## Most odd degree hyperelliptic curves have only one rational point

##### Author(s)

Poonen, Bjorn; Stoll, Michael
DownloadPoonen_Most odd.pdf (398.0Kb)

OPEN_ACCESS_POLICY

# Open Access Policy

Creative Commons Attribution-Noncommercial-Share Alike

##### Terms of use

##### Metadata

Show full item record##### 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.

##### Date issued

2014-11##### Department

Massachusetts Institute of Technology. Department of Mathematics##### 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

##### ISSN

0003-486X