The Algebraic Equation of Type E_{6} and Chebotarev Density

Tetsuji Shioda

1 Introduction

Suppose we have an algebraic equation:

F (X, λ) = X^{n} + a_{1}X^{n−1} + ⋯ + a_{n} = 0  (1)

where a_{1}, …, a_{n} belong to a field k_{0} = Q(λ) which is a purely transcendental extension of Q generated by some independent parameters λ. Let G be the Galois group of this algebraic equation, i.e. that of the splitting field K over k_{0}.

When we specialize the parameters λ to some rational numbers λ′, (1) becomes an algebraic equation over Q. By Hilbert irreducibility theorem, its Galois group over Q is equal to G for most choice of λ′ (see e.g. [4]). Take and fix such a λ′, and let K/Q be the corresponding Galois extension.

According to Chebotarev density theorem, the density of prime numbers p such that the Frobenius transformation ϕ_{p} of K at p belongs to a given conjugacy class C ⊂ G is equal to |C|/|G|. (Here |A| denotes the cardinality of a set A). In particular, the prime numbers p for which ϕ_{p} is the identity (such a p is called completely split) has the density 1/|G|.

Thus it might be of some interest to ask if one can arrange K/Q in the above situation in such a way that there is a small p, compared with the size of |G|, which is completely split.

In this note, we give an example of this sort for the algebraic equation of of degree 27 of type E_{6}, where G = W (E_{6}) is the Weyl group of type E_{6} with order |G| = 51840. This equation arises naturally from the Mordell-Weil lattices of type E_{6}, which can be also interpreted as the algebraic equation of 27 lines on a smooth cubic surface.

A numerical example of Galois extension of Q with Galois group W (E_{6}) has been constructed in our previous paper [6]. In the present note, we construct a new example where a completely split p occurs as early as p = 13. The idea is to combine the construction for (a) big Galois and (b) small Galois discussed in [6, Part II].

2 Algebraic equations of type E_{6}

The polynomial F (X) of degree 27 below is what we called the universal polynomial of type E_{6} in [5]. The coefficients belong to the ring R = Z[p_{0}, p_{1}, p_{2}, q_{0}, q_{1}, q_{2}]. It arises naturally from the Mordell-Weil lattice of the elliptic curve E_{λ} over k(t) defined by the equation:

y^{2} = x^{3} + (p_{0} + p_{1}t + p_{2}t^{2})x + (q_{0} + q_{1}t + q_{2}t^{2} + t^{4})  (2)

where we set λ = (p_{0}, p_{1}, p_{2}, q_{0}, q_{1}, q_{2}), k_{0} = Q(λ) = Q(p_{0}, p_{1}, p_{2}, q_{0}, q_{1}, q_{2}), and k denotes the algebraic closure of k_{0}.

For general λ, the Mordell-Weil lattice E_{λ}(k(t)) is isomorphic to the dual lattice E_{6}^{∗} of the root lattice E_{6}. As a consequence, there are exactly 27 rational points P = (x, y) ∈ E_{λ}(k(t)) of the form:

x = at + b, y = t^{2} + dt + e (a, b, d, e ∈ k).  (3)

The algebraic equation F (X) = 0 is obtained as follows. First, substituting (3) into (2) and comparing the coefficients of t^{i}, we get several relations among a, b, d, e. Then elimination of d, e, b from them yields the desired equation F (a) = 0. In other words, the 27 roots of the polynomial F (X) = F (X, λ) below are precisely the t-coefficient a of the x-coordinate of such P. We know that a is the most important parameter on which other b, d, e depend rationally (cf. also [8]).

By the coordinate change y′ = y − t^{2}, (x : y′ : t : 1) = (X, Y, Z, W), the equation (2) is transformed to the equation of a cubic surface V_{λ}:

Y^{2}W +2Y Z^{2} = X^{3}+(p_{0}W^{2}+p_{1}ZW +p_{2}Z^{2})X+(q_{0}W^{2}+q_{1}ZW +q_{2}Z^{2})W (4)

The 27 lines on V_{λ} are then given by the equations

X = aZ + bW, Y = dZ + eW.  (5)

Therefore the algebraic equation F (X) = 0 becomes that of 27 lines on the cubic surface (4) which contains 6 essential parameters p_{i}, q_{j}. (For a historical account of the algebraic equation of 27 lines on a cubic surface, we refer to [1], [2] and [3].)

F (X) = X^{27} +X^{25}(12p_{2}) +X^{23}(60p_{2}^{2}) +X^{22}(−48p_{1}) +X^{21} (96q_{2} + 168p_{2}^{3}) +X^{20}(−336p_{1}p_{2}) +X^{19} (480p_{0} + 294p_{2}^{4} + 528p_{2}q_{2}) +X^{18} (−1344q_{1} − 1008p_{1}p_{2}^{2}) +X^{17} (144p_{1}^{2} + 2304p_{0}p_{2} + 336p_{2}^{5} + 1152p_{2}^{2}q_{2}) +X^{16} (−1680p_{1}p_{2}^{3} − 5568p_{2}q_{1} − 768p_{1}q_{2}) +X^{15} (17280q_{0} + 608p_{1}^{2}p_{2} + 4768p_{0}p_{2}^{2} + 252p_{2}^{6} + 1200p_{2}^{3}q_{2} − 1248q_{2}^{2}) +X^{14} (2304p_{0}p_{1} − 1680p_{1}p_{2}^{4} − 8832p_{2}^{2}q_{1} − 2688p_{1}p_{2}q_{2}) +X^{13} (976p_{1}^{2}p_{2}^{2} + 5696p_{0}p_{2}^{3} + 120p_{2}^{7} + 43776p_{2}q_{0} + 3264p_{1}q_{1} + 12288p_{0}q_{2}

+480p_{2}^{4}q_{2} − 4800p_{2}q_{2}^{2}) +X^{12} (832p_{1}^{3} + 5888p_{0}p_{1}p_{2} − 1008p_{1}p_{2}^{5} − 6528p_{2}^{3}q_{1} − 3072p_{1}p_{2}^{2}q_{2} + 10752q_{1}q_{2}) +X^{11} (−34560p_{0}^{2} + 704p_{1}^{2}p_{2}^{3} + 4384p_{0}p_{2}^{4} + 33p_{2}^{8} + 41472p_{2}^{2}q_{0} + 2688p_{1}p_{2}q_{1}

+4224p_{1}^{2}q_{2} + 34560p_{0}p_{2}q_{2} − 144p_{2}^{5}q_{2} − 6720p_{2}^{2}q_{2}^{2}) +X^{10} (2560p_{1}^{3}p_{2} + 3584p_{0}p_{1}p_{2}^{2} − 336p_{1}p_{2}^{6} + 64512p_{1}q_{0} − 70656p_{0}q_{1}

−2112p_{2}^{4}q_{1} − 768p_{1}p_{2}^{3}q_{2} + 23040p_{2}q_{1}q_{2} + 8448p_{1}q_{2}^{2}) +X^{9} (−18944p_{0}p_{1}^{2} − 39936p_{0}^{2}p_{2} + 176p_{1}^{2}p_{2}^{4} + 2176p_{0}p_{2}^{5} + 4p_{2}^{9} + 22528p_{2}^{3}q_{0}

−5504p_{1}p_{2}^{2}q_{1} − 40704q_{1}^{2} + 8960p_{1}^{2}p_{2}q_{2} + 32768p_{0}p_{2}^{2}q_{2} − 192p_{2}^{6}q_{2} +110592q_{0}q_{2} − 3840p_{2}^{3}q_{2}^{2} + 5120q_{2}^{3})

+X^{8} (2688p_{1}^{3}p_{2}^{2} − 1536p_{0}p_{1}p_{2}^{3} − 48p_{1}p_{2}^{7} + 82944p_{1}p_{2}q_{0} + 4608p_{1}^{2}q_{1} −64512p_{0}p_{2}q_{1} − 192p_{2}^{5}q_{1} − 73728p_{0}p_{1}q_{2} + 768p_{1}p_{2}^{4}q_{2} + 13824p_{2}^{2}q_{1}q_{2} +16128p_{1}p_{2}q_{2}^{2})

+X^{7} (−2560p_{1}^{4} − 16384p_{0}p_{1}^{2}p_{2} − 20992p_{0}^{2}p_{2}^{2} − 32p_{1}^{2}p_{2}^{5} + 608p_{0}p_{2}^{6} −165888p_{0}q_{0} + 9600p_{2}^{4}q_{0} − 6144p_{1}p_{2}^{3}q_{1} − 38400p_{2}q_{1}^{2} + 5376p_{1}^{2}p_{2}^{2}q_{2} +10752p_{0}p_{2}^{3}q_{2} − 48p_{2}^{7}q_{2} + 156672p_{2}q_{0}q_{2} − 15360p_{1}q_{1}q_{2} − 56832p_{0}q_{2}^{2} −480p_{2}^{4}q_{2}^{2} + 9984p_{2}q_{2}^{3})

+X^{6} (49152p_{0}^{2}p_{1} + 1024p_{1}^{3}p_{2}^{3} − 1792p_{0}p_{1}p_{2}^{4} + 21504p_{1}p_{2}^{2}q_{0} +10240p_{1}^{2}p_{2}q_{1} − 19456p_{0}p_{2}^{2}q_{1} − 110592q_{0}q_{1} − 10240p_{1}^{3}q_{2} −57344p_{0}p_{1}p_{2}q_{2} + 384p_{1}p_{2}^{5}q_{2} + 1536p_{2}^{3}q_{1}q_{2} + 6912p_{1}p_{2}^{2}q_{2}^{2} − 21504q_{1}q_{2}^{2})

+X^{5} (−1536p_{1}^{4}p_{2} − 4608p_{0}p_{1}^{2}p_{2}^{2} − 3072p_{0}^{2}p_{2}^{3} − 16p_{1}^{2}p_{2}^{6} + 64p_{0}p_{2}^{7} +27648p_{1}^{2}q_{0} − 110592p_{0}p_{2}q_{0} + 2304p_{2}^{5}q_{0} − 9216p_{0}p_{1}q_{1} − 1344p_{1}p_{2}^{4}q_{1} −12288p_{2}^{2}q_{1}^{2} + 73728p_{0}^{2}q_{2} + 768p_{1}^{2}p_{2}^{3}q_{2} + 55296p_{2}^{2}q_{0}q_{2} +10752p_{1}p_{2}q_{1}q_{2} − 19200p_{1}^{2}q_{2}^{2} − 46080p_{0}p_{2}q_{2}^{2} + 192p_{2}^{5}q_{2}^{2} + 4608p_{2}^{2}q_{2}^{3})

+X^{4} (8192p_{0}p_{1}^{3} + 16384p_{0}^{2}p_{1}p_{2} + 64p_{1}^{3}p_{2}^{4} − 256p_{0}p_{1}p_{2}^{5} + 3072p_{1}p_{2}^{3}q_{0} −512p_{1}^{2}p_{2}^{2}q_{1} − 1024p_{0}p_{2}^{3}q_{1} − 36864p_{2}q_{0}q_{1} − 39936p_{1}q_{1}^{2} − 4096p_{1}^{3}p_{2}q_{2} −8192p_{0}p_{1}p_{2}^{2}q_{2} + 73728p_{1}q_{0}q_{2} + 24576p_{0}q_{1}q_{2} − 768p_{1}p_{2}^{3}q_{2}^{2} − 3072p_{2}q_{1}q_{2}^{2} −18432p_{1}q_{2}^{3})

+X^{3} (−16384p_{0}^{3} + 256p_{1}^{4}p_{2}^{2} − 1024p_{0}p_{1}^{2}p_{2}^{3} + 256p_{0}^{2}p_{2}^{4} −6144p_{1}^{2}p_{2}q_{0} − 18432p_{0}p_{2}^{2}q_{0} + 256p_{2}^{6}q_{0} − 110592q_{0}^{2} + 15360p_{1}^{3}q_{1} −14336p_{0}p_{1}p_{2}q_{1} − 128p_{1}p_{2}^{5}q_{1} − 2560p_{2}^{3}q_{1}^{2} + 8192p_{0}p_{1}^{2}q_{2} + 24576p_{0}^{2}p_{2}q_{2} +128p_{1}^{2}p_{2}^{4}q_{2} − 256p_{0}p_{2}^{5}q_{2} + 9216p_{2}^{3}q_{0}q_{2} + 2048p_{1}p_{2}^{2}q_{1}q_{2} − 30720q_{1}^{2}q_{2} −2560p_{1}^{2}p_{2}q_{2}^{2} − 7680p_{0}p_{2}^{2}q_{2}^{2} + 55296q_{0}q_{2}^{2} − 256p_{2}^{3}q_{2}^{3} − 6912q_{2}^{4})

+X^{2} (−1024p_{1}^{5} + 4096p_{0}p_{1}^{3}p_{2} − 12288p_{1}p_{2}q_{1}^{2} + 24576p_{1}^{2}q_{1}q_{2}) +X (−4096p_{0}^{2}p_{1}^{2} − 3072p_{1}^{2}p_{2}^{2}q_{0} + 2048p_{1}^{3}p_{2}q_{1} − 1024p_{0}p_{1}p_{2}^{2}q_{1}

−36864p_{1}q_{0}q_{1} + 12288p_{0}q_{1}^{2} − 256p_{2}^{4}q_{1}^{2} − 2048p_{1}^{4}q_{2} + 4096p_{0}p_{1}^{2}p_{2}q_{2} +512p_{1}p_{2}^{3}q_{1}q_{2} − 6144p_{2}q_{1}^{2}q_{2} − 256p_{1}^{2}p_{2}^{2}q_{2}^{2} + 9216p_{1}q_{1}q_{2}^{2})

+1024 (4p_{1}^{3}q_{0} − 4p_{0}p_{1}^{2}q_{1} − p_{1}p_{2}^{2}q_{1}^{2} − 4q_{1}^{3} + 2p_{1}^{2}p_{2}q_{1}q_{2} − p_{1}^{3}q_{2}^{2}) .

Let K_{λ} denote the splitting field of F (X) = F (X, λ) over k_{0} = Q(λ) = Q(p_{i}, q_{j}). The fundamental theorems for the algebraic equation of type E_{6} ([5], Th.10.3, 10.4) state that, for λ generic, K_{λ} is a Galois extension of k_{0} with the Galois group W (E_{6}), the Weyl group of type E_{6}; the latter is a finite reflection group of order 51840 and has a simple subgroup of index 2. Moreover, if P_{1}, …, P_{6} are 6 rational points of the form (3) which are linearly independent in the Mordell-Weil lattice, then the corresponding a_{1}, …, a_{6} are algebraically independent and generate K_{λ} over Q, i.e. K_{λ} = Q(a_{1}, …, a_{6}). Furthermore p_{i}, q_{j} are polynomials of a_{1}, …, a_{6} with rational coefficients which form a set of the fundamental invariants of W (E_{6}) with weights wt(p_{i}) = 8 − 3i, wt(q_{i}) = 12 − 3i(i = 0, 1, 2).

The fundamental theorems have the standard applications (a) and (b) below, in perfect analogy to the familiar case of generic algebraic equation of degree n (corresponding to type A_{n−1}).

(a) big Galois: By specializing the parameters p_{i}, q_{j} to suitable rational numbers, we get a Galois extension of Q with Galois group W (E_{6}). By Hilbert irreducibility theorem, this is the case for most choice of specialization. Moreover every extension of Q with Galois group W (E_{6}) is obtained this way (see [6]).

(b) small Galois: By specializing a_{1}, …, a_{6} to rational numbers in such a way that the Mordell-Weil lattice does not degenerate, we get the trivial extension of Q, which means that the Mordell-Weil group E(Q(t)) has rank 6 (without making any constant field extension) (see [5]).

3 Construction method of examples

It follows from (a) that for most choice of p_{i}, q_{j} ∈ Q, we get a Galois extension of Q with Galois group W (E_{6}). As in [6], there is a criterion to verify this for given p_{i}, q_{j} by considering F (X) mod p for various primes p (see below). But this does not guarantee if the trivial Frobenius transformation ϕ_{p} = 1 occurs for a small p. So we start from (b).

\underline{Step 1} Choosing a_{1}, …, a_{6} ∈ Q as in (b), determine λ′ = (p′_{i}, q′_{j}) ∈ Q^{6} by the explicit formula of the fundamental invariants and let E′ = E_{λ′}. Then E′(Q(t)) ≃ E_{6}^{∗} has rank 6 and has 27 rational points of the form (3) with rational coefficients. The algebraic equation F (X, λ′) = 0 splits into 27 linear factors over Q.

\underline{Step 2} Consider a non-degenerate reduction of E′ modulo a suitable prime p (in the sense of Mordell-Weil lattices). Namely, reducing E′ modulo p, we get an elliptic curve Ē′ = E′_{p} over F_{p}(t) such that the Mordell-Weil group Ē′(F_{p}(t)) has again rank 6. In fact, this is the case for almost all p. Fix one such p.

\underline{Step 3} Next lift Ē′ back to characteristic 0. Namely consider E = E_{λ} with λ = (p_{i}, q_{j}) ∈ Q^{6} such that p_{i} ≡ p′_{i}, q_{j} ≡ q′_{j} mod p. By (a), we can expect that for a suitable choice of λ, the algebraic equation F (X, λ) = 0 over Q has the Galois group W (E_{6}). In that case, the chosen p is clearly a completely split prime in the corresponding extension, since E and E′ have the same reduction Ē′ mod p.

4 Examples

Let us write down some explicit examples by the above method.

Example 1 Take p_{0} = 13, p_{1} = p_{2} = q_{2} = 0, q_{0} = 8, q_{1} = 13. The elliptic curve E over Q(t) is given by

E : y^{2} = x^{3} + 13x + 8 + 13t + t^{4},  (6)

and we have

F (X) = X^{27} + 6240X^{19} − 17472X^{18} + 138240X^{15} − 5840640X^{11}

−11940864X^{10} − 6878976X^{9} − 17252352X^{7} − 11501568X^{6}

−43073536X^{3} + 26996736X − 8998912.  (7)

The splitting field K of this equation F (X) = 0 over Q has the Galois group W (E_{6}), and p = 13 is completely split in K.

This can be verified as follows. As the auxiliary curve E′ in Step 1, we take

E′ : y^{2} = x^{3} + (−59475 − 78t^{2})x + 2848750 + 18226t^{2} + t^{4}  (8)

which has been studied in detail in [8, §8]. The corresponding cubic surface V′ is given by the equation:

Y^{2}W +2Y Z^{2} = X^{3} +X(−59475W^{2} −78Z^{2})+2848750W^{3} +18226Z^{2}W, (9)

and all the 27 lines are defined over Q. This is constructed from the data a_{i} = 2(9 − i) (1 ≤ i ≤ 6) such that the 6 lines L_{i} on V′ corresponding to a_{i} are mutually disjoint. The 27 roots of F (X, λ′) = 0 are as follows:

0^{3}, ±2^{2}, ±4^{2}, ±6^{2}, ±8^{2}, ±10, ±12, ±14, ±16,  (10)

where for instance ±4^{2} means that each of ±4 appears with multiplicity 2. Reducing (8) mod p = 13, we have

Ē′ : y^{2} = x^{3} + 8 + t^{4}  (11)

over F_{p}(t), which is obviously non-degenerate and of type E_{6}.

The equation (6) is a lifting of (11), so p = 13 is completely split (note that F (X) and F (X, λ′) are equal mod p = 13).

It remains to show that F (X) = 0 has the full W (E_{6}) as the Galois group. For this, we look at F_{ℓ} = F (X) mod ℓ for various prime number ℓ. We find that for ℓ = 5 (resp. 51), F_{ℓ} is decomposed into 3 irreducible factors of degree 9 (resp. 5 irreducible factors of degrees 2,5,5,5,10). This information suffices to conclude that the Galois group is equal to W (E_{6}) by [6, Lem.7.5].

Taking the same E′ in Step 1 as above and choosing the prime number p = 17 in Step 2, we have the reduced curve

Ē′ : y^{2} = x^{3} + (8 + 7t^{2})x + 9 + 2t^{2} + t^{4}.  (12)

When we lift it back to characteristic 0, this equation (12) itself viewed over Q(t) is not appropriate for our purpose, since p_{1} = q_{1} = 0 implies F (X) = X^{3} · (X^{24} + ⋯) which is reducible and cannot have the big Galois group W (E_{6}). So modifying a little, we try (13) below, and we obtain a second example:

Example 2 Consider the elliptic curve E over Q(t) given by

E : y^{2} = x^{3} + (8 + 7t^{2})x + 9 + 17t + 2t^{2} + t^{4}.  (13)

Then we have

F (X) = X^{27} + 84X^{25} + 2940X^{23} + 57816X^{21} + 717126X^{19} − 22848X^{18}
+5889072X^{17} − 662592X^{16} + 32490332X^{15} − 7357056X^{14}

+119580040X^{13} − 37699200X^{12} + 288236801X^{11} − 90331200X^{10} +471155868X^{9} − 93242688X^{8} + 618051168X^{7} − 130124800X^{6}

+535579136X^{5} − 82026496X^{4} − 44281088X^{3} − 174084352X
−20123648.  (14)

By the same argument as above, this equation is shown to have the Galois group W (E_{6}), for which p = 17 is a completely split prime.

This example is mentioned for the following reason. Recall that there are 25 conjugacy classes in the group W (E_{6}) (cf.[3], [9]). In Example 2, not only the identity class occurs for a small p, but also 23 (out of 25) conjugacy classes occur as the classes containing the Frobenius transformations ϕ_{p} already for p < 1000. The 2 missing classes are No.3 and 11 in the notation of [3] with density 1/648 or 1/1152. The verification can be done by the same method as in [7, §4].

References

[1] Henderson, A. : The twenty-seven lines upon the cubic surface, Cambridge Univ. Press (1911).

[2] Jordan, C. : Traité des substitutions et des équations algébriques, Gauthier-Villars, Paris (1870, 1957).

[3] Manin, Ju. : Cubic Forms, North-Holland (1974).

[4] Serre, J-P. : Lectures on the Mordell-Weil Theorem. Vieweg (1989).

[5] Shioda, T. : Construction of elliptic curves with high rank via the invariants of the Weyl groups, J. Math. Soc. Japan 43, 673- 719 (1991).

[6] — : Theory of Mordell-Weil lattices, Proc. ICM Kyoto-1990, Springer, vol.I, 473-489 (1991).

[7] — : Some remarks on elliptic curves over function fields, Proc. Journée Arithmetique 91, Genève, Astérisque 209, 99-114 (1992).

[8] — : Weierstrass transformations and cubic surfaces, Preprint 1994.

[9] Swinnerton-Dyer, H.P.F.: The zeta function of a cubic surface over a finite field, Proc.Cambridge Phil.Soc. 63, 55-71 (1967).

Department of Mathematics Rikkyo University Nishi-Ikebukuro, Tokyo 171 Japan