Family 019Number theoryNo LeanNot yet digested
The local p-adic section conjecture and global consequences
Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚ correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves and of genus at least two.
Manuscripts in this family · 2
Étale covers with a prescribed exterior sheet
Sep 24, 2026·main.pdf
Let X be a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.
The p-adic section conjecture
Oct 6, 2026·main.pdf
We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚ. Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes and of genus at least two over ℚ.