Reducibility and rational torsion in modular abelian varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908577303625728 |
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| author | Agashe, Amod Winters, Matthew |
| author_facet | Agashe, Amod Winters, Matthew |
| contents | Let N be a square-free positive integer and let f be a newform of weight 2 on Γ_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has residue characteristic r such that r does not divide 6N. We show that if either A[m] or the canonical representation ρ_m over T/m associated to m is reducible, then r divides the order of the cuspidal subgroup of J_0(N) and A[m] has a nontrivial rational point. We mention some applications of this result, including an application to the second part of the Birch and Swinnerton-Dyer conjecture for A. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reducibility and rational torsion in modular abelian varieties Agashe, Amod Winters, Matthew Number Theory Numerical Analysis Let N be a square-free positive integer and let f be a newform of weight 2 on Γ_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has residue characteristic r such that r does not divide 6N. We show that if either A[m] or the canonical representation ρ_m over T/m associated to m is reducible, then r divides the order of the cuspidal subgroup of J_0(N) and A[m] has a nontrivial rational point. We mention some applications of this result, including an application to the second part of the Birch and Swinnerton-Dyer conjecture for A. |
| title | Reducibility and rational torsion in modular abelian varieties |
| topic | Number Theory Numerical Analysis |
| url | https://arxiv.org/abs/2510.04323 |