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Restriction of Scalars, Chabauty's method, and P1 minus 3 points

Nicholas Triantafillou, University of Georgia
Tuesday, November 19, 2019 - 11:00am to 11:50am
PDL C-401

For a number field K and a curve C/K, Chabauty's method is a powerful p-adic tool for bounding/enumerating the set C(K). The method typically requires that dimension of the Jacobian J of C is greater than the rank of J(K). Since this condition often fails, especially when [K: Q] is large, several techniques have been proposed to augment Chabauty's method. For proper curves, Wetherell/Siksek introduced an analogue of Chabauty's method for the restriction of scalars Res_K/Q C that can succeed when the rank of J(K) is as large as [K: Q] * (dim J - 1). Using an analogue of Chabauty's method for restrictions of scalars in the non-proper case, we study the power of this approach together with descent for computing C = (P^1 \ {0,1,∞})(O_{K,S}). As an application, we show that if 3 splits completely in K and [K:Q] is prime to 3, then there are no solutions to the unit equation x + y = 1 with x,y both units in O_K.

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