CURVE OF GENUS 2 WITH AN AUTOMORPHISM OF ORDER 3

Abstract. We exhibit curves C of genus 2 with a nontrivial automorphism of order 3 both defined over ℚ having good reduction outside a prime p. The Jacobian of C is isogenous to the product of two isogenous conjugate elliptic curves over ℚ(\sqrt{p}) and is conjecturally isogenous to a factor of J_{1}(p).

The purpose of this brief note is to exhibit certain curves C of genus 2 with a nontrivial automorphism σ of order 3 both defined over ℚ and having good reduction outside a prime p. An example is X_{1}(13), where X_{1}(N) denotes the modular curve parametrizing elliptic curves with an N-torsion point. We will show that J(C), the Jacobian of C, is isogenous over \overline{ℚ} to the product of two elliptic curves E × E′, where E, E′ are defined over ℚ(\sqrt{p}), E′ is the conjugate of E under the nontrivial Galois automorphism of ℚ(\sqrt{p})/ℚ, and E′ is isomorphic over \overline{ℚ} to the quotient of E by a subgroup of order 3. In particular, E is a ℚ-curve (i.e. it is isogenous to all of its Galois conjugates); conjecturally J(C) is a factor of the Jacobian of X_{1}(p). Our construction depends on solving the diophantine equation t^{2} ± 108 = p. The question of how many primes can be so represented is a difficult unsolved problem, though it is expected that there are infinitely many.

Acknowledgments I would like to thank B. Poonen for several helpful conversations.

1. Let us fix an algebraic closure \overline{ℚ} of ℚ. Let F ⊂ \overline{ℚ} be a subfield. It is known (see [Me §2] for example) that any curve of genus 2 over F is the normalization of the affine curve given by a Weierstrass equation

y^{2} = r(x), r ∈ F [x], deg(r) = 6 ,

with r separable. We will say that such an equation determines C. If we homogenize r so that r(x) = f(x, 1) with f ∈ F [x_{1}, x_{2}] homogeneous of degree 6 (a sextic) then curves corresponding to two sextics f, f′ ∈ F [x_{1}, x_{2}] are isomorphic over F if and only if f′(x_{1}, x_{2}) = u^{2}f(ax_{1} + bx_{2}, cx_{1} + dx_{2}) for some a, b, c, d, u ∈ F with u and ad − bc nonzero.

Let (C, σ) be a curve of genus 2 together with a nontrivial automorphism σ of C of order 3 both defined over ℚ. Over \overline{ℚ} it is easy to show that (C, σ) is isomorphic (in a natural sense) to a pair (C_{λ}, τ) for some λ ∈ \overline{ℚ}, where C_{λ} is the curve of genus 2 determined by the Weierstrass equation

C_{λ} : y^{2} = x^{6} + λx^{3} + 1 ,

and τ : (x, y) ↦ (ζx, y) with ζ ∈ k a primitive cube root of 1 (both ζ and ζ^{−1} give isomorphic pairs via (x, y) ↦ (1/x, y/x^{3})). Conversely, any such (C_{λ}, τ) for λ ∈ \overline{ℚ} \ \{±2\} is a curve of genus 2 with a nontrivial automorphism of order 3.

Key words and phrases. Curves of genus 2.

1991 AMS Classification ***

Given a curve C defined over \overline{ℚ} its field of moduli is the field F ⊂ \overline{ℚ} characterized by the property: For every τ ∈ Gal(\overline{ℚ}/ℚ), C^{τ} is isomorphic to C if and only if τ is the identity on F. Clearly isomorphic curves have the same field of moduli. Notice that F is the smallest field over which a curve isomorphic to C could be defined, but it is not in general a field over which it can be defined. In fact, for example, Shimura showed that no generic hyperelliptic curve of even genus has a model over its field of moduli [Sh Thm 3]. See [Me] for a discussion of this issue for curves of genus 2. For elliptic curves the situation is simple: the field of moduli is generated by the j-invariant of the curve and there always is an isomorphic curve defined over it.

Lemma 1. Let C and C′ be two curves of genus 2 determined by Weierstrass equations y^{2} = h(x^{3}), y^{2} = h′(x^{3}) respectively, where h, h′ ∈ \overline{ℚ}[x] have degree 2. Then C and C′ are isomorphic if and only if h_{2}(x) = αh_{1}(βx) for some α, β ∈ \overline{ℚ}^{∗}. In particular, for λ_{j} ∈ \overline{ℚ} \ \{±2\}, j = 1, 2, C_{λ_{1}} is isomorphic to C_{λ_{2}} if and only if λ_{1}^{2} = λ_{2}^{2}.

Proof. It is easy to verify that the element of GL_{2}(\overline{ℚ}) giving the isomorphism must be diagonal. □

Lemma 2. Let λ be in \overline{ℚ} \ \{±2\}. Then the field of moduli of C_{λ} is ℚ(λ^{2}).

Proof. For every τ ∈ Gal(\overline{ℚ}/ℚ) we have that C_{λ}^{τ} = C_{λ^{τ}}. Hence if C_{λ}^{τ} is isomorphic to C_{λ} we must have, with the notation of the previous lemma, α = 1, β^{6} = 1; but then β^{3} = ±1 so λ^{τ} = ±λ and we are done. □

We conclude that the isomorphism class of a curve C of genus 2 defined over \overline{ℚ} with a nontrivial automorphism σ of order 3 is uniquely determined by its associated modular invariant μ(C) := λ^{2} ∈ \overline{ℚ} and that its field of moduli is ℚ(μ(C)). Certainly C can be defined over its field of moduli, since C_{λ} is isomorphic to the curve determined by y^{2} = λ^{6}x^{6} + λ^{4}x^{3} + 1, however this may not be the case for the pair (C, σ). More precisely, we have the following.

Proposition 1. Let (C, σ) be as above, F = ℚ(μ(C)) the field of moduli of C, and K = F (ζ) with ζ ∈ \overline{ℚ} a primitive cube root of 1. Then there is a pair (C_{1}, σ_{1}) defined over F isomorphic over \overline{ℚ} to (C, σ) if and only if μ(C) = ℕ_{K/F} (α) for some α ∈ K.

Proof. First notice that we have already proved our claim when K = F. Assume then that [K : F] = 2 and let w ∈ K be a primitive 6^{th} root of 1 and α ↦ α′ denote the nontrivial automorphism of K/F.

It is easy to check that the map τ : (x_{1}, x_{2}) ↦ (x_{1} − w′x_{2}, −x_{1} + wx_{2}) sends the sextic f(x_{1}, x_{2}) = ax_{1}^{6} + bx_{1}^{3}x_{2}^{3} + a′x_{2}^{6} with a ∈ K, b ∈ F to a sextic g = f^{τ} with coefficients in F satisfying g(x_{1} − x_{2}, x_{1}) = g(x_{1}, x_{2})

If μ(C) is a norm write it as μ(C) = b^{2}/aa′ for some a ∈ K^{∗} and b ∈ F and let f and g = f^{τ} be as above for these choices of a and b. Then it is easy to verify that C is isomorphic over \overline{ℚ} to the curve determined by y^{2} = f(x, 1) and hence also to C_{1} determined by y^{2} = g(1, x). Clearly C_{1} is defined over F and σ_{1} : (x, y) ↦ ((x − 1)/x, y/x^{3}) gives a nontrivial automorphism of C_{1} of order 3 also defined over F.

Conversely, let (C_{1}, σ_{1}) be defined over F and let y^{2} = g(x, 1) with g(x_{1}, x_{2}) ∈ F [x_{1}, x_{2}] homogeneous of degree 6 be a hyperelliptic equation of C_{1}. Now σ must be given by a fractional linear transformation of x. Since σ is nontrivial of order 3 it is not hard to show that we may assume that g(x_{1} − x_{2}, x_{1}) = g(x_{1}, x_{2}). Now applying τ^{−1} is not hard to see that we obtain a sextic f(x_{1}, x_{2}) = ax_{1}^{6} + bx_{1}^{3}x_{2}^{3} + a′x_{2}^{6} for some a ∈ K^{∗}, b ∈ F and find that μ(C) = b^{2}/aa′, showing it is a norm from K. □

2. From now on we will only consider pairs (C, σ) defined over ℚ. We let J(C) be its Jacobian. By proposition 1 we may write μ(C) = b^{2}/aa′ for some b ∈ ℚ and a ∈ K^{∗} where K = ℚ(w) with w ∈ \overline{ℚ} a fixed primitive 6^{th} root of unity. We let f(x_{1}, x_{2}) = ax_{1}^{6} + bx_{1}^{3}x_{2}^{3} + a′x_{2}^{6} as before and let J_{2k} = J_{2k}(f) for k = 1, …, 5 be its Igusa invariants (see [Ig]). Using Mestre’s [Me] implementation of classical algorithms to compute invariants of sextics, we obtained the following expressions for J_{2k} as polynomials in a, a′ and b.

(1)  \begin{aligned}J_{2} & = 2^{−2} · 3^{4} · (40aa′ − b^{2}) \\ J_{4} & = 2^{−7} · 3^{7} · (220aa′ − b^{2})(4aa′ − b^{2}) \\ J_{6} & = −2^{−10} · 3^{9} · (320aa′ + b^{2})(4aa′ − b^{2})^{2} \\ J_{8} & = 2^{−16} · 3^{13} · (b^{4} + 2440b^{2}aa′ − 196400a^{2}a′^{2})(4aa′ − b^{2})^{2} \\ J_{10} & = 2^{−12} · 3^{21} · (4aa′ − b^{2})^{3}a^{2}a′^{2} \end{aligned}

Proposition 2. Let a ∈ ℤ[w] and b ∈ ℤ be such that aa′ = N_{1} and b^{2} −4aa′ = 2^{4} ·3^{3} ·N_{2} with N_{1}N_{2} ≠ 0 and (N_{1}N_{2}, 6) = 1. Let m = gcd(N_{1}, N_{2}). Then

(1) There is an equation

y^{2} + Q(x)y + P (x) = 0, P, Q ∈ ℤ[x], deg(P) ≤ 6, deg(Q) ≤ 3 ,

which determines a pair (C, σ) defined over ℚ with modular invariant μ(C) = b^{2}/aa′ and good reduction outside N_{1}N_{2}.

(2) Fix a prime p and let C be a stable model of C over the ring of integers of some

finite extension of ℚ_{p} and let C_{s} be its special fiber. Then:

(a) If p divides N_{1}/m then C_{s} consist of two elliptic curves with j-invariant equal to 0 crossing at a point;

(b) if p divides N_{2}/m then C_{s} consists of two copies of ℙ^{1} crossing at 3 points;

(c) otherwise C_{s} is smooth.

Proof. Let f(x_{1}, x_{2}) = ax_{1}^{6} + bx_{1}^{3}x_{2}^{3} + a′x_{2}^{6} and g = f^{τ} as before. Let a = r + sw with r, s ∈ ℤ. By multiplying by a power of w if necessary we may assume that 3|s. We may then choose the sign of b so that b ≡ 2r + s mod 27 holds. A tedious verification shows that g = 27h for some h ∈ ℤ[x_{1}, x_{2}]. It is also straightforward to check that we may solve ±h(x, 1) = Q^{2} − 4P with P, Q ∈ ℤ[x] for some choice of sign proving (1).

To prove (2) one verifies that J_{10}(h) = N_{1}^{2} · N_{2}^{3}. As for (3), the claims follow by applying Theorem 1 of [Liu] using the invariants calculated in (1); we leave the details to the reader. □

The cases (a) and (b) correspond respectively to the values μ = ∞, 4 of the modular invariant. By Raynaud, in case (a) the special fiber of the connected component of the Neron model of J(C) (over some finite extension of ℚ_{p}) is isomorphic to E × E, where E is an elliptic curve with j(E) = 0 (see for example [Li §5]). In particular, J(C) in that case has potentially good reduction at p whereas C itself does not.

The proposition implies that to any solution to t^{2} ± 2^{4} · 3^{3} = p with t ∈ ℤ and p prime we may associate a pair (C, σ) defined over ℚ with C having good reduction outside p. Indeed, in that case p ≡ 1 mod 3 and we may take a ∈ ℤ[w] so that aa′ = p and b = 2t (clearly the choice of a is immaterial). As we mentioned earlier, it is not known if there are infinitely many primes p of this form. Here is a list of the first few cases with equations determining the corresponding curves.

For p = 13 we recover the curve X_{1}(13). Indeed, if we let r(x) = h(x, 1) = x^{6} − 2x^{5} + x^{4} − 2x^{3} + 6x − 4x+ = (x^{3} − x^{2} − 1)^{2} + 4x(x − 1) we find the equation y^{2} = r(x) for this curve obtained by Fricke [Fr]. Similarly, with a = w, b = 10 (a case not quite covered by the proposition) and r_{1}(x) = 3h(x, 1) = x^{6} − 2x^{5} + 5x^{4} − 10x^{3} + 10x^{2} − 4x + 1 = (x^{3} − x^{2} − 1)^{2} − 4x(x − 1)(x^{2} − x + 1) we get Fricke’s equation y^{2} = r_{1}(x) for X_{1}(18). (There are only 3 values of N for which X_{1}(N) has genus 2: N = 13, 16, 18; only the first two have a nontrivial automorphism of order 3.)

Is not hard to check that t^{2} ± 2^{4} · 3^{3} = p can be solved for both choices of sign only for p = 733, 2917. For example, for p = 733 we get the following two curves with good reduction outside p. Their Jacobians are not isogenous over \overline{ℚ} as is easily checked by computing their zeta functions modulo various primes.

y^{2} = 4x^{5} + 15x^{4} − 50x^{3} + 35x^{2} − 4x; y^{2} = 4x^{6} − 16x^{5} + 9x^{4} + 22x^{3} − 11x^{2} − 8x + 4 .

We should add that by a theorem of Iwaniec there are infinitely many pairs of primes p, q > 3 such that t^{2} + 108 = pq for some t ∈ ℤ. In that case we must have p, q ≡ 1 mod 3 and we may hence define a pair (C, σ) over ℚ with C with good reduction outside pq.

3. The full group of automorphisms G of C_{λ} for λ ∈ \overline{ℚ} \ \{0, ±2\} is isomorphic to ℤ/2ℤ × D_{6}. Here ℤ/2ℤ corresponds to the center of G, generated by the hyperelliptic involution ι : (x, y) ↦ (x, −y), and D_{6} (the dihedral group of order 6) to the subgroup of G generated by ρ : (x, y) ↦ (1/x, y/x^{3}) and σ : (x, y) ↦ (ζx, y) with ζ ∈ \overline{ℚ} a fixed primitive cube root of unity. (For λ = 0 the group is isomorphic to ℤ/2ℤ × D_{12} with ρ replaced by (x, y) ↦ (−ζx, y).)

It is not hard to work out how G must act on the stable models described in proposition 2. In case (a) (with the common point as the identity element for the group structure of each elliptic curve): ρ permutes the two elliptic curves, σ acts as an nontrivial automorphism of order 3 on one of the elliptic curves and as its inverse on the other, and ι acts as −1 on each elliptic curve; in case (b): ρ and σ act by permuting the 3 singular points and ι by permuting the two ℙ^{1}′s.

If C has genus 2 and an involution other than the hyperelliptic one (e.g. C_{λ} and ρ above) then J(C) is isogenous to a product of elliptic curves (see for example [Ig Lemma 9]). We will now make this precise in our case.

Lemma 4. Let the notation be as above and let F = ℚ(λ). Then:

(1) The quotient of C_{λ} by the involution ρ is isomorphic over F to the elliptic curve

E_{λ} : y^{2} = x^{3} + (3x − λ + 2)^{2} ;

(2) there is an isogeny J(C) ⟶ E_{λ} × E_{−λ} defined over F with kernel isomorphic to ℤ/2ℤ × ℤ/2ℤ;

(3) the F -rational point P = (0, 2 − λ) of E_{λ} is a nontrivial point of order 3;

(4) the quotient of E_{λ} by the subgroup generated by P is isomorphic over F to the F (\sqrt{−3})/F -twist of E_{−λ}.

Proof. Let T = x + 1/x and U = y(1 + 1/x^{3}). Clearly ρ fixes U and T; in fact, these functions generate the fixed field of ρ in the function field of C_{λ}. We have

U^{2} = (T − 1)^{2}(T + 2)(T^{3} − 3T + λ) .

We replace U/(T − 1) by U_{1} and T by T − 2 and obtain

U_{1}^{2} = T_{1}(T_{1}^{3} − 6T_{1}^{2} + 9T_{1} + λ − 2) .

Then we replace U_{1} by T_{1}U_{2} and obtain

T_{1}U_{2}^{2} = T_{1}^{3} − 6T_{1}^{2} + 9T_{1} + λ − 2 ,

from which (1) follows easily.

To prove (2) note that the quotient of C_{λ} by ιρ is E_{−λ} and hence we may identify E_{±λ} with the ±1 eigenspaces of ρ acting on J(C_{λ}). The proofs of (3) and (4) amount to simple verifications that we leave to the reader. □

The lemma implies that if μ(C)) is not a square in ℚ then the elliptic curve E_{λ} with λ^{2} = μ(C) is a ℚ-curve. Specifically, E_{λ} is defined over the quadratic extension ℚ(λ)/ℚ and E_{λ} has an isogeny of degree 3 (defined over \overline{ℚ}) to its Galois conjugate E_{−λ}. It follows that conjecturally J(C) is isogenous to a factor of the Jacobian of a modular curve.

For the curves C considered at the end of the last section it is not difficult to see that J(C) should be isogenous to a factor of X_{1}(p). Also, in this case the involution ρ and the splitting of J(C) are defined over the unique abelian extension of ℚ of degree 6 unramified outside p.

References

[Fr] R. Fricke, Elliptische Funktionen und ihre Anwendungen, Teil 2.

[Ig] J-I. Igusa, Arithmetic variety of moduli for genus two, Ann. Math. 72, 612-649 (1960)

[Li] Q. Liu, Courbes stables de genre 2 et leur schéma de modules, Math. Ann. 295, 201-222 (1993)

[Me] J-F. Mestre, Construction de courbes de genre 2 à partir de leurs modules

[Sh] G. Shimura, On the field of rationality of an abelian variety, Nagoya Math. J. 45, 167-178 (1971)

Department of Mathematics University of Texas at Austin Austin, TX 78712