Most odd degree hyperelliptic curves have only one rational point

Titelangaben

Poonen, Bjorn ; Stoll, Michael:
Most odd degree hyperelliptic curves have only one rational point.
In: Annals of Mathematics. Bd. 180 (2014) Heft 3 . - S. 1137-1166.
ISSN 0003-486X
DOI: https://doi.org/10.4007/annals.2014.180.3.7

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Abstract

Consider the smooth projective models C of curves y² = 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.