p-adic approaches to rational points on curves

Bjorn Poonen

In these four lectures, I will describe Chabauty's p-adic method for determining the rational points on a curve whose Jacobian has rank less than the genus, hint at Kim's nonabelian generalization, and finally discuss the recent paper of Lawrence and Venkatesh that uses p-adic period maps to give a new proof of Faltings\'s theorem.

References:

• McCallum-Poonen, The method of Chabauty and Coleman
• Corwin, From Chabauty's method to Kim's non-abelian Chabauty's method
• Lawrence-Venkatesh, Diophantine problems and p-adic period mappings

Notes:

pdf file