Restriction of scalars, the Chabauty-Coleman Method, and P¹ \ {0, 1, |}

Abstract

We extend Siksek's development of Chabauty's method for restriction of scalars of curves to give a method to compute the set of S-integral points on certain O [subscript capital K,S⁻ models C of punctured genus g curves C over a number field K. Our assumptions on C guarantee that it carries a morphism j : C --> J to a commutative group scheme J over O[subscript K,S] which is analogous to the Abel-Jacobi map from a proper curve of positive genus to its Jacobian. While Chabauty's method (generally) requires that rank J(O[subscript K,S]) </_ J[subscript K] - 1 in order to compute a finite subset p-adic points on C containing C(O[subscript K,S]), Chabauty's method for restriction of scalars computes a subset [sigma]-subscript C] of p-adic points of Res C which contains C(O[subscript K,S]). Naïvely, one might expect that [sigma]C is finite whenever the RoS inequality rank J(O[subscript K,S]) <_ [K : Q](dim J[subscript K] - 1) is satisfied.

However, even if this inequality is satisfied, [sigma]C can be infinite for geometric reasons, which we call base change obstructions and full Prym obstructions. When attempting to compute the O[subscript K,S]⁻points of C = P1 x {0, 1, |o}, we show that C can be replaced with a suitable descent set 2 of covers D, such that for each D [epsilon] T the RoS Chabauty inequality holds for D. Although we do not prove that the [sigma][subscript D] are finite, we do prove that the [sigma][subscript D] are not forced to be infinite for any of the known geometric reasons. In other words, there are no base change or full Prym obstructions to RoS Chabauty for D. We also give several examples of the method. For instance, when both 3 splits completely in K and [K : Q] is prime to 3 we show that (P¹ \ {0, 1, |})(O[subscript K) = [phi]. We also give new proofs that (P¹ \ {0, 1, |})(O[subscript K]) is finite for several classes of number fields K of low degree.

These results represent the first infinite class of cases where Chabauty's method for restrictions of scalars is proved to succeed where the classical Chabauty's method does not.