M., Shankar A., “Binary Quartic Forms Having Bounded Invariants, and the Boundedness of the Average Rank of Elliptic Curves”, Ann. Math., 181:1 (2015), 191–242  crossref  mathscinet  zmath  isi
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