Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929682959564800 |
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| author | Shi, Zhan Weng, Lin |
| author_facet | Shi, Zhan Weng, Lin |
| contents | As a continuation of our earlier paper, we offer a new approach to murmurations and Sato-Tate laws for higher rank zetas of elliptic curves. Our approach here does not depend on the Riemann hypothesis for the so-called a-invariant in rank n>2 even for the Sato-Tate law, rather, on a much refined structure, a similar version of which was already observed by Zagier and the senior author when the rank n Riemann hypothesis was established. Namely, instead of the rank n Riemann hypothesis bounds, we use much stronger asymptotic bounds. Accordingly, rank n Sato-Tate law can be established and rank n murmuration can be formulated equally well, similar to the corresponding structures in the abelian framework for Artin zetas of elliptic curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis Shi, Zhan Weng, Lin Number Theory Algebraic Geometry As a continuation of our earlier paper, we offer a new approach to murmurations and Sato-Tate laws for higher rank zetas of elliptic curves. Our approach here does not depend on the Riemann hypothesis for the so-called a-invariant in rank n>2 even for the Sato-Tate law, rather, on a much refined structure, a similar version of which was already observed by Zagier and the senior author when the rank n Riemann hypothesis was established. Namely, instead of the rank n Riemann hypothesis bounds, we use much stronger asymptotic bounds. Accordingly, rank n Sato-Tate law can be established and rank n murmuration can be formulated equally well, similar to the corresponding structures in the abelian framework for Artin zetas of elliptic curves. |
| title | Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2501.10220 |