Superelliptic jacobians and central simple representations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909206396796928 |
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| author | Zarhin, Yuri G. |
| author_facet | Zarhin, Yuri G. |
| contents | Let f(x) be a polynomial of degree at least 5 with complex coefficients and without repeated roots. Let p be an odd prime. Suppose that all the coefficients of f(x) lie in a subfield K such that:
1) K contains a primitive p-th root of unity;
2) f(x) is irreducible over K;
3) the Galois group \Gal(f) of f(x) acts doubly transitively on the set of roots of f(x);
4) the index of every maximal subgroup of Gal(f) does not divide deg(f)-1.
Then the endomorphism ring of the Jacobian of the superelliptic curve y^p=f(x) is isomorphic to the pth cyclotomic ring for all primes p>deg(f). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_12022 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Superelliptic jacobians and central simple representations Zarhin, Yuri G. Number Theory Algebraic Geometry 14H40, 14K05, 11G30 Let f(x) be a polynomial of degree at least 5 with complex coefficients and without repeated roots. Let p be an odd prime. Suppose that all the coefficients of f(x) lie in a subfield K such that: 1) K contains a primitive p-th root of unity; 2) f(x) is irreducible over K; 3) the Galois group \Gal(f) of f(x) acts doubly transitively on the set of roots of f(x); 4) the index of every maximal subgroup of Gal(f) does not divide deg(f)-1. Then the endomorphism ring of the Jacobian of the superelliptic curve y^p=f(x) is isomorphic to the pth cyclotomic ring for all primes p>deg(f). |
| title | Superelliptic jacobians and central simple representations |
| topic | Number Theory Algebraic Geometry 14H40, 14K05, 11G30 |
| url | https://arxiv.org/abs/2305.12022 |