Most odd degree hyperelliptic curves have only one rational point

By Bjorn Poonen, Michael Stoll. 2013

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Consider the smooth projective models C of curves y 2 = f(x) with f(x) 2 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 innity. We prove a lower bound on this fraction that tends to 1 as g! 1. 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 satised for p = 2.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/10f7096032e8709d9a0ed7b671ab32a97a756357

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Citations in the corpus

Squarefree values of polynomial discriminants I (2016, background): “erm (−1)iai represents the sum of the i-fold products of the roots of f. It is thus natural to order monic polynomials f(x) = xn + a1xn−1 +··· + an by the height H(f) := max{|ai|1/i} (see, e.g., [3], [13], [17]). We determine the density of monic integer polynomials having squarefree discriminant wi”
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, background): “ ≥ 3) a positive proportion of such curves have the point at inﬁni ty as their only rational point; moreover, the density of such curves having only this one rational point approaches 1 as n → ∞. See [39] for details and further such results. Note that Corollary 1.4(a) is also true when n = 1: it fo”
Most hyperelliptic curves over Q have no rational points (2013, methodology): “group of the family of hyperelliptic curves with a marked rational Weierstrass point is 3. By developing a beautiful reﬁnement of the method of Chabauty, Coleman, and McCallum [ 28], Poonen and Stoll [35] use this result on Selmer groups to show that a positiveproportion of odd degree hyperelliptic ”