Finite field elements of high order arising from modular curves

In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate...

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Bibliographic Details
Published inarXiv.org
Main Authors Burkhart, Jessica F, Calkin, Neil J, Gao, Shuhong, Hyde-Volpe, Justine C, James, Kevin, Maharaj, Hiren, Manber, Shelly, Ruiz, Jared, Smith, Ethan
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.10.2012
Subjects
Fields (mathematics)
Mathematics - Number Theory
Modular construction
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Summary:In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result.
ISSN:2331-8422
DOI:10.48550/arxiv.1210.3504