Fields of rationality of cusp forms
Abstract In this paper, we prove that for any totally real field F, weight k, and nebentypus character χ, the proportion of Hilbert cusp forms over F of weight k and character χ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmativ...
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| Format: | Article |
| Language: | English |
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The Hebrew University Magnes Press
2018
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| Online Access: | http://hdl.handle.net/1721.1/114641 |
| _version_ | 1811085688193417216 |
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| author | Binder, John |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Binder, John |
| author_sort | Binder, John |
| collection | MIT |
| description | Abstract
In this paper, we prove that for any totally real field F, weight k, and nebentypus character χ, the proportion of Hilbert cusp forms over F of weight k and character χ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmative, a question of Serre. The proof has three main inputs: first, a lower bound on fields of rationality for admissible GL2 representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for GL2; and third, a Plancherel equidistribution theorem for cusp forms with fixed central character. The equidistribution theorem is the key intermediate result and builds on earlier work of Shin and Shin–Templier and mirrors work of Finis–Lapid–Mueller by introducing an explicit bound for certain families of orbital integrals. |
| first_indexed | 2024-09-23T13:13:23Z |
| format | Article |
| id | mit-1721.1/114641 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:13:23Z |
| publishDate | 2018 |
| publisher | The Hebrew University Magnes Press |
| record_format | dspace |
| spelling | mit-1721.1/1146412024-06-27T19:12:32Z Fields of rationality of cusp forms Binder, John Massachusetts Institute of Technology. Department of Mathematics Abstract In this paper, we prove that for any totally real field F, weight k, and nebentypus character χ, the proportion of Hilbert cusp forms over F of weight k and character χ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmative, a question of Serre. The proof has three main inputs: first, a lower bound on fields of rationality for admissible GL2 representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for GL2; and third, a Plancherel equidistribution theorem for cusp forms with fixed central character. The equidistribution theorem is the key intermediate result and builds on earlier work of Shin and Shin–Templier and mirrors work of Finis–Lapid–Mueller by introducing an explicit bound for certain families of orbital integrals. 2018-04-09T19:48:27Z 2018-04-09T19:48:27Z 2017-11 2018-02-14T04:56:47Z Article http://purl.org/eprint/type/JournalArticle 0021-2172 1565-8511 http://hdl.handle.net/1721.1/114641 Binder, John. “Fields of Rationality of Cusp Forms.” Israel Journal of Mathematics 222, no. 2 (October 2017): 973–1028. doi:10.1007/s11856-017-1610-z. en http://dx.doi.org/10.1007/s11856-017-1610-z Israel Journal of Mathematics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Hebrew University of Jerusalem text/xml The Hebrew University Magnes Press Springer |
| spellingShingle | Binder, John Fields of rationality of cusp forms |
| title | Fields of rationality of cusp forms |
| title_full | Fields of rationality of cusp forms |
| title_fullStr | Fields of rationality of cusp forms |
| title_full_unstemmed | Fields of rationality of cusp forms |
| title_short | Fields of rationality of cusp forms |
| title_sort | fields of rationality of cusp forms |
| url | http://hdl.handle.net/1721.1/114641 |
| work_keys_str_mv | AT binderjohn fieldsofrationalityofcuspforms |