$\ell$-adic local systems and Higgs bundles: the generic case
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909262488272896 |
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| author | Yu, Hongjie |
| author_facet | Yu, Hongjie |
| contents | Let $X$ be a projective smooth geometrically connected curve defined over a finite field $\mathbb{F}_q$ of cardinality $q$. Let $S$ be a finite set of closed points of $X$. Let $\bar{X}$ and $\bar{S}$ be the base change of $X$, $S$ to an algebraic closure. We consider the set of $\ell$-adic ($\ell\nmid q$) local systems of rank $n$ over $\bar{X}-\bar{S}$ with prescribed tame regular semisimple and generic ramifications in $\bar{S}$. The genericity ensures that such an $\ell$-adic local system is automatically irreducible. We show that the number of these $\ell$-adic local systems fixed by Frobenius endomorphism equals the number of stable logarithmic Higgs bundles of rank $n$ and degree $e$ coprime to $n$, with a fixed residue, up to a power of $q$. In the split case, this number is equal to the number of stable parabolic Higgs bundles (with full flag structures) fixed by $\mathbb{G}_m$-action with generic parabolic weights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_06637 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $\ell$-adic local systems and Higgs bundles: the generic case Yu, Hongjie Algebraic Geometry Number Theory Let $X$ be a projective smooth geometrically connected curve defined over a finite field $\mathbb{F}_q$ of cardinality $q$. Let $S$ be a finite set of closed points of $X$. Let $\bar{X}$ and $\bar{S}$ be the base change of $X$, $S$ to an algebraic closure. We consider the set of $\ell$-adic ($\ell\nmid q$) local systems of rank $n$ over $\bar{X}-\bar{S}$ with prescribed tame regular semisimple and generic ramifications in $\bar{S}$. The genericity ensures that such an $\ell$-adic local system is automatically irreducible. We show that the number of these $\ell$-adic local systems fixed by Frobenius endomorphism equals the number of stable logarithmic Higgs bundles of rank $n$ and degree $e$ coprime to $n$, with a fixed residue, up to a power of $q$. In the split case, this number is equal to the number of stable parabolic Higgs bundles (with full flag structures) fixed by $\mathbb{G}_m$-action with generic parabolic weights. |
| title | $\ell$-adic local systems and Higgs bundles: the generic case |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2304.06637 |