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 ${X_1}(25)$.References
- Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193–237. MR 434947, DOI 10.1112/plms/s3-33.2.193
- Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603
- Odile Lecacheux, Unités d’une famille de corps cycliques réeles de degré $6$ liés à la courbe modulaire $X_1(13)$, J. Number Theory 31 (1989), no. 1, 54–63 (French, with English summary). MR 978099, DOI 10.1016/0022-314X(89)90051-6 —, private communication.
- Emma Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), no. 182, 535–541. MR 929551, DOI 10.1090/S0025-5718-1988-0929551-0
- B. Heinrich Matzat, Rationality criteria for Galois extensions, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 361–383. MR 1012171, DOI 10.1007/978-1-4613-9649-9_{6}
- Andrew Ogg, Survey of modular functions of one variable, Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Mathematics, Vol. 320, Springer, Berlin, 1973, pp. 1–35. Notes by F. van Oystaeyen. MR 0337785
- Emma Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), no. 182, 535–541. MR 929551, DOI 10.1090/S0025-5718-1988-0929551-0
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 56 (1991), 795-800
- MSC: Primary 11R20; Secondary 11G16
- DOI: https://doi.org/10.1090/S0025-5718-1991-1068821-5
- MathSciNet review: 1068821