[Paper Review] Patching and local-global principles for homogeneous spaces over function fields of p-adic curves

Jean-Louis Colliot-Thélène, R. Parimala+1 more|ArXiv.org|Dec 16, 2008

Algebraic Geometry and Number Theory23 references3 citations

TL;DR

This paper establishes local-global principles for rational points on homogeneous spaces over function fields of curves over p-adic fields, using patching techniques from Harbater, Hartmann, and Krashen. It proves that smooth quadrics (in odd characteristic) and principal homogeneous spaces under reductive groups with certain base extensions have rational points if they have points over all completions, resolving key cases of the Hasse principle in this context.

ABSTRACT

This is the final version, to appear in Commentarii Mathematici Helvetici.

Motivation & Objective

• To establish a local-global principle for rational points on homogeneous spaces over function fields F = K(X), where K is a p-adic field and X is a smooth projective curve.
• To investigate whether the existence of F_v-points for all discrete valuations v implies the existence of an F-rational point.
• To extend known results on isotropy of quadratic forms and triviality of Galois cohomology classes using modern patching methods.
• To provide a classification-free proof of the triviality of the kernel of the Rost invariant for split simply connected groups over function fields in one variable over p-adic fields.

Proposed method

• Applies the patching technique developed by Harbater, Hartmann, and Krashen to function fields of curves over p-adic fields.
• Uses the fact that the function field F has cohomological dimension 3, which allows control over Galois cohomology groups.
• Relies on the Rost invariant and Katô's theorem to analyze the kernel of the restriction map in Galois cohomology.
• Applies Bruhat-Tits theory to study reductive groups over discrete valuation rings and their base changes.
• Reduces the problem to analyzing the kernel of the residue map in cohomology, particularly in H^3 and H^2 groups.
• Employs a descent argument via the exact sequence involving G_m, M', and M to relate cohomological obstructions to local points.

Experimental results

Research questions

• RQ1 Does the existence of F_v-points for all discrete valuations v imply the existence of an F-rational point on a smooth projective quadric over F?
• RQ2 Is the restriction map H^1(F, G) → ∏_v H^1(F_v, G) injective for reductive groups G over F = K(X)?
• RQ3 Can the kernel of the Rost invariant for split simply connected groups over F be trivial when all local invariants vanish?
• RQ4 Does the Hasse principle hold for principal homogeneous spaces under reductive groups that are base extensions of groups over the ring of integers of K?
• RQ5 Can the isotropy of quadratic forms in ≥3 variables over F be deduced from their isotropy over all completions F_v?

Key findings

• For any nondegenerate quadratic form q over F = K(X) in at least 3 variables, if q is isotropic over F_v for all discrete valuations v, then q is isotropic over F, provided the residue characteristic of K is not 2.
• The restriction map H^1(F, G) → ∏_v H^1(F_v, G) has trivial kernel for any fibrewise connected reductive A-group G over the ring of integers A of K.
• The Hasse principle holds for principal homogeneous spaces under reductive groups G that are base extensions of a reductive group over the ring of integers of K.
• The kernel of the Rost invariant for split simply connected groups over F is trivial if all local invariants vanish, and this holds without assuming the absence of E_8 factors.
• The paper provides a classification-free proof of the triviality of the Rost invariant kernel, extending previous results that required case-by-case analysis.