On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring
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- by King Cheong Fung and Ben Kane PDF
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Abstract:
Chevyrev and Galbraith recently devised an algorithm which inputs a maximal order of the quaternion algebra ramified at one prime and infinity and constructs a supersingular elliptic curve whose endomorphism ring is precisely this maximal order. They proved that their algorithm is correct whenever it halts, but did not show that it always terminates. They did however prove that the algorithm halts under a reasonable assumption which they conjectured to be true. It is the purpose of this paper to verify their conjecture and in turn prove that their algorithm always halts.
More precisely, Chevyrev and Galbraith investigated the theta series associated with the norm maps from primitive elements of two maximal orders. They conjectured that if one of these theta series “dominated” the other in the sense that the $n$th (Fourier) coefficient of one was always larger than or equal to the $n$th coefficient of the other, then the maximal orders are actually isomorphic. We prove that this is the case.
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Additional Information
- King Cheong Fung
- Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong
- Email: mrkcfung@hku.hk
- Ben Kane
- Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong
- MR Author ID: 789505
- Email: bkane@maths.hku.hk
- Received by editor(s): November 3, 2015
- Received by editor(s) in revised form: July 21, 2016, and August 15, 2016
- Published electronically: May 1, 2017
- Additional Notes: The research of the second author was supported by grant project numbers 27300314 and 17302515 of the Research Grants Council.
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 501-514
- MSC (2010): Primary 11E20, 11E45, 11F37, 11G05, 16H05, 68W40
- DOI: https://doi.org/10.1090/mcom/3206
- MathSciNet review: 3716203