Most odd degree hyperelliptic curves have only one rational point

• Autores: Bjorn Poonen, Michael Stoll
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 180, Nº 3, 2014, págs. 1137-1166
• Idioma: inglés
• DOI: 10.4007/annals.2014.180.3.7
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• Resumen

  • Consider the smooth projective models C of curves y 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?8 . 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 .