A Tate-Type Theorem for Crystalline Classes in the 1-Motivic Category
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917337598263296 |
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| author | Mohajer, Mohammadreza |
| author_facet | Mohajer, Mohammadreza |
| contents | The Tate conjecture predicts that Galois-invariant classes in $\ell$-adic cohomology, and Frobenius-invariant classes in crystalline cohomology, arise from algebraic cycles. We prove an unconditional p-adic analogue of this principle in the 1-motivic range. Our starting point is a full-faithfulness theorem for Deligne 1-motives: after p-adic scalar extension, the Barsotti-Tate crystal functor identifies Hom-groups of 1-motives with Hom-groups in the category of filtered Dieudonne modules. Using the equivalence between the derived category of 1-motives up to isogeny and the 1-motivic part of Voevodsky's triangulated category of effective motives with rational coefficients, we extend this full-faithfulness result to the entire 1-motivic thick subcategory. More precisely, over a finite field $k = F_q$, we show that every Frobenius-compatible morphism between Barsotti-Tate crystalline realizations is induced by a unique 1-motivic morphism. Consequently, the Frobenius-invariant classes that occur in this range are already motivic and therefore algebraic. This yields an explicit linear-algebraic description of motivic morphisms and extension classes in level at most 1 in terms of Frobenius-equivariant maps preserving the Hodge filtration, giving a crystalline analogue of the principle that Tate classes are algebraic without invoking cycle conjectures in codimension at least 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12407 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Tate-Type Theorem for Crystalline Classes in the 1-Motivic Category Mohajer, Mohammadreza Algebraic Geometry Number Theory 14F30, 14L05, 14K15, 14F35, 14F42, 14G15 The Tate conjecture predicts that Galois-invariant classes in $\ell$-adic cohomology, and Frobenius-invariant classes in crystalline cohomology, arise from algebraic cycles. We prove an unconditional p-adic analogue of this principle in the 1-motivic range. Our starting point is a full-faithfulness theorem for Deligne 1-motives: after p-adic scalar extension, the Barsotti-Tate crystal functor identifies Hom-groups of 1-motives with Hom-groups in the category of filtered Dieudonne modules. Using the equivalence between the derived category of 1-motives up to isogeny and the 1-motivic part of Voevodsky's triangulated category of effective motives with rational coefficients, we extend this full-faithfulness result to the entire 1-motivic thick subcategory. More precisely, over a finite field $k = F_q$, we show that every Frobenius-compatible morphism between Barsotti-Tate crystalline realizations is induced by a unique 1-motivic morphism. Consequently, the Frobenius-invariant classes that occur in this range are already motivic and therefore algebraic. This yields an explicit linear-algebraic description of motivic morphisms and extension classes in level at most 1 in terms of Frobenius-equivariant maps preserving the Hodge filtration, giving a crystalline analogue of the principle that Tate classes are algebraic without invoking cycle conjectures in codimension at least 2. |
| title | A Tate-Type Theorem for Crystalline Classes in the 1-Motivic Category |
| topic | Algebraic Geometry Number Theory 14F30, 14L05, 14K15, 14F35, 14F42, 14G15 |
| url | https://arxiv.org/abs/2603.12407 |