Note on a Polynomial of Emma Lehmer
نویسنده
چکیده
Recently, Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor p = t + 5t + I5t + 25/ + 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",(25) —► ,Y0(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 -Y](25). 1. Lehmer's polynomial Throughout the discussion, we fix a choice {(„} of primitive «th roots of unity for each n , say by Çn = e2ni/n . Let P5(Y, T) = r5 + 72r4-2(73 + 372 + 57 + 5)73 (1) +(74 + 573+1172 + 157 + 5)y2 + (73 + 472 +107+10)7+1 be the quintic polynomial constructed in [5]. The discriminant of P5(Y, 7), viewed as a polynomial in Y with coefficients in Q(7), is D(T) = (73 + 572 + 107 + 7)2(74 + 573 + 1572 + 257 + 25)4 . The projective curve C in P2 defined by the affine equation (1) has three nodal singularities whose 7-coordinates are the roots of the first factor of D(T). The points (y, t), where t is a root of the second factor, are branch points for the covering of C onto the 7-line. As shown in [5], the polynomial P5(Y, 7) defines a regular Galois extension of Q(7) with Galois group Z/5Z. By the analysis above, it is ramified at the four conjugate points 7 = -\/5i5, V5ÇS, -VSÇJ , V5ÇJ , the zeros of the Received September 18, 1989; revised February 12, 1990, March 22, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 11F11, 11R20, 11R32, 11Y40, 12F10. Partially supported by a Doctoral Fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC). ©1991 American Mathematical Society 0025-5718/91 $1.00+ $.25 per page
منابع مشابه
Note on a polynomial of Emma Lehmer Henri Darmon September 9 , 2007
In [Leh], Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor p = t + 5t + 15t + 25t+ 25, as translates of Gaussian periods. Later, Schoof and Washington [SW] showed that these units were fundamental units. In this note, we observe that Lehmer’s family comes from the covering of modular curves X1(25) −→ X0(25). This gives a conceptual explanation for t...
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