Note on a polynomial of Emma Lehmer

Henri Darmon

May 23, 2000

1 Abstract

In [Leh], Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor p = t^{4} + 5t^{3} + 15t^{2} + 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 X_{1}(25) ⟶ X_{0}(25). This gives a conceptual explanation for the existence of Lehmer’s units: they are modular units (which have been studied extensively, for example in [K-L]). By relating Lehmer’s construction with ours, one finds expressions for certain Gauss sums as values of modular units on X_{1}(25).

2 Lehmer’s polynomial

Throughout the discussion, we fix a choice {ζ_{n}} of primitive nth roots of unity for each n, say by ζ_{n} = e^{2πi/n}.

Let

P_{5}(Y, T) = Y^{5} + T^{2}Y^{4} − 2(T^{3} + 3T^{2} + 5T + 5)Y^{3} +
(T^{4} + 5T^{3} + 11T^{2} + 15T + 5)Y^{2} +  (1)
(T^{3} + 4T^{2} + 10T + 10)Y + 1

1Partially supported by a Doctoral Fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC).

2I wish to thank Dan Abramovich and Noam Elkies for some interesting discussions.

be the quintic polynomial constructed in [Leh]. The discriminant of P_{5}(Y, T), viewed as a polynomial in Y with coefficients in Q(T), is

D(T) = (T^{3} + 5T^{2} + 10T + 7)^{2}(T^{4} + 5T^{3} + 15T^{2} + 25T + 25)^{4}.

The projective curve C in P_{2} defined by the affine equation (1) has three nodal singularities whose T -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 T -line.

As shown in [Leh] the polynomial P_{5}(Y, T) defines a regular Galois extension of Q(T) with Galois group Z/5Z. By the analysis above, it is ramified at the four conjugate points T = −\sqrt{5}ζ_{5}, \sqrt{5}ζ_{5}^{2}, −\sqrt{5}ζ_{5}^{−1}, \sqrt{5}ζ_{5}^{−2}, satisfying the minimal polynomial

T^{4} + 5T^{3} + 15T^{2} + 25T + 25.

(Here \sqrt{5} denotes the positive square root.) If t ∈ Z is chosen so that

p = t^{4} + 5t^{3} + 15t^{2} + 25t + 25

is prime, (hence in particular p ≡ 1 mod 5), then the roots r_{1}, …, r_{5} of P_{5}(Y, t) are translates of Gaussian periods:

r_{i} = (\frac{t}{5})η_{i} + [(\frac{t}{5}) − t^{2}]/5,

where

η_{j} = ∑_{x∈Γ_{j}} ζ_{p}^{x},

and Γ_{j} denotes the jth coset of (Z/pZ)^{∗5} in (Z/pZ)^{∗}.

Since C admits a five-to-one map to P_{1} which is totally ramified at four points, the geometric genus of C is 4 by the Riemann-Hurwitz theorem. On the other hand, C is realized as a plane curve of degee d = 6, and its arithmetic genus is (d − 1)(d − 2)/2 = 10. Let C′ denote the normalization of C; it is a smooth projective curve of genus 4. The covering C′ ⟶ P_{1} defines a Galois covering of P_{1} with Galois group Z/5Z, and has the following properties:

1. It is ramified only over the four closed points in R = \{−\sqrt{5}ζ_{5}, \sqrt{5}ζ_{5}^{2}, −\sqrt{5}ζ_{5}^{−1}, \sqrt{5}ζ_{5}^{−2}\}.

2. The closed points of the fiber above ∞ ∈ P_{1} are rational.

Proposition 2.1 The properties 1 and 2 determine the covering C′ uniquely up to Q-isomorphism. Proof: Let (P_{1} − R) be the projective line with the points of R removed, viewed as a curve over Q̄. The space V = H_{et}^{1}(P_{1}−R, Z/5Z) is a vector space of dimension 3 over F_{5}, and is endowed with a natural action of Gal(Q̄/Q). In fact, one has

V = H_{et}^{1}(P_{1} − R, μ_{5}) ⊗ μ_{5}^{−1},

where μ_{5} denotes the group scheme of 5th roots of unity. By Kummer theory, H_{et}^{1}(P_{1} − R, μ_{5}) is identified with the subspace of Q̄(T)^{∗}/Q̄(T)^{∗5} spanned by the elements

\begin{matrix}(T + ζ_{5}\sqrt{5})/(T − ζ_{5}^{2}\sqrt{5}), & (T − ζ_{5}^{2}\sqrt{5})/(T + ζ_{5}^{−1}\sqrt{5}), \\ (T + ζ_{5}^{−1}\sqrt{5})/(T − ζ_{5}^{−2}\sqrt{5}), & (T − ζ_{5}^{−2}\sqrt{5})/(T + ζ_{5}\sqrt{5}), \end{matrix}

whose product is 1. Hence the action of Gal(Q̄/Q) on H_{et}^{1}(P_{1}−R, μ_{5}) factors through Gal(Q(ζ_{5})/Q), and is isomorphic to the regular representation of Gal(Q(ζ_{5})/Q) minus the trivial representation. It follows that V decomposes as a direct sum of three irreducible one-dimensional Galois representations,

V = V_{0} ⊕ V^{ω} ⊕ V^{ω^{2}},

where V_{0} is the trivial representation, and V^{ω}, V^{ω^{2}} denote one dimensional spaces on which Gal(Q(ζ_{5})/Q) acts via the Teichmuller character ω and the square of the Teichmuller character ω^{2} respectively. In particular, V_{0} is the unique one-dimensional subspace of V which is fixed by Gal(Q̄/Q). But the cyclic quintic coverings of P_{1} which are Galois over Q and unramified outside R correspond exactly to such subspaces. Hence the property 1 determines C′ uniquely as a curve over Q̄. (Alternately, one could use the “rigidity criterion” of Matzat, cf. [M, p. 368].) It is not hard to see that there is a unique rational form of the covering C′ such that the closed points above ∞ ∈ P_{1} are all rational (twisting this rational form by a cocycle c in H^{1}(Q, Aut (C′/P_{1})) will cause these points to be defined over the larger extension “cut out” by c). Thus, property 2 determines C′ ⟶ P_{1} up to Q-isomorphism.

3 A modular covering interpretation of Lehmer’s quintic

We assume in this section some basic facts about modular forms and the geometry of modular curves. A good reference for this material is [Ogg].

Let X_{0}(25) and X_{1}(25) denote the modular curves of level 25, compactified by adjoining a finite set of cusps. The curve X_{0}(25) is of genus 0 and is isomorphic to P_{1} over Q. The covering X_{1}(25) ⟶ X_{0}(25) is Galois with Galois group canonically isomorphic to G = (Z/25Z)^{∗}/ < ±1 >. The quotient X of X_{1}(25) by the involution 7 ∈ G gives a cyclic covering of X_{0}(25) of degree 5.

Let T_{5} = η(z)/η(25z), F_{5} = (η(z)/η(5z))^{6} be Hauptmoduls for X_{0}(25) and X_{0}(5) respectively. One has:

F_{5} = T_{5}^{5}/(T_{5}^{4} + 5T_{5}^{3} + 15T_{5}^{2} + 25T_{5} + 25).

The curve X_{0}(5) has two cusps C_{1} and C_{2} corresponding to the values F_{5} = 0 and F_{5} = ∞ respectively. Hence X_{0}(25) has six cusps: a unique one lying above C_{1}, corresponding to T_{5} = 0; and five cusps above C_{2}, given by T_{5} = ∞, −\sqrt{5}ζ_{5}, \sqrt{5}ζ_{5}^{2}, −\sqrt{5}ζ_{5}^{−1}, \sqrt{5}ζ_{5}^{−2} (cf. [K]). The covering X ⟶ X_{0}(25) is ramified at the four non-rational cusps, and the fiber above the cusp T_{5} = ∞ is composed of rational points (cf. [K, p. 226]). By proposition 2.1, X can be described by Lehmer’s quintic; the roots r_{1}, …, r_{5} of P_{5}(Y, T_{5}) are modular functions on X_{1}(25) (in fact, on X) with divisor supported at the P_{i}, where P_{1}, …, P_{5} are the closed points of X which lie above the cusp T_{5} = ∞ of X_{0}(25). By using Hensel’s lemma to solve explicitly the equation P_{5}(Y, T_{5}) = 0, one obtains the following q-expansions for the r_{i}:

r_{1} = −q^{3} + q^{4} + q^{10} − q^{11} − q^{12} + q^{13} − q^{15} + q^{17} + ⋯
r_{2} = q^{−1} + 1 + q^{6} + q^{7} − q^{10} − q^{11} + ⋯
r_{3} = −q − q^{3} + q^{4} + q^{6} − q^{12} − q^{14} + q^{18} + q^{20} ⋯  (2)
r_{4} = −q^{−2} − q − q^{2} − q^{5} + q^{15} + q^{17} + q^{18} ⋯
r_{5} = q^{−1} + q^{5} + q^{7} − q^{8} − q^{12} + q^{13} − q^{14} + ⋯

By [SW, p. 548], the transformation

r ↦ \frac{(T_{5} + 2) + T_{5}r − r^{2}}{1 + (T_{5} + 2)r}

permutes the roots of P (Y, T_{5}) cyclically; one can thus label the r_{i} in such a way that a generator of Gal(X/X_{0}(25)) ≃ Z/5Z sends r_{i} to r_{i+1}, where the subscripts are taken modulo 5. The five cusps of X lying above the cusp T_{5} = ∞ are permuted cyclically by the Galois group of X over X_{0}(25). By considering the q-expansions above, we may fix a labelling of the cusps P_{1}, …, P_{5} so that a generator of Gal(X/X_{0}(25)) sends P_{i} to P_{i+1} and such that:

Divisor(r_{1}) = 3P_{1} − P_{2} + P_{3} − 2P_{4} − P_{5}.

Now, let a belong to Z/25Z, and define

℘_{a}(τ) = ℘(a/25; τ),

where

℘(z; τ) = \frac{1}{z^{2}} + ∑_{(m,n)∈Z^{2}−0} \frac{1}{(z − n − mτ)^{2}} − \frac{1}{(n + mτ)^{2}}

is the Weierstrass ℘-function. It is well-known that the functions

℘_{a,b}(τ) = ℘_{a}(τ) − ℘_{b}(τ)

are modular units on X_{1}(25). The divisors of these functions are computed in [K]. In particular, we find that

Divisor (\frac{℘_{7,9}℘_{6,3}℘_{1,12}℘_{8,4}}{℘_{1,3}℘_{7,4}℘_{6,7}℘_{8,1}} ) = 3P_{1} − P_{2} + P_{3} − 2P_{4} − P_{5},

where the P_{i} denote the cusps on X which are above the cusp ∞ of X_{0}(25). By expressing the function on the left in terms of so-called Klein forms t_{(a_{1},a_{2})} (cf. [K-L]), the above simplifies to give:

Divisor (\frac{t_{(0,1)}t_{(0,7)}}{t_{(0,9)}t_{(0,12)}} ) = 3P_{1} − P_{2} + P_{3} − 2P_{4} − P_{5}.

Let us abbreviate t_{(0,a)} to t_{a}. By comparing divisors and q-expansions, one finds the following infinite product expressions for the r_{i}:

r_{1} = \frac{t_{1}t_{7}}{t_{9}t_{12}} (25τ) = −q^{3} ∏_{n≡±1,±7(25)}(1 − q^{n})/ ∏_{n≡±9,±12(25)}(1 − q^{n})

r_{2} = \frac{t_{2}t_{11}}{t_{1}t_{7}} (25τ) = q^{−1} ∏_{n≡±2,±11(25)}(1 − q^{n})/ ∏_{n≡±1,±7(25)}(1 − q^{n}),
r_{3} = \frac{t_{4}t_{3}}{t_{11}t_{2}} (25τ) = −q ∏_{n≡±4,±3(25)}(1 − q^{n})/ ∏_{n≡±11,±2(25)}(1 − q^{n})
r_{4} = \frac{t_{8}t_{6}}{t_{3}t_{4}} (25τ) = −q^{−2} ∏_{n≡±8,±6(25)}(1 − q^{n})/ ∏_{n≡±3,±4(25)}(1 − q^{n})
r_{5} = \frac{t_{9}t_{12}}{t_{6}t_{8}} (25τ) = q^{−1} ∏_{n≡±9,±12(25)}(1 − q^{n})/ ∏_{n≡±6,±8(25)}(1 − q^{n})

The Galois group Gal(X_{1}(25)/X_{0}(25)) = (Z/25Z)^{∗}/ < ±1 > acts on the t_{a} by multiplying the subscripts (which are viewed as belonging to (Z/25Z)^{∗}/ < ±1 >). Hence to go from r_{i} to r_{i+1}, one applies the Galois automorphism 2 ∈ Gal(X/X_{0}(25)) = (Z/25Z)^{∗}/ < ±1, ±7 >.

4 Gauss sums

Given a prime p ≡ 1( mod 5), let Ψ_{p} : F_{p} ⟶ C^{∗} be the additive character sending 1 to ζ_{p}. We consider the Gauss sum:

g(p) = ∑_{x∈F_{p}} χ(x)Ψ_{p}(x),

where χ is a character of F_{p}^{∗} of order 5. The value of g(p) is independent of χ, up to the action of Gal(Q(ζ_{5})/Q).

By combining Lehmer’s explicit determination of the roots of her polynomial as Gaussian periods, and our identification of these roots with certain modular forms of level 25, we obtain: Theorem 4.1 If η(τ)/η(25τ) = n ∈ Z, and η(5τ)^{6}/(η(τ)η(25τ)^{5}) = p is prime, then:

∏_{i=1}^{4}(η(τ)/η(25τ) − σ_{i}^{−1}(ζ_{5}\sqrt{5}))^{i/5} = (\frac{n}{5})g(p),

where σ_{i} ∈ Gal(Q(ζ_{5})/Q) sends ζ_{5} to ζ_{5}^{i}. There is some ambiguity in the formula, since the value of g(p) depend on the choice of a multiplicative character χ, and the left hand side is really only defined up to a fifth root of 1. We are asserting that there is a way of making these choices so that the formula holds.

Observe that the left hand side is a modular unit (i.e., a unit for the covering X_{1}(25) ⟶ X_{0}(1)). Thus the above expresses Gauss sums as values of certain modular units on X_{1}(25). It seems that the other coverings of lower degree studied by Lehmer yield similar results. It would be interesting to obtain such formulas a priori: this might provide a justification for the fact that translates of Gaussian period polynomials yield cyclic units for extensions of small degree.

Note: The idea of studying families of units in cyclic extensions of Q arising from the modular covering X_{1}(N) ⟶ X_{0}(N) has been explored by Odile Lecacheux (see, for example, the paper [Le1] which studies units in sextic extensions which arise from the modular covering X_{1}(13) ⟶ X_{0}(13)). Independently of the author, Lecacheux has also observed the connection between Lehmer’s quintic and the modular curve X_{1}(25) [Le2].

References

[K] Daniel S. Kubert. Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976) 193-237.

[K-L] Daniel S. Kubert and Serge Lang. Modular Units. Springer-Verlag, 1981, New York.

[Le1] Odile Lecacheux, Unités d’une famille de corps cycliques réels de degré 6 liés à la courbe modulaire X_{1}(13), Journal of Number Theory 31, 54-63 (1989).

[Le2] Odile Lecacheux, private communication.

[Leh] Emma Lehmer. Connection between Gaussian periods and cyclic units, Math. Comp., vol. 50, No. 182, April 1988, pp. 535-541.

[M] B.H. Matzat, Rationality Criteria for Galois Extensions, in Galois Groups over Q, Poceedings of a Workshop held March 23-27, 1987, MSRI publications, pp. 361-384, 1989.

[Ogg] A. Ogg, Survey of modular functions of one variable, in Modular functions of one variable, Proceedings, Antwerp 1972, SLN 320.

[SW] Rene Schoof and Lawrence C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. vol. 50, no. 182, April 1988, pp. 543-556.

Henri Darmon - Harvard Univ. Math. Dept., 1 Oxford St., Cambridge MA 02138.