Constructing abelian varieties from rank 3 Galois representations with real trace field
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916179592871936 |
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| author | Krishnamoorthy, Raju Lam, Yeuk Hay Joshua |
| author_facet | Krishnamoorthy, Raju Lam, Yeuk Hay Joshua |
| contents | Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$ extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking $U$, there exists an abelian scheme $f\colon B_U\rightarrow U$ such that $L$ is a summand of $R^2f_*\overline{\mathbb Q}_{\ell}(1)$.
The key ingredients are: (1) the totally real assumption implies $L$ admits a square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over $U_{\bar K}$ geometrically realizing $L$; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18138 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing abelian varieties from rank 3 Galois representations with real trace field Krishnamoorthy, Raju Lam, Yeuk Hay Joshua Algebraic Geometry Number Theory Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$ extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking $U$, there exists an abelian scheme $f\colon B_U\rightarrow U$ such that $L$ is a summand of $R^2f_*\overline{\mathbb Q}_{\ell}(1)$. The key ingredients are: (1) the totally real assumption implies $L$ admits a square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over $U_{\bar K}$ geometrically realizing $L$; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration. |
| title | Constructing abelian varieties from rank 3 Galois representations with real trace field |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2403.18138 |