Grothendieck's section conjecture for the modular curves X0(N) and X1(N) over Q
Claims Theorem 8.2: the global section conjecture holds for every smooth proper geometrically connected curve of genus at least two over a number field whose finite-cover descent locus equals its rational points; by Stoll's criteria this includes curves mapping nonconstantly to an abelian variety with finitely many rational points and trivial divisible Sha, and the modular curves , of genus at least two over . It does NOT prove the section conjecture for arbitrary curves over number fields; the descent-locus equality remains a separate hypothesis in general.
Exact FrontierDelta
Scope and record
Occurred: Sep 24, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: Grothendieck's section conjecture for the modular curves X0(N) and X1(N) over Q · Section conjecture for X0(N), X1(N)
Confidence: Not scored
Registry verification: unreviewed · announcement · partial
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VibeMathed
registry · event recorded by
Unreleased internal OpenAI model
model · ai model contributor · OpenAI
Lineage and corrections
Claims Theorem 8.2: the global section conjecture holds for every smooth proper geometrically connected curve of genus at least two over a number field whose finite-cover descent locus equals its rational points; by Stoll's criteria this includes curves mapping nonconstantly to an abelian variety with finitely many rational points and trivial divisible Sha, and the modular curves , of genus at least two over . It does NOT prove the section conjecture for arbitrary curves over number fields; the descent-locus equality remains a separate hypothesis in general. parent of this event
This event attributed to Unreleased internal OpenAI model
Companion: Etale covers with a prescribed exterior sheet evidence for this event
Grothendieck, letter to Faltings (27 June 1983), in Geometric Galois Actions 1 evidence for this event
Grothendieck's section conjecture for the modular curves X0(N) and X1(N) over Q parent of this event
Earlier edition (September 24, 2026) evidence for this event
The local p-adic section conjecture for curves of genus at least two evidence for this event
VibeMathed record: Grothendieck's section conjecture for the modular curves X0(N) and X1(N) over Q evidence for this event
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