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Curves of every genus with many points, II: Asymptotically good families

Noam D. Elkies·Everett W. Howe·Andrew Kresch·Bjorn Poonen·Joseph L. Wetherell·Michael E. Zieve
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Abstract

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every non-negative integer g, there is a genus-g curve over F_q with at least c_q * g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q = d * (log q). We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over F_q that has at least c*g/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r > c*g/n.

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Noam D. Elkies, Everett W. Howe, Andrew Kresch, Bjorn Poonen, Joseph L. Wetherell, Michael E. Zieve. 2002-08-08. Curves of every genus with many points, II: Asymptotically good families. https://doi.org/10.1215/s0012-7094-04-12224-9

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