Section conjecture for p-adic curves

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Section conjecture for p-adic curves

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Recursos

Take a branched cover of a p-adic curve and a collar around one of its branch points, and decide whether the paper's patching step can swap that branched cover for a finite étale one. Then follow how that one check proves the p-adic section conjecture.

The claim and where it gets hard

For a smooth proper curve X of genus at least two over a finite extension k of Q_p, every rational point gives a section of π₁(X) → G_k. Theorem 1.1 says the map from X(k) to conjugacy classes of sections is bijective [0:16]. Most of the difficulty sits in one inclusion: K^sep ⊂ H, where H is the completion of the universal étale field K̃. With it, a section localized at a p-adic valuation lifts to a birational section, and Koenigsmann's theorem leads to a contradiction. The remaining valuations lead back to a rational point.

The patching check

Over a branch point b, with ramification indices e_c above it [1:18]:

the auxiliary cover must give a root of the local coordinate t of order m_b, a multiple of every e_c (take lcm), and
that root must exist on every sheet of the inverse collar, not only on one.

When both hold, each Z^e − t splits into coprime unit factors. The cover becomes trivial over the collar, extends as the trivial cover across the disk, and gluing plus GAGA gives a finite étale V → Y that maps to C outside the disks. Tameness is not needed: p may divide e. In the practice case (indices 2 and 4 with a sixth root) the check fails, because 4 ∤ 6. A 4th or 12th root works [2:50].

Where the roots come from, and what follows

Stage 1 [3:17]: ℓ-th roots (ℓ ≠ p, an odd prime) on node annuli from stable reduction and the Amini–Baker–Brugallé–Rabinoff lifting theorem. Mochizuki–Tsujimura moves branch collars into the skeleton.
Stage 2 [4:06]: roots of any order m from a cyclic cover with ℓ parallel skeleton edges. The Jacobian character has slope one; pull back by [m], then use p-adic log/exp on a middle collar.
Global [5:02]: when the finite-cover descent locus equals C(K), the global section map is bijective. Stoll's results give this for X₀(N) and X₁(N) over Q of genus at least two.

Use the check whenever you need étale covers to dominate a ramified one [5:33].

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