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Article: On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring
Title | 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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Authors | |
Keywords | Halting of algorithms Quaternion algebras Sign changes of cusp forms Supersingular elliptic curves Ternary quadratic forms |
Issue Date | 2018 |
Publisher | American Mathematical Society. |
Citation | Mathematics of Computation, 2018, v. 87 n. 309, p. 501-514 How to Cite? |
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 nth (Fourier) coefficient of one was always larger than or equal to the nth coefficient of the other, then the maximal orders are actually isomorphic. We prove that this is the case. © 2017 American Mathematical Society. |
Persistent Identifier | http://hdl.handle.net/10722/231990 |
ISSN | 2023 Impact Factor: 2.2 2023 SCImago Journal Rankings: 1.460 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | FUNG, KC | - |
dc.contributor.author | Kane, B | - |
dc.date.accessioned | 2016-09-20T05:26:50Z | - |
dc.date.available | 2016-09-20T05:26:50Z | - |
dc.date.issued | 2018 | - |
dc.identifier.citation | Mathematics of Computation, 2018, v. 87 n. 309, p. 501-514 | - |
dc.identifier.issn | 0025-5718 | - |
dc.identifier.uri | http://hdl.handle.net/10722/231990 | - |
dc.description.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 nth (Fourier) coefficient of one was always larger than or equal to the nth coefficient of the other, then the maximal orders are actually isomorphic. We prove that this is the case. © 2017 American Mathematical Society. | - |
dc.language | eng | - |
dc.publisher | American Mathematical Society. | - |
dc.relation.ispartof | Mathematics of Computation | - |
dc.rights | First published in [Mathematics of Computation] in [2018, v. 87 n. 309], published by the American Mathematical Society | - |
dc.subject | Halting of algorithms | - |
dc.subject | Quaternion algebras | - |
dc.subject | Sign changes of cusp forms | - |
dc.subject | Supersingular elliptic curves | - |
dc.subject | Ternary quadratic forms | - |
dc.title | On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring | - |
dc.type | Article | - |
dc.identifier.email | Kane, B: bkane@hku.hk | - |
dc.identifier.authority | Kane, B=rp01820 | - |
dc.description.nature | postprint | - |
dc.identifier.doi | 10.1090/mcom/3206 | - |
dc.identifier.scopus | eid_2-s2.0-85038967128 | - |
dc.identifier.hkuros | 263624 | - |
dc.identifier.volume | 87 | - |
dc.identifier.issue | 309 | - |
dc.identifier.spage | 501 | - |
dc.identifier.epage | 514 | - |
dc.identifier.isi | WOS:000413773100017 | - |
dc.publisher.place | United States | - |
dc.identifier.issnl | 0025-5718 | - |