{"id":"openai-math/The-p-adic-section-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-p-adic-section-conjecture-October-6-2026","tags":["number-theory","arithmetic-geometry","p-adic-analysis"],"score":0,"title":"The p-adic section conjecture","domain":"Number theory","family":"019","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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 p . 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 $`X_0(N)`$ and $`X_1(N)`$ of genus at least two over ℚ.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-06","reactions":{},"formalized":false,"family_title":"The local p -adic section conjecture and global consequences"}