Note on a polynomial of Emma Lehmer

Darmon, Henri

doi:10.1090/S0025-5718-1991-1068821-5

Note on a polynomial of Emma Lehmer
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by Henri Darmon PDF

Math. Comp. 56 (1991), 795-800 Request permission

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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