Essential dimension via prismatic cohomology
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866913244849897472 |
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| author | Farb, Benson Kisin, Mark Wolfson, Jesse |
| author_facet | Farb, Benson Kisin, Mark Wolfson, Jesse |
| contents | For $X$ a smooth, proper complex variety we show that for $p\gg 0$, the restriction of the mod $p$ cohomology $H^i(X,\mathbb{F}_p)$ to any Zariski open has dimension at least $h^{0,i}_X$. The proof uses the prismatic cohomology of Bhatt-Scholze.
We use this result to obtain lower bounds on the $p$-essential dimension of covers of complex varieties. For example, we prove the $p$-incompressibility of the mod $p$ homology cover of an abelian variety, confirming a conjecture of Brosnan for sufficiently large $p.$ By combining these techniques with the theory of toroidal compactifications of Shimura varieties, we show that for any Hermitian symmetric domain $X,$ there exist $p$-congruence covers that are $p$-incompressible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_05534 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Essential dimension via prismatic cohomology Farb, Benson Kisin, Mark Wolfson, Jesse Algebraic Geometry Algebraic Topology Number Theory For $X$ a smooth, proper complex variety we show that for $p\gg 0$, the restriction of the mod $p$ cohomology $H^i(X,\mathbb{F}_p)$ to any Zariski open has dimension at least $h^{0,i}_X$. The proof uses the prismatic cohomology of Bhatt-Scholze. We use this result to obtain lower bounds on the $p$-essential dimension of covers of complex varieties. For example, we prove the $p$-incompressibility of the mod $p$ homology cover of an abelian variety, confirming a conjecture of Brosnan for sufficiently large $p.$ By combining these techniques with the theory of toroidal compactifications of Shimura varieties, we show that for any Hermitian symmetric domain $X,$ there exist $p$-congruence covers that are $p$-incompressible. |
| title | Essential dimension via prismatic cohomology |
| topic | Algebraic Geometry Algebraic Topology Number Theory |
| url | https://arxiv.org/abs/2110.05534 |