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On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring


Authors: King Cheong Fung and Ben Kane
Journal: Math. Comp. 87 (2018), 501-514
MSC (2010): Primary 11E20, 11E45, 11F37, 11G05, 16H05, 68W40
DOI: https://doi.org/10.1090/mcom/3206
Published electronically: May 1, 2017
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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
Email: bkane@maths.hku.hk

DOI: https://doi.org/10.1090/mcom/3206
Keywords: Sign changes of cusp forms, supersingular elliptic curves, quaternion algebras, theta series, ternary quadratic forms, halting of algorithms
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.
Article copyright: © Copyright 2017 American Mathematical Society