Average Analytic Ranks of Elliptic Curves over Number Fields
We give a conditional bound 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, we show that the average...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001270/type/journal_article |
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| author | Tristan Phillips |
| author_facet | Tristan Phillips |
| author_sort | Tristan Phillips |
| collection | DOAJ |
| description | We give a conditional bound 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, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by
$(9\deg (K)+1)/2$
, 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 | Article |
| id | doaj-art-f9296afe00644e47afa5168b18cdaefe |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-f9296afe00644e47afa5168b18cdaefe2025-02-11T06:21:42ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.127Average Analytic Ranks of Elliptic Curves over Number FieldsTristan Phillips0https://orcid.org/0000-0001-6503-3039Departement of Mathematics, Dartmouth College, 29 North Main Street, Hanover, NH, 03755-3551, United StatesWe give a conditional bound 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, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$ , 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.https://www.cambridge.org/core/product/identifier/S2050509424001270/type/journal_article11G0511G0711G3511G5011D4514G05 |
| spellingShingle | Tristan Phillips Average Analytic Ranks of Elliptic Curves over Number Fields Forum of Mathematics, Sigma 11G05 11G07 11G35 11G50 11D45 14G05 |
| title | Average Analytic Ranks of Elliptic Curves over Number Fields |
| title_full | Average Analytic Ranks of Elliptic Curves over Number Fields |
| title_fullStr | Average Analytic Ranks of Elliptic Curves over Number Fields |
| title_full_unstemmed | Average Analytic Ranks of Elliptic Curves over Number Fields |
| title_short | Average Analytic Ranks of Elliptic Curves over Number Fields |
| title_sort | average analytic ranks of elliptic curves over number fields |
| topic | 11G05 11G07 11G35 11G50 11D45 14G05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001270/type/journal_article |
| work_keys_str_mv | AT tristanphillips averageanalyticranksofellipticcurvesovernumberfields |