MATH Seminar

Title: Restriction of Scalars, Chabauty's Method, and $\mathbb P^1\smallsetminus \{0,1 \infty\}$.
Seminar: Algebra
Speaker: Nicholas Triantafillou of Massachusetts Institute of Technology
Contact: David Zurick-Brown, dzb@mathcs.emory.edu
Date: 2019-03-26 at 4:00PM
Venue: MSC W201
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Abstract: For a number field $K$ and a curve $C/K$, the 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 $[K:\mathbb Q]$ is large, several techniques have been proposed to augment Chabauty's method. For proper curves, Siksek introduced an analogue of Chabauty's method for the restriction of scalars ${Res}_{K/\mathbb Q} C$ that can succeed when the rank of $J(\mathcal O_{K,S})$ is as large as $[K:\mathbb Q]\cdot (\dim J - 1)$. 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 $C = (\mathbb P^1\smallsetminus \{0,1,\infty\})(\mathcal O_{K,S})$. As an application, we show that if $3$ splits completely in $K$ then there are no solutions to the unit equation $x + y = 1$ with $x,y \in \mathcal O_{K}^{\times}$.