Most odd degree hyperelliptic curves have only one rational point
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Poonen_Most odd.pdf
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Author(s)
Poonen, Bjorn
Stoll, Michael
Date Issued
November 2014
Journal
Annals of Mathematics
Publisher
Princeton University Press
Citation
Poonen, Bjorn, and Michael Stoll. “Most Odd Degree Hyperelliptic Curves Have Only One Rational Point.” Ann. Math. (November 1, 2014): 1137–1166.
Version
Author's final manuscript
Abstract
Consider the smooth projective models C of curves y [superscript 2] = f(x) with f(x) ∈Z[x] monic and separable of degree 2g+1. We prove that for g ≥ 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g→∞. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty’s method that shows that certain computable conditions imply #C(Q)=1; on the other hand, using further p-adic analysis, the theory of arithmetic surfaces, a new result on torsion points on hyperelliptic curves, and crucially the Bhargava–Gross theorems on the average number and equidistribution of nonzero 2-Selmer group elements, we prove that these conditions are often satisfied for p=2.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.4007/annals.2014.180.3.7
