Title data
Poonen, Bjorn ; Stoll, Michael:
Most odd degree hyperelliptic curves have only one rational point.
In: Annals of Mathematics.
Vol. 180
(2014)
Issue 3
.
 pp. 11371166.
ISSN 0003486X
DOI: https://doi.org/10.4007/annals.2014.180.3.7
Project information
Project title: 



Project financing: 
Deutsche Forschungsgemeinschaft 
Abstract in another language
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 padic 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 padic analysis, the theory of arithmetic surfaces, a new result on torsion points on hyperelliptic curves, and crucially the BhargavaGross theorems on the average number and equidistribution of nonzero 2Selmer group elements, we prove that these conditions are often satisfied for p = 2.
Further data
Item Type:  Article in a journal 

Refereed:  Yes 
Institutions of the University:  Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II  Univ.Prof. Dr. Michael Stoll Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics 
Result of work at the UBT:  Yes 
DDC Subjects:  500 Science > 510 Mathematics 
Date Deposited:  20 Nov 2014 10:44 
Last Modified:  20 Nov 2014 10:44 
URI:  https://eref.unibayreuth.de/id/eprint/3748 