Wednesday, September 4 2019, 3:45pm Boyd Room 303 Title: Restriction of Scalars, Chabauty's Method, and P1 - {0,1 ∞}. Abstract: For a number field K and a curve X/K, Chabauty's method is a powerful p-adic tool for bounding/enumerating the set X(K). The method typically requires that dimension of the Jacobian J of X is greater than the rank of $(K). Since this condition often fails, especially when the degree of K is large, several techniques have been proposed to augment Chabauty's method. For proper curves, Wetherell introduced an analogue of Chabauty's method for the restriction of scalars Res_K/Q( X ) that can succeed when the rank of J(O_{K,S}) is as large as deg(K)*(dim J - 1). In this talk, we will discuss how to adapt Chabuaty's method for restrictions of scalars to compute integral points on not-necessarily proper curves. We mostly focus on the power of this approach together with descent for computing the set (P1 - {0,1 ∞})(O_{K,S}) -- also known as the set of solutions to the S-unit equation. As an application, we give a Chabauty-inspired proof that if 3 splits completely in K and deg(K) is prime to 3, then there are no solutions to the unit equation x + y = 1 with x,y both units in O_K. Although this talk will elaborate on material mentioned in my Oberseminar from last week, this talk is independent and will not assume prior knowledge of last week's talk.
