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Number Theory Seminar

Date and time:
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.

Postdoctoral Research and Teaching Associate

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