Chabauty’s Method Proves That Most Odd Degree Hyperelliptic Curves Have Only One Rational Point
نویسنده
چکیده
Consider the smooth projective models C of curves y = 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, via Chabauty’s method at the prime 2.
منابع مشابه
Most Odd Degree Hyperelliptic Curves Have Only One Rational Point
Consider the smooth projective models C of curves y = 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 b...
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