Geometric local systems on the projective line minus four points
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912458985177088 |
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| author | Lam, Yeuk Hay Joshua Litt, Daniel |
| author_facet | Lam, Yeuk Hay Joshua Litt, Daniel |
| contents | Let $J(m)$ be an $m\times m$ Jordan block with eigenvalue $1$. For $λ\in \mathbb{C}\setminus\{0,1\}$, we explicitly construct all rank $2$ local systems of geometric origin on $\mathbb{P}^1\setminus\{0,1,λ, \infty\}$, with local monodromy conjugate to $J(2)$ at $0,1,λ$ and conjugate to $-J(2)$ at $\infty$. The construction relies on Katz's middle convolution operation. We use our construction to prove two conjectures of Sun-Yang-Zuo (one of which was proven earlier by Lin-Sheng-Wang; the other was proven independently from us by Yang-Zuo). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11314 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometric local systems on the projective line minus four points Lam, Yeuk Hay Joshua Litt, Daniel Algebraic Geometry Number Theory Let $J(m)$ be an $m\times m$ Jordan block with eigenvalue $1$. For $λ\in \mathbb{C}\setminus\{0,1\}$, we explicitly construct all rank $2$ local systems of geometric origin on $\mathbb{P}^1\setminus\{0,1,λ, \infty\}$, with local monodromy conjugate to $J(2)$ at $0,1,λ$ and conjugate to $-J(2)$ at $\infty$. The construction relies on Katz's middle convolution operation. We use our construction to prove two conjectures of Sun-Yang-Zuo (one of which was proven earlier by Lin-Sheng-Wang; the other was proven independently from us by Yang-Zuo). |
| title | Geometric local systems on the projective line minus four points |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2305.11314 |