Decomposing Jacobians of Hyperelliptic Curves

Home , Hyperelliptic curve, Prym variety

Jennifer Paulhus University of Illinois at Urbana-Champaign [email protected] First Version: December 16, 2005 Revised Version: September 10, 2006

1 Introduction

Many interesting questions can be asked about the decomposition of Jacobians of curves. For instance, we may want to know which curves have completely decomposable Jacobians (Jacobians which are the product of g elliptic curves) [4]. We may ask about number theoretic properties of the elliptic curves that show up in the decomposition of Jacobians of curves [2]. We would also like to know how many isogenous elliptic curve factors can occur in the decomposition of some curve for a given genus. Our goal in this paper is to decompose the Jacobians of hyperelliptic curves using information about groups of automorphisms acting on the curves. The decomposition in the genus 2 case is well known [6], [7]. We determine the decomposition of many hyperelliptic curves of genus 3 and 4.

2 Convention

We denote the cyclic group and dihedral group of order n as Cn and Dn, respectively. Dn is generated by elements r and s of orders n/2 and 2, respectively. The groups Un, Vn and Gn are as in [12]. Throughout, any ﬁeld k will be of characteristic 0, ζn denotes a primitive n-th root of unity, Ei denotes an elliptic curve and Ai denotes a genus 2 curve.

3 Jacobian Decomposition

Given a hyperelliptic curve, C, we can use knowledge of its automorphism group to ﬁnd a decomposition of its Jacobian JC . We focus on hyperelliptic curves since their automorphism groups are well studied. See, for instance, [1] or [12]. In both these papers, the automorphism groups of all hyperelliptic curves are computed over an algebraically closed ﬁeld of characteristic zero. For each individual case below, the automorphism of the curve will be the same over

1 the field of definition, F , of the automorphisms. The Jacobian decompositions we find in this paper are decompositions over F . The techniques we employ to find the decompositions will still work over arbitrary fields, however the automorphisms of the curves may be different.

3.1 Technique We ﬁrst create an isogeny relation between the Jacobian of the curve and the Ja- cobians of quotients of the curve. Then we evaluate these quotients to determine how the original curve’s Jacobian decomposes. To generate isogeny relations, we use the following theorem: Theorem 1 (Kani and Rosen, [9]). Given a curve C, let G ≤Aut(C) be a ﬁnite group such that G = H1 ∪ · · · ∪ Ht where the subgroups Hi ≤ G satisfy Hi ∩ Hj = 1G if i 6= j. Then we have the following isogeny relation:

J t−1 × J g ∼ J h1 × · · · × J ht C C/G C/H1 C/Ht

m where g = |G| and hi = |Hi| and J means the product of J with itself m times. A group G satisfying the assumptions of Theorem 1 is said to have a par- tition. One word of caution: we cannot apply the previous theorem to curves whose automorphism groups have only cyclic subgroups and so our results ex- clude some hyperelliptic curves.

Once we have an isogeny relation between the Jacobian of the curve and the product of Jacobians of some of its quotient curves, we use the following results to determine the structure of these factors.

Theorem 2. If C is a curve of genus g then JC has dimension g. Theorem 3 (Hurwitz). Suppose we have a non-constant separable map φ : C1 → C2 of smooth curves over k of genus g1 and g2, respectively, and let eφ(P ) be the ramiﬁcation index of φ at P , then X 2g1 − 2 = (deg φ)(2g2 − 2) + (eφ(P ) − 1).

P ∈C1 Suppose a curve C has an automorphism group that contains the group G. Let hσi ⊂ G be some subgroup which appears on the right side of the isogeny relation. We apply Theorem 3 to the map φ : C 7→ C/hσi to determine the genus of C/hσi which gives us, by Theorem 2, the dimension of one factor of the Jacobian of C. In particular, if the factor has dimension 1, we have an elliptic curve as a factor. In order to apply Theorem 3 we must be able to determine eφ(P ) for every point P at which φ is ramified. We use the fixed points of the automorphism φ to determine these values. See Hartshorne ([8], ex.4.2.5) for the relation between ramification and fixed points.

2 Sometimes, we may have an isogeny relation from Theorem 1 involving a power of the Jacobian we would like to decompose. For instance, in the case below where the automorphism group of a curve contains the group ha, bi =∼ C2 × C2, Theorem 1 produces the following isogeny: J 2 × J 4 ∼ J 2 × J 2 × J 2 . (3.1) C C/C2×C2 C/hai C/hbi C/habi However we are interested in how the Jacobian of the curve itself decomposes. To rectify this situation, we note that the relation in the statement of Theorem 1 is derived from a particular relation among idempotents in the group ring Q[G] [9] and so we can use a slightly diﬀerent relation to translate (3.1) into:

J × J 2 ∼ J × J × J . (3.2) C C/C2×C2 C/hai C/hbi C/habi (Note also that we can produce (3.2) from (3.1) by applying Poincare Duality.)

Finally, we may be able to consolidate some of the quotient Jacobians by using a result on conjugacy groups.

Proposition 1. Suppose H1 and H2 are subgroups of G that are conjugates of ∼ each other. Then C/H1 = C/H2.

4 Results

Below we provide the decompositions of the Jacobians of hyperelliptic curves of genus 3 and 4 based on their automorphism group, G. We summarize our genus 3 results in Table 1 and our genus 4 results in Table 2. We let δ denote the dimension of the family of curves whose automorphism group contains the group in question.

4.1 Genus 3 In most genus 3 cases, we will get the best possible decomposition by looking at a subgroup of the automorphism group isomorphic to C2 ×C2 (ha, bi). Applying Theorem 1 to this group gives us:

2 JC × JC/ha,bi ∼ JC/hai × JC/hbi × JC/habi. (4.1) If b is the hyperelliptic involution and a any other involution (4.1) can be reduced to:

JC ∼ JC/hai × JC/habi (4.2)

Theorem 2 tells us that JC has dimension 3 which means that one of the factors on the right side of (4.2) must be a genus one curve while the other must be the Jacobian of a genus 2 curve. Since this is true for any involution a other than the hyperelliptic involutions we have proved the following:

3 Proposition 2. If C is a hyperelliptic curve of genus 3 with extra involutions, half the quotients of involutions are elliptic curves and the other half are genus 2 curves (excepting the hyperelliptic involution).

4.1.1 C2 × C2

Any curve C who’s full automorphism group G is isomorphic to C2 × C2 has only two non-hyperelliptic involutions. By Proposition 2 we know one must be of genus one and one must be of genus two. (We can also see this by applying Theorem 3 and information about the ﬁxed points of each automorphism.) Thus JC ∼ E × JA for some elliptic curve E and a genus 2 curve A.

4.1.2 D4 × C2

D4 × C2 has subgroups isomorphic to C2 × C2, ha, ci. Unlike our previous case, however, there are subgroups of this form which do not contain the hyperelliptic involution and so we are able to get more information about the Jacobian of this curve. Theorem 1 produces:

2 JC × JC/ha,ci ∼ JC/hai × JC/hci × JC/haci. (4.3) Considering ﬁxed points and using Theorem 3, we may conclude that each quotient on the right has genus one and so JC ∼ E1 × E2 × E3 for three elliptic curves.

4.1.3 C4 × C2

Both U2 and H2 are isomorphic to C4×C2. This group has subgroups isomorphic to C2 × C2 which all contain the hyperelliptic involution. Thus the best we can conclude is that JC ∼ E × JA for some elliptic curve E and a genus 2 curve A.

4.1.4 D8 × C2

D8 × C2 has a subgroup, not containing the hyperelliptic involution, isomorphic to C2 × C2 which gives us the following isogenous relation (1C2 is the identity element in C2):

2 JC × J 2 ∼ JC/h(s,1 )i × JC/h(r2,1 )i × JC/h(sr2,1 )i. (4.4) C/h(s,1C2 ),(r ,1C2 )i C2 C2 C2 All three curves on the right side of (4.4) are of genus 1. Thus by Theorem 2, 2 C/h(s, 1C2 ), (r , 1C2 )i must be of genus 0. Furthermore, two of the curves on the right are isomorphic to each other, since their automorphisms are in the 2 same conjugacy class (Proposition 1). Thus JC ∼ E1 × E2.

4 4.1.5 D12 2 The group D12 has a subgroup isomorphic to S3 generated by s and r . Theorem 1 then gives us:

3 6 3 2 2 2 JC × JC/hr2,si ∼ JC/hr2i × JC/hsi × JC/hsr2i × JC/hsr4i. (4.5) The last 3 curves on the right are isogenous by Proposition 1 and so we may rewrite (4.5) as: 3 6 3 6 JC × JC/hr2,si ∼ JC/hr2i × JC/hsi. (4.6) By applying Poincare duality to (4.6) we can reduce the exponents:

2 2 JC × JC/hr2,si ∼ JC/hr2i × JC/hsi. (4.7)

Both curves on the right side of (4.7) are genus 1 and so C/hr2, si is genus 0. Thus JC is the product of three elliptic curves, two of which are isogenous. We can explicitly compute these elliptic curves. Any curve of genus 3 with automorphism group containing D12 is isomorphic to a curve of the form y2 = x (x6 + αx3 + 1) for some α. C/hsi is isomorphic to the curve y2 = x3 − 3x + α which has j invariant 6912/(4 − α2) while C/hr2i is isomorphic to y2 = x3 + αx2 + x which has j-invariant 256(α2 − 3)3/(α2 − 4). The corresponding quotient maps are given by

x y (x, y) → , . (4.8) (x + 1)2 (x + 1)4

(x, y) → (x3, xy). (4.9)

4.1.6 U6 Up to isomorphism, there is only one curve with automorphism group isomor- 2 12 7 phic to U6 = ha, b | a , b , abab i. As with D12, U6 has a subgroup generated 4 by a and b which is isomorphic to S3. We use similar computations as those in D12 above to get: 2 2 JC × JC/ha,b4i ∼ JC/hb4i × JC/hai. (4.10) 2 We conclude from (4.10) that JC ∼ E1 × E2. We compute these elliptic curves and ﬁnd they are isomorphic to y2 = x3 + 3x and y2 = x3 − x, both of which have j-invariant 1728 and so, if we consider the curves over C, these curves are 3 isomorphic. Hence JC ∼ E over C. The corresponding quotient maps for these curves are given by 1 y (x, y) → x − , . (4.11) x x2 (x, y) → (x3, xy). (4.12)

5 Jacobian j-invariant of Elliptic Curves G δ Decomposition in Decomposition of Special Curves C2 × C2 3 E × JA D4 × C2 2 E1 × E2 × E3 H2 1 E × JA U2 1 E × JA 2 D12 1 E1 × E2 2 D8 × C2 1 E1 × E2 2 U6 0 E1 × E2 j(Ei) = 1728 2 V8 0 E1 × E2 j(E1) = 1728 j(E2) = 8000 3 35152 S4 × C2 0 E j(E) = 9 Table 1: Genus 3 Jacobian Decomposition

4.1.7 V8 4 8 2 −1 2 V8 = ha, b | a , b , (ab) , (a b) i and once more we look at a certain subgroup isomorphic to C2 × C2:

2 JC × JC/ha3b,b4i ∼ JC/ha3bi × JC/ha3b5i × JC/hb4i. (4.13) Each of the quotient curves on the right side of (4.13) has genus 1 and therefore 3 3 5 JC must be isogenous to three elliptic curves. In fact, since a b and a b are in the same conjugacy class in V8, the elliptic curves which are the quotients of the 2 original curve by each of these are isogenous. Hence JC ∼ E1 × E2. When we quotient out by b4 we get the genus 1 curve y2 = x4 − 1 which is isomorphic to the curve y2 = x3 +4x. The j-invariant of this curve is 1728. When we quotient 3 2 4 7 2 out by a b we end up with the curve y = x − 4ζ8 x − 2i which has j-invariant 8000.

4.1.8 S4 × C2

There is only one curve, up to isomorphism, with automorphism group S4 × C2, the curve C : y2 = x8 +14x4 +1. Applying Theorem 1 to one particular subgroup of S4 × C2 isomorphic to C2 × C2 we get:

J ∼ J × J × J . (4.14) C C/h((12)(34),1C2 )i C/h((13)(24),1C2 )i C/h((14)(23),1C2 )i But all the subgroups on the right side of (4.14) are conjugates and so by 3 Proposition 1 we get JC ∼ E . This elliptic curve may be given by the equation 2 4 2 35152 y = x + 14x + 1 which has j-invariant 9 .

4.2 Genus 4 As with the genus 3 case, all the groups we will consider have subgroups isomorphic to C2 × C2. Again, if b is the hyperelliptic involution and a any other

6 involution we can reduce (4.1) to:

JC ∼ JC/hai × JC/habi (4.15)

By Theorem 2, JC has dimension 4 which means that both of the factors on the right side of (4.15) must be Jacobians of genus 2 curves (using Theorem 3). This is true for any involution a other than the hyperelliptic involutions and so: Proposition 3. If C is a hyperelliptic curve of genus 4 with extra involutions and W a subgroup of Aut(C), then C/W is a genus 2 curve if and only if, W is generated by a non-hyperelliptic involution. The “only if” part of the above proposition follows from exhaustively considering quotients formed by all non-involution automorphisms and observing that they are genus 0 curves. Unfortunately, all the genus 2 curves we get from Proposition 3 have cyclic automorphism groups and so we cannot decompose them further (at least using our methods) into the product of two elliptic curves.

4.2.1 C2 × C2

If a curve has an automorphism group containing C2 × C2, Theorem 1 gives us the following isogeny relation for the Jacobian:

JC ∼ JC/hai × JC/hbi × JC/habi. (4.16) From Proposition 3 we know that the quotient of C by either a or ab must be a genus 2 curve and so (4.16) gives us that JC is the product of the Jacobians of two genus 2 curves.

4.2.2 D8 and D16

Let C be a curve whose automorphism group contains D8 or D16. Let n = 8 or 16 (the order of the group). In either case, we can form the following isogeny relation from Theorem 1:

JC × JC/hrn/4,si ∼ JC/hrn/4i × JC/hsi × JC/hsrn/4i (4.17) In both cases rn/4 is the hyperelliptic involution and so C/hrn/4i has genus 0. n/4 Also, s and sr are in the same conjugacy class so JC/hsi and JC/hsrn/4i (both genus 2 curves) are isogenous. So, from (4.17) we can conclude that JC is the square of the Jacobian of a genus 2 curve.

4.2.3 D10 × C2 ' D20

As with the previous cases, there are quite a few subgroups of D20 which are isomorphic to C2 ×C2 and contain the hyperelliptic involution. However, unlike the previous case, none of these subgroups contain two elements from the same conjugacy class. The best we can conclude, from the equation below, is that

7 Jacobian G δ Decomposition

C2 × C2 4 JA1 × JA2 ∼ 2 V2 = D8 2 JA 2 D8 2 JA 2 D16 1 JA

D10 × C2 1 JA1 × JA2 2 U8 0 JA 2 V10 0 JA Table 2: Genus 4 Jacobian Decomposition

the Jacobian of curves in this family is the product of two Jacobians of genus 2 curves.

JC × JC/hr5,si ∼ JC/hr5i × JC/hsi × JC/hsr5i (4.18)

4.2.4 U8

Only one curve of genus 4 has automorphism group isomorphic to U8, the curve y2 = x(x8 − 1). This curve is, in particular, in the same family as curves whose automorphism group contains D16. Since we already concluded that curves in this family have Jacobians isogenous to the square of the Jacobian of a genus 2 curve and since, by Proposition 3, no quotient of the curve by a nonhyperelliptic involution produces a genus 1 curve, the best we can conclude for this curve is that its Jacobian is the square of the Jacobian of a genus 2 curve.

4.2.5 V10

Here, too, only one curve of genus 4 has automorphism group isomorphic to V10: y2 = x10 − 1. We apply Theorem 1 to the subgroup generated by a2 and ab to get:

JC × JC/ha2,b5i ∼ JC/ha2i × JC/hb5i × JC/ha2b5i (4.19) The automorphism a2 is the hyperelliptic involution. The automorphisms a2b5 and b5 are in the same conjugacy class and so C/hb5i and C/ha2b5i are isomorphic. So, from (4.19) we conclude that the Jacobian of the curve is isogenous to the square of the Jacobian of a genus 2 curve. This genus 2 curve is isomorphic 2 5 to the curve y = x − 1 which has automorphism group isomorphic to C10.

8 4.3 General Cases

4.3.1 C2 × C2 2 2g+2 2g 2g−2 2 Any hyperelliptic curve of the form y = x +a1x +a2x +···+agx +1 where g is the genus of the curve, has automorphism group containing C2 × C2. We can use Theorem 1 to give us a decomposition of the Jacobian of curves of this form for any genus, g. Theorem 4. Any curve C of the form above has a Jacobian that decomposes as JC ∼ JC1 × JC2 .

• If g ≡ 0 (mod 2) then gC1 = gC2 = g/2.

• If g ≡ 1 (mod 2) then gC1 = (g − 1)/2 and gC2 = (g + 1)/2.

Proof: Applying Theorem 1 to the group C2 × C2 gives us the following isogeny:

2 2 2 2 JC ∼ JC/hai × JC/hbi × JC/habi (4.20) The three automorphisms of this curve send y to −y and fix x (b) , send x to −x and fix y (a), and send both x and y to their negatives (ab). In both cases, the first automorphism is the hyperelliptic involution and so the quotient of C by this automorphism is a genus 0 curve so we may disregard it in (4.20). Hence (4.20) gives us:

JC ∼ JC1 × JC2 (4.21) where C1 = C/hai and C2 = C/habi. When g ≡ 0 (mod 2), the automorphism a has two fixed points (0, ±1) as does the automorphism ab (the two points at infinity are fixed). If we apply Theorem 3 to either automorphism, we see that:

2g − 2 = 2(2gCi − 2) + 2

g = 2gCi

Since g is even, g/2 is a positive integer and so we get that gCi = g/2. When g ≡ 1 (mod 2), the automorphism a has four fixed points (0, ±1) as well as the two points at infinity. However, the automorphism ab has no fixed points. In these cases Theorem 3 gives:

2g − 2 = 2(2gC1 − 2) + 4

g − 1 = 2gC1 and

2g − 2 = 2(2gC2 − 2) + 0

g + 1 = 2gC2 .

Since g is odd, g − 1 and g + 1 both are even and so we get gC1 = (g − 1)/2 and gC2 = (g + 1)/2. 2

9 4.3.2 D2m m 2 2 Suppose that we have a curve C such that Aut(C) ⊆ D2m = hr, s| r , s , (rs) i. We consider two cases, m odd and m even.

• m odd:

In this case, all involutions in D2m are in the same conjugacy class. Ap- plying Theorem 1 gives us

2 JC × JC/D2m ∼ JC/hri × JC/hsi. (4.22)

∼ 1 We let P (A/B) denote the Prym variety of A over B. If JC/D2m = P then we know that

2 JC/hri × P (C/C/hri) ∼ JC ∼ JC/hri × JC/hsi.

∼ 2 And so by Poincare Duality we have that P (C/C/hri) = JC/hsi. This particular result is stated in [11] with a diﬀerent proof. More general results involving Jacobian decompositions and Prym varieties may also be found in [3]. We can obtain several of their de-

compositions using our techniques by replacing JC/hri with JC/D2m ×

P (C/hri / C/D2m) and replacing JC/hsi with JC/D2m ×P (C/hsi / C/D2m) in (4.22). • m even:

Similar to the D8, D16, and D20 examples for genus 4, in this case we have, from Theorem 1, the decomposition:

2 J × J ∼ J m/4 × J × J m/4 (4.23) C C/D2m C/hr i C/hsi C/hsr i

and in the case when m is a power of two, s and srm/4 are conjugates of each other which yields:

2 2 J × J ∼ J m/4 × J . (4.24) C C/D2m C/hr i C/hsi

m/4 In the cases when D2m is the full automorphism group of the curve, r is the hyperelliptic involution and so (4.23) becomes

JC ∼ JC/hsi × JC/hsrm/4i (4.25)

while (4.24) is: 2 JC ∼ JC/hsi (4.26)

10 References

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