The Hilbert symbol in the Hodge standard conjecture
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909459197984768 |
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| author | Ancona, Giuseppe Marmora, Adriano |
| author_facet | Ancona, Giuseppe Marmora, Adriano |
| contents | We study the Hodge standard conjecture for varieties over finite fields admitting a CM lifting, such as abelian varieties or products of K3 surfaces. For those varieties we show that the signature predicted by the conjecture holds true modulo $4$. This amounts to determining the discriminant and the Hilbert symbol of the intersection product. The first is obtained by $\ell$-adic arguments whereas the second needs a careful computation in $p$-adic Hodge theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_14001 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Hilbert symbol in the Hodge standard conjecture Ancona, Giuseppe Marmora, Adriano Algebraic Geometry Number Theory We study the Hodge standard conjecture for varieties over finite fields admitting a CM lifting, such as abelian varieties or products of K3 surfaces. For those varieties we show that the signature predicted by the conjecture holds true modulo $4$. This amounts to determining the discriminant and the Hilbert symbol of the intersection product. The first is obtained by $\ell$-adic arguments whereas the second needs a careful computation in $p$-adic Hodge theory. |
| title | The Hilbert symbol in the Hodge standard conjecture |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2210.14001 |