p-adic Section Conjecture

The -adic section conjecture asserts that rational points on a smooth proper geometrically connected curve of genus at least 2 over a finite extension correspond bijectively to continuous sections of its arithmetic étale fundamental-group sequence. If is the absolute Galois group of , the sequence is

Sections are taken modulo conjugation by the full geometric etale fundamental group . A point determines a section through its induced map of fundamental groups, and geometric conjugation removes the dependence on the chosen path between base points.