Average analytic ranks of elliptic curves over number fields
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866915158562963456 |
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| author | Phillips, Tristan |
| author_facet | Phillips, Tristan |
| contents | A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $(9\ \text{deg}(K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_09527 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Average analytic ranks of elliptic curves over number fields Phillips, Tristan Number Theory 11G05 (Primary), 11D45, 11G07, 11G35, 11G40, 11G50, 14G05 (Secondary) A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $(9\ \text{deg}(K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest. |
| title | Average analytic ranks of elliptic curves over number fields |
| topic | Number Theory 11G05 (Primary), 11D45, 11G07, 11G35, 11G40, 11G50, 14G05 (Secondary) |
| url | https://arxiv.org/abs/2205.09527 |