Non-isogenous superelliptic jacobians II
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909359146008576 |
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| author | Zarhin, Yuri G. |
| author_facet | Zarhin, Yuri G. |
| contents | Let $\ell$ be an odd prime and $K$ a field of characteristic different from $\ell$. Let $\bar{K}$ be an algebraic closure of $K$. Assume that $K$ contains a primitive $\ell$th root of unity. Let $n \ne \ell$ be another odd prime. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots.
Let us consider superelliptic curves $C_{f,\ell}: y^{\ell}=f(x)$ and $C_{h,\ell}: y^{\ell}=h(x)$ of genus $(n-1)(\ell-1)/2$, and their jacobians $J^{(f,\ell)}$ and $J^{(h,\ell)}$, which are $(n-1)(\ell-1)/2$-dimensional abelian varieties over $\bar{K}$.
Suppose that one of the polynomials is irreducible and the other reducible over $K$. We prove that if $J^{(f,\ell)}$ and $J^{(h,\ell)}$ are isogenous over $\bar{K}$ then both endomorphism algebras $\mathrm{End}^{0}(J^{(f,\ell)})$ and $\mathrm{End}^{0}(J^{(h,\ell)})$ contain an invertible element of multiplicative order $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01365 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-isogenous superelliptic jacobians II Zarhin, Yuri G. Number Theory Algebraic Geometry 14H40, 14K05, 11G30, 11G10 Let $\ell$ be an odd prime and $K$ a field of characteristic different from $\ell$. Let $\bar{K}$ be an algebraic closure of $K$. Assume that $K$ contains a primitive $\ell$th root of unity. Let $n \ne \ell$ be another odd prime. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider superelliptic curves $C_{f,\ell}: y^{\ell}=f(x)$ and $C_{h,\ell}: y^{\ell}=h(x)$ of genus $(n-1)(\ell-1)/2$, and their jacobians $J^{(f,\ell)}$ and $J^{(h,\ell)}$, which are $(n-1)(\ell-1)/2$-dimensional abelian varieties over $\bar{K}$. Suppose that one of the polynomials is irreducible and the other reducible over $K$. We prove that if $J^{(f,\ell)}$ and $J^{(h,\ell)}$ are isogenous over $\bar{K}$ then both endomorphism algebras $\mathrm{End}^{0}(J^{(f,\ell)})$ and $\mathrm{End}^{0}(J^{(h,\ell)})$ contain an invertible element of multiplicative order $n$. |
| title | Non-isogenous superelliptic jacobians II |
| topic | Number Theory Algebraic Geometry 14H40, 14K05, 11G30, 11G10 |
| url | https://arxiv.org/abs/2401.01365 |