Supersingular K3 surfaces for large primes
Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height), then its Picard rank is 22. Along with work of Nygaard–Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p≥5. We prove Artin’s conjec...
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| Format: | Article |
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Duke University Press
2018
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| Online Access: | http://hdl.handle.net/1721.1/115923 https://orcid.org/0000-0002-7525-318X |
| _version_ | 1826215870613946368 |
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| author | Maulik, Davesh |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Maulik, Davesh |
| author_sort | Maulik, Davesh |
| collection | MIT |
| description | Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height), then its Picard rank is 22. Along with work of Nygaard–Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p≥5. We prove Artin’s conjecture under the additional assumption that X has a polarization of degree 2d with p>2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime to p when p≥5.
The argument uses Borcherds’s construction of automorphic forms on O(2,n) to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov–Pinkham–Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors. |
| first_indexed | 2024-09-23T16:38:21Z |
| format | Article |
| id | mit-1721.1/115923 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T16:38:21Z |
| publishDate | 2018 |
| publisher | Duke University Press |
| record_format | dspace |
| spelling | mit-1721.1/1159232024-06-27T18:56:28Z Supersingular K3 surfaces for large primes Maulik, Davesh Massachusetts Institute of Technology. Department of Mathematics Maulik, Davesh Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height), then its Picard rank is 22. Along with work of Nygaard–Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p≥5. We prove Artin’s conjecture under the additional assumption that X has a polarization of degree 2d with p>2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime to p when p≥5. The argument uses Borcherds’s construction of automorphic forms on O(2,n) to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov–Pinkham–Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors. 2018-05-29T13:33:21Z 2018-05-29T13:33:21Z 2014-10 2018-05-24T18:31:12Z Article http://purl.org/eprint/type/JournalArticle 0012-7094 1547-7398 http://hdl.handle.net/1721.1/115923 Maulik, Davesh. “Supersingular K3 Surfaces for Large Primes.” Duke Mathematical Journal, 163, 13 (October 2014): 2357–2425 https://orcid.org/0000-0002-7525-318X http://dx.doi.org/10.1215/00127094-2804783 Duke Mathematical Journal Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Duke University Press arXiv |
| spellingShingle | Maulik, Davesh Supersingular K3 surfaces for large primes |
| title | Supersingular K3 surfaces for large primes |
| title_full | Supersingular K3 surfaces for large primes |
| title_fullStr | Supersingular K3 surfaces for large primes |
| title_full_unstemmed | Supersingular K3 surfaces for large primes |
| title_short | Supersingular K3 surfaces for large primes |
| title_sort | supersingular k3 surfaces for large primes |
| url | http://hdl.handle.net/1721.1/115923 https://orcid.org/0000-0002-7525-318X |
| work_keys_str_mv | AT maulikdavesh supersingulark3surfacesforlargeprimes |