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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Note on a polynomial of Emma Lehmer


Author: Henri Darmon
Journal: Math. Comp. 56 (1991), 795-800
MSC: Primary 11R20; Secondary 11G16
MathSciNet review: 1068821
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Abstract: Recently, Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor $ p = {t^4} + 5{t^3} + 15{t^2} + 25t + 25$ as translates of Gaussian periods. Later, Schoof and Washington showed that these units were fundamental units. In this note, we observe that Lehmer's family comes from the covering of modular curves $ {X_1}(25) \to {X_0}(25)$. This gives a conceptual explanation for the existence of Lehmer's units: they are modular units (which have been studied extensively). By relating Lehmer's construction with ours, one finds expressions for certain Gauss sums as values of modular units on $ {X_1}(25)$.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1991-1068821-5
PII: S 0025-5718(1991)1068821-5
Article copyright: © Copyright 1991 American Mathematical Society