DIHEDRAL GROUPS ACTING ON JACOBIANS

ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

Abstract. We study curves with action of a dihedral group, giving an equivariant decomposition of their Jacobians into geometric factors, on each of which acts precisely one of the rational irreducible representations of the dihedral group.

Contents

1. Introduction 1 2. Curves with D_{n}-action 2 3. Irreducible rational representations of the dihedral group 4 4. The case n odd 6 5. The decomposition for n odd 8 5.1. The case n = p, an odd prime 10 6. The case n is a power of 2 13 7. The case n even and not a power of two 14 7.1. The case n = 2p, with p an odd prime 17 Appendix A. 24 References 29

1. Introduction

In this paper we give a geometric equivariant decomposition for the Jacobian of curves where a dihedral group acts.

The case when the dihedral group D_{p} with p an odd prime acts on a curve W such that the quotient curve W/D_{p} is the Riemann sphere ℙ^{1} and such that the number of fixed points on W of any element r of order p in D_{p} is zero or two was studied by Ries in [Ri].

There he concluded that the Prym variety P (W/W_{r}) is isomorphic to twice the Jacobian JW_{k}, where W_{r} is the quotient of W by the cyclic

Date: September 14, 2001.

This research was partially supported by CONACYT 27962–E, Fondecyt 1000623 and Presidential Science Chair on Geometry.

2 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

group generated by r and W_{k} is the quotient of W by any involution k in D_{p}.

We prove that the Jacobian JW of a curve W with D_{n}-action is isogenous to a product of abelian sub-varieties where the group acts on each factor by a suitable multiple of precisely one of the rational irreducible representations of G.

For the particular case of n = p and n = 2p, where p is a prime, we also compute the kernel of the isogeny, obtaining Ries result as a special case.

It is interesting to mention here that some of the factors we obtain may be described as Jacobians or Prym varieties of intermediate covers, while others are new kinds of sub-varieties; namely, orthogonal complements of Pryms inside Pryms.

In Section 2 we set up the notation and collect some general results which will be useful later on.

In Section 3 we describe the irreducible representations of the dihedral group over the rational numbers.

In Section 4 we describe the action for the case n odd and in Section 5 we give the decomposition for JW in this case.

In Section 5.1 we study the special case n an odd prime in detail, computing the kernel of the equivariant isogeny and re-obtaining Ries’ result.

In Section 6 and in Section 7 we give the decomposition for the cases when n is a power of two and when is even but not a power of two, respectively.

In Section 7.1 we describe the case n = 2p with p an odd prime in detail, including the kernel for a special isogeny.

2. Curves with D_{n}-action

Throughout this paper, W will denote be a smooth connected complete curve over the complex numbers ℂ which admits the dihedral group D_{n} as group of automorphisms.

For any subgroup H of D_{n} we will denote by W_{H} the quotient curve W/H and by f_{H} : W → W_{H} the natural quotient map. If g is any element of D_{n}, we will abbreviate W_{⟨g⟩} (the quotient by the cyclic group generated by g) by W_{g} and f_{⟨g⟩} by f_{g}.

If H is a subgroup of a subgroup N of D_{n}, then f_{H,N} : W_{H} → W_{N} will denote the natural projection between the corresponding quotient curves.

JW_{H} will denote the Jacobian of W_{H} and P (W_{H}/W_{N}) the Prym variety of f_{H,N}; namely, the orthogonal component of f_{H,N}^{∗}(JW_{N}) in JW_{H} with respect to the natural Hermitian form on JW_{H}.

Also, JW^{H} will denote the subvariety of JW fixed by H. For any natural number s, JW [s] will denote the subgroup of JW formed by the points of order s.

If f : W_{H} → W_{N} is an unramified cyclic cover of curves of degree s, we will denote by η_{f} a point in JW_{N}[s] which determines the cover f; equivalently, a generator for the kernel of f^{∗} : JW_{N} → JW_{H} (see [L–B]).

We have the following diagram of curves and covers.

(2.1)  \xymatrix{ & W \ar[dl]_{f_{r^{j} s}} \ar[dr]^{f_{r}} \ar[dd]_{γ} \\ W_{r^{j}s} \ar[dr]_{f_{r^{j} s,Dn}} &  & W_{r} \ar[dl]^{f_{r,Dn}} \\  & W_{D_{n}}}

The cases n odd and n even will be treated separately in the next sections.

We now collect some general results that will be used later.

The following result appeared in [SA] for G = A_{5} and was generalized as follows in [C–R].

Proposition 2.1. Let G be a finite group and let {W_{j}}_{j=1}^{r} denote its irreducible representations over the rational numbers.

Assume there exist H ≤ N ≤ G subgroups of G and 1 ≤ ℓ ≤ r such that

1. ρ_{H}^{G} − ρ_{N}^{G} = ∑_{j=1}^{ℓ} W_{j}, and

2. ⟨ρ_{H}^{G}, W_{j}⟩_{G} = 1 for all 1 ≤ j ≤ ℓ, where ρ_{H}^{G} denotes the permutation representation of G induced by the trivial representation of H and ⟨, ⟩_{G} denotes the inner product between representations of G.

Let W denote any rational representation of G containing as different isotypical components W_{1}^{n_{1}} ⊕⊕ … W_{ℓ}^{n_{ℓ}} and denote by W^{H} the subspace of W fixed by H.

Then

W^{H} = W^{N} ⊕ (W_{1}^{n_{1}})^{H} ⊕ … ⊕ (W_{ℓ}^{n_{ℓ}})^{H}

Furthermore, if for every 1 ≤ j ≤ ℓ we consider the (complex) dimension v_{j} of a complex irreducible representation V_{j} associated to W_{j}, then, under condition 2) above, there are v_{j} elements g_{1}^{j}, …, g_{v_{j}}^{j} of G such that

W_{j} = ⨁_{k=1}^{v_{j}} W_{j}^{g_{k}^{j} H (g_{k}^{j})^{−1}}

Corollary 2.2. Let G, H and N be as in Proposition 2.1 and let W be a curve with G-action.

Denote by f_{H} : W → W_{H} = W/H the natural projection.

Then

(df_{H}^{∗})(P (W_{H}/W_{N})) = (W_{1}^{n_{1}})^{H} ⊕ … ⊕ (W_{ℓ}^{n_{ℓ}})^{H}

where the sum is orthogonal with respect to any Hermitian product invariant under G.

Furthermore, if ℓ = 1, then W_{1}^{n_{1}} acts on

f_{H}^{∗}(P (W_{H}/W_{N}))^{s_{1}}

Proof. Consider the rational representation of G on T_{0}(JW) and apply the Proposition.

The last assertion is proved by observing that, for each 1 ≤ k ≤ v_{1}, g_{k}^{1} gives an isomorphism

g_{k}^{1} : P (W_{H}/W_{N}) → P (W_{g_{k}^{1} H (g_{k}^{1})^{−1}}/W_{g_{k}^{1} N (g_{k}^{1})^{−1}})

3. Irreducible rational representations of the dihedral group

We denote by D_{n} = ⟨r, s : r^{n} = s^{2} = (rs)^{2} = 1⟩ the dihedral group of order 2n.

It is known that this group has two complex irreducible representations of degree one, χ_{0} and χ_{1}, if n is odd, and four, χ_{0}, χ_{1}, χ_{2} and χ_{3}, if n is even.

They are given as follows.

Representation r  s Case

χ_{0}  1  1 any

χ_{1}  1 −1 any

χ_{2}  −1 1 n even

χ_{3}  −1 −1 n even

The other complex irreducible representations of D_{n} all have degree

two, and there are \frac{n − 1}{2} of them for n odd, respectively \frac{n − 2}{2} for n even, given as follows.

V_{j}(r) = ( \begin{matrix}ω^{j} & 0 \\ 0 & ω^{−j} \end{matrix} ) and V_{j}(s) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

where ω = ω_{n} = exp \frac{2πi}{n} and 1 ≤ j < \frac{n}{2}.

We are interested in the irreducible rational representations of D_{n}, which are described in the next result; we were unable to find them explicitly in the literature.

Remark 3.1. Let D_{n} denote the dihedral group of order 2n, and let

Ω(n) = {d : d is a divisor of n}

For each d in Ω(n) and d < n/2, consider the complex irreducible representation of degree two V_{d} given above.

Its field of definition coincides with the field generated by its characters

K_{d} = ℚ(w_{n}^{d} + w_{n}^{−d}) = ℚ(w\frac{n}{d} + w−1_{\frac{n}{d}})

and furthermore

[K_{d} : ℚ] = \frac{φ(\frac{n}{d})}{2}

where φ denotes the Euler function.

Hence the irreducible rational representation W_{d} of D_{n} generated by V_{d} is given by

K_{d} ⊗_{ℚ} W_{d} = ⨁_{σ∈Gal(K_{d}/ℚ)}(K_{d} ⊗_{σ} V_{d})

and has degree

deg(W_{d}) = dim_{K_{d}} V_{d} · [K_{d} : ℚ] = 2 · \frac{φ(\frac{n}{d})}{2} = φ (\frac{n}{d})

Theorem 3.2. Let D_{n} denote the dihedral group of order 2n and let Ω(n) = {d : d is a divisor of n}.

Then the rational irreducible representations of D_{n} are given as follows.

If n is odd, they are χ_{0} and χ_{1}, of degree one, and {W_{d}} with d in Ω(n) and d ≠ n, of respective degrees deg(W_{d}) = φ (\frac{n}{d}).

If n is even, they are χ_{0}, χ_{1}, χ_{2} and χ_{3}, of degree one, and {W_{d}} with d in Ω(n) and d ≠ n, n/2, of respective degrees deg(W_{d}) = φ (\frac{n}{d}).

Proof. We already know that the given representations are irreducible over the rationals and that they have the given degrees.

All that is left to show is that there are no other irreducible representations.

For n odd, just observe that

∑_{\begin{matrix}d∈Ω(n) \\ d≠n \end{matrix}} deg(W_{d}) = ∑_{\begin{matrix}d∈Ω(n) \\ d≠n \end{matrix}} φ (\frac{n}{d}) = n − 1 = \frac{n − 1}{2} · 2 = ∑_{j=1}^{\frac{n−1}{2}} deg(V_{j})

A similar argument for n even finishes the proof.

4. The case n odd

When n is odd, all curve W_{r^{j}s} are isomorphic to each other since the involutions r^{j}s are conjugate in D_{n}. Hence the (common) number of fixed points of each of these involutions on W must be even, since it corresponds to the number of ramification points of the double cover W_{r} → W_{D_{n}}; we will denote it by 2β.

Furthermore, the fixed points of r are permuted among themselves by s, say, and therefore the number of fixed points of r must also be even; we will denote it by 2α.

For the case of n = p an odd prime, there are no other fixed points. A calculation proves the following result.

Proposition 4.1. Let n be an odd number and let W be a curve with D_{n} ⊆ Aut(W) and associated Diagram (2.1).

We denote by 2β the number of fixed points of any involution of the form r^{j}s in D_{n} and by 2α the number of fixed points of any element of order n in D_{n}.

Then the following table describes the genera of the corresponding curves and the total ramification order for the corresponding covers, where g denotes the genus of W_{D_{n}}.

genus  order of ramification |B|

g_{W} = 2ng + nβ + (n − 1)α + 1 − 2n

g_{W_{r}} = 2g − 1 + β  |B(W → W_{r})| = 2(n − 1)α
|B(W_{r} → W_{D_{n}})| = 2β

g_{W_{r^{j} s}} = ng + \frac{n − 1}{2} (α + β − 2)  |B(W → W_{r^{j}s})| = 2β
|B(W_{r^{j}s} → W_{D_{n}})| = (n − 1) (α + β)

For the particular case when n = p is an odd prime number, the curves in Diagram (2.1) give all the possible quotient curves. In this case, we can also prove the following results.

Proposition 4.2. Let p denote an odd prime and let W be a curve such that D_{p} ⊆ Aut(W), with associated Diagram (2.1).

Let k denote any involution in D_{p}. Then the following results hold.

i) f_{k,D_{n}}^{∗} : JW_{D_{n}} → JW_{k} is injective.

ii) ker f_{r,D_{n}}^{∗} = ker γ^{∗}. iii) ker f_{k}^{∗} = f_{k,D_{n}}^{∗}(ker f_{r,D_{n}}^{∗}). iv) f_{r,D_{n}}^{∗}(JW_{D_{n}}) ∩ ker f_{r}^{∗} = {0}.

Proof. i) holds because the cover f_{k,D_{n}} : JW_{k} → JW_{D_{n}} is non-cyclic of prime order (see [L–B, p. 337]).

As for ii), it is clear that ker f_{r,D_{n}}^{∗} ⊆ ker γ^{∗}. For every P ∈ ker γ^{∗} we have that f_{k,D_{n}}^{∗}(P) ∈ ker f_{k}^{∗} ⊆ JW_{k}[2]. By i) f_{k,D_{n}}^{∗} is injective, and therefore 2P = 0.

But also f_{r,D_{n}}^{∗}(P) ∈ ker f_{r}^{∗} ⊆ JW_{r}[p]; since p is odd we must have f_{r,D_{n}}^{∗}(P) = 0 and therefore P ∈ ker f_{r,D_{n}}^{∗}.

To prove iii) just note that ker f_{r,D_{n}}^{∗} = ker γ^{∗} = (f_{k,D_{n}}^{∗})^{−1}(ker f_{k}^{∗}), where the first equality holds from ii) and the second one from the commutativity of Diagram (2.1).

8 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

The proof of iv) is direct from ii): if Q ∈ f_{r,D_{n}}^{∗}(JW_{D_{n}}) ∩ ker f_{r}^{∗} then Q = f_{r,D_{n}}^{∗}(P) for some P ∈ JW_{D_{n}} and 0 = f_{r}^{∗}(Q) = γ^{∗}(P). Hence P must be in ker f_{r,D_{n}}^{∗} and therefore Q = 0.

We also have the following result.

Corollary 4.3. Let W be as in Proposition 4.1 and let k = r^{j}s denote an involution in D_{n}.

Then f_{k}^{∗} is injective if and only if f_{r,D_{n}}^{∗} is injective.

Proof. The result follows immediately from Proposition 4.2 when n is an odd prime.

For the general case of n any odd natural number, note that since both f_{k} and f_{r,D_{n}} are of degree two, then f_{k}^{∗} (respectively f_{r,D_{n}}^{∗}) is injective if and only if f_{k} (respectively f_{r,D_{n}}) is ramified.

But the involution k : W → W descends to the involution σ : W_{r} → W_{r} that gives the morphism f_{r,D_{n}} : W_{r} → W_{D_{n}}.

Therefore, if w ∈ W is a ramification point of f_{k}, or, equivalently, if the stabilizer of w in D_{n} is the group generated by k, then f_{r}(w) is a fixed point of σ, or, equivalently, it is a ramification point of f_{r,D_{n}}.

Conversely, a ramification point of f_{r,D_{n}} is a fixed point of σ of the form f_{r}(w), and hence w is a fixed point of some conjugate r^{l}s of k. Then the image of w under an appropriate power of r produces a fixed point of k.

The next result will be very useful later on.

Corollary 4.4. Let W be as in Proposition 4.1 and let k = r^{j}s denote an involution in D_{n}.

Then f_{k}^{∗}|_{P (W_{k}/W_{Dn})} is injective.

Proof. If f_{k}^{∗} is not injective on JW_{k}, then

ker f_{r,D_{n}}^{∗} = {0, η_{f_{r,Dn}}} ⊆ JW_{D_{n}}[2]

with η_{f_{r,Dn}} ≠ 0.

Furthermore, f_{k,D_{n}}^{∗}(η_{f_{r,Dn}}) = η_{f_{k}} ∈ f_{k,D_{n}}^{∗}(JW_{D_{n}}[2]) is the non zero element of ker f_{k}^{∗}, because ker f_{k,D_{n}}^{∗} ⊆ JW_{D_{p}}[n] consists of points of odd order.

But f_{k,D_{n}}^{∗}(JW_{D_{n}}) ∩ P (W_{k}/W_{D_{n}}) ⊆ JW_{k}[n] also consists of points of odd order, and therefore η_{f_{k}} ∉ P (W_{k}/W_{D_{n}}).

5. The decomposition for n odd

Theorem 5.1. Let n = p_{1}^{α_{1}} · . . . · p_{r}^{α_{r}} be the prime decomposition of the odd natural number n and let W be a curve with D_{n} action.

For each d ∈ Ω(n) different from n, write d = _{\frac{n}{p_{1}^{i_{1}}·...·p_{r}^{ir}}} and let D_{d} be a dihedral subgroup of D_{n} of order 2d, which are all conjugate among themselves for fixed d.

Also let

B_{d} = ⋂_{\begin{matrix}j∈\{1,... ,r\} \\ i_{j}≥1 \end{matrix}} P (W_{D_{d}}/W_{D_{p_{j} ·d}}) ⊆ JW_{D_{d}} .

Then the isogeny decomposition of JW with respect to D_{n} is the following.

JW ∼ JW_{D_{n}} × P (W_{r}/W_{D_{n}}) × ∏_{\begin{matrix}d∈Ω(n) \\ d≠n \end{matrix}} 2B_{d}

where the D_{n}−action on each factor on the right hand side is given by (appropriate multiples of) the trivial action χ_{0} on JW_{D_{n}}, the alternating action χ_{1} on P (W_{r}/W_{D_{n}}), and W_{d} on 2B_{d} for each d ∈ Ω(n) with d ≠ n.

Furthermore, for each j such that i_{j} ≥ 1

2P (D_{d}/D_{p_{j}·d}) is isogenous to P (W_{r^{\frac{n}{d}}} /W_{r^{\frac{n}{p_{j} ·d}}}) .

Proof. For each subgroup H of D_{n} we denote by ρ_{H} the representation of D_{n} induced by the trivial representation of H.

Then calculations show that

ρ_{D_{d}} = χ_{0} + ∑_{\begin{matrix}k∈Ω(n),k≠n \\ k·p_{1}^{i_{1}}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{k}

and that if i_{1} ≥ 1 then

ρ_{D_{p_{1}·d}} = χ_{0} + ∑_{\begin{matrix}ℓ∈Ω(n),ℓ≠n \\ ℓ·p_{1}^{i_{1}−1}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{ℓ}

Hence if i_{1} ≥ 1 then

(5.2)  ρ_{D_{d}} − ρ_{D_{p_{1}·d}} = ∑_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} W_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

and therefore (by 2.2)

P (W_{D_{d}}/W_{D_{p_{1}·d}}) ∼ ∏_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} B_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

with similar results for the other p′_{j}s for which i_{j} ≥ 1.

Furthermore, it follows from Corollary 2.2 that then

(5.3) (an appropriate multiple of) W_{d} acts on

2B_{d} = 2 ⋂_{\begin{matrix}j∈\{1,... ,r\} \\ i_{j}≥1 \end{matrix}} P (W_{D_{d}}/W_{D_{p_{j} ·d}})

We also compute that

ρ_{⟨r^{\frac{n}{d}} ⟩} = χ_{0} + χ_{1} + 2 ∑_{\begin{matrix}k/n,k≠n \\ k·p_{1}^{i_{1}}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{k} = ρ_{⟨r⟩} + 2 ∑_{\begin{matrix}k/n,k≠n \\ k·p_{1}^{i_{1}}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{k}

from where we obtain

ρ_{⟨r⟩} − ρ_{D_{n}} = χ_{1} (5.4)

and

ρ_{⟨r^{\frac{n}{d}} ⟩} − ρ_{⟨r^{\frac{n}{p_{1}·d}} ⟩} = 2 ∑_{\begin{matrix}s_{j}∈\{0,... ,i_{j}\} \\ j∈\{2,... ,r\} \end{matrix}} W_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}} (5.5)

Now we apply Corollary 2.2 to conclude from (5.4) that

(5.6) (an appropriate multiple of) χ_{1} acts on P (W_{r}/W_{D_{n}})

and also, by comparing (5.2) and (5.5), that if i_{1} ≥ 1 then

2P (W_{D_{d}}/W_{D_{p_{1}·d}} is isogenous to P (W_{r^{\frac{n}{d}}} /W_{r^{\frac{n}{p_{1}·d}}})

The results (5.3) and (5.6) give the isogeny decomposition, thus completing the proof.

5.1. The case n = p, an odd prime. In this case, we can be more precise.

Let 2α denote the number of fixed points in W of any element of order p in D_{p} and let k denote an involution in D_{p} .

If α ≥ 2, let {P_{1}, …, P_{α}} denote the set of total ramification points of f_{k,D_{p}} : W_{k} → W_{D_{p}}. For each j ∈ {2, …, α} we choose m_{j} ∈ Pic^{0}(W_{D_{p}}) such that m_{j}^{⊗p} = O_{W_{Dp}}(f_{k,D_{p}}(P_{1}) − f_{k,D_{p}}(P_{j})) and we define

F_{j} = O_{W_{k}}(P_{j} − P_{1}) ⊗ f_{k,D_{p}}^{∗}(m_{j}) . (5.7)

For p an odd prime, D_{p} has exactly three irreducible representations over the rational numbers, given by χ_{0}, χ_{1} and W_{1}, which is the sum of the (p − 1)/2 irreducible complex representations of degree two of D_{p}.

We obtain the next result, which will be proved in Appendix A.

Proposition 5.2. The morphism

(5.8)  P (W_{k}/W_{D_{p}}) × P (W_{k}/W_{D_{p}}) ⟶^{φ_{k}} P (W/W_{r})
(z_{1}, z_{2}) −−→ f_{k}^{∗}(z_{1}) + rf_{k}^{∗}(z_{2})

is a D_{p}-equivariant isogeny whose kernel is isomorphic to

(5.9) K = {z ∈ P (W_{k}/W_{D_{p}})[p] : f_{k}^{∗}(z) is D_{p} invariant}

More precisely, K may be described as follows.

i) f_{k,D_{p}}^{∗}(JW_{D_{p}})[p] ⊆ K.

ii) If α = 0 or 1 then K = f_{k,D_{p}}^{∗}(JW_{D_{p}})[p]. iii) If α ≥ 2 then

K = f_{k,D_{p}}^{∗}(JW_{D_{p}})[p] ⊕ ⨁_{j=2}^{α} F_{j}ℤ/pℤ

where the F_{j} are defined by (5.7). and therefore the cardinality of K is given as follows, if g denotes the genus of W_{D_{p}} and 2β denotes the number of fixed points of the involution k in D_{p}.

|K| = \begin{cases}p^{2g} & if α = 0; \\ p^{2g+α−1} & if α ≥ 1 . \end{cases} (5.10)

We obtain as a corollary the result in [Ri].

Corollary 5.3. Consider a curve W with D_{p}-action such that r has two or no fixed points in W and such that W_{D_{p}} = ℙ^{1}.

Then P (W/W_{r}) is isomorphic (as a polarized abelian variety) to JW_{k} × JW_{k}.

Proof. Under the hypothesis in the corollary, φ_{k} is an isomorphism and the polarization induced on P (W/W_{r}) by the natural principal polarization on JW is twice a principal polarization.

We now refine Theorem 5.1 for this case as follows.

Theorem 5.4. Let p denote an odd prime and let W be a curve such that D_{p} ⊆ Aut(W), with associated Diagram (2.1).

Also, denote by k an involution in D_{p}.

Then the isogeny decomposition of JW with respect to D_{p} is the following.

R : JW_{D_{p}} × P (W_{r}/W_{D_{p}}) × 2P (W_{k}/W_{D_{p}}) ∼ JW (5.11)

12 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

with R(w_{1}, w_{2}, w_{3}, w_{4}) = f_{D_{p}}^{∗}(w_{1})+f_{r}^{∗}(w_{2})+f_{k}^{∗}(w_{3})+rf_{k}^{∗}(w_{4}), where the D_{p}−action on each factor on the left hand side is given by (appropriate multiples of) the trivial action χ_{0} on JW_{D_{p}} , the alternating action χ_{1} on P (W_{r}/W_{D_{p}}), and W_{1} on 2P (W_{k}/W_{D_{p}}).

Furthermore, if we denote by 2α the number of fixed points in W of the element r (of order p) in D_{p} , if g denotes the genus of W_{D_{p}} and 2β denotes the number of fixed points of an involution in D_{p}, then the isogeny R has kernel with cardinality given as follows.

| ker R| = \begin{cases}2^{2g−1}p^{6g−3+α} & if β = 0; \\ 2^{2g}p^{6g−3+α+2β} & if β ≥ 1 . \end{cases}

Proof. The decomposition (5.11) follows immediately from Theorem 5.1 once we observe that d in Ω(p) and d ≠ p imply that d = 1 and therefore W_{1} acts on 2B_{1}, where

B_{1} = P (W_{D_{1}}/W_{D_{p}}) = P (W_{k}/W_{D_{p}}) .

To prove the last part of the theorem, first observe that the following diagram is commutative

\xymatrix{JW_{D_{p}} × P (W_{r}/W_{D_{p}}) × 2P (W_{k}/W_{D_{p}}) \ar[dd]_{R} \ar@{-->}[dr]^{(id_{JW_{Dp} ×P (Wr /W_{Dp})}, φ_{k})} \\  & JW_{D_{p}} × P (W_{r}/W_{D_{p}}) × P (W/W_{r}) \ar@{-->}[dl]^{ψ} \\ JW} (5.12)

where φ_{k} is defined in Proposition 5.2 and

ψ(z_{1}, z_{2}, z_{3}) = f_{D_{p}}^{∗}(z_{1}) + f_{r}^{∗}(z_{2}) + z_{3}

has kernel with cardinality as follows (see [Rec–Ro2]).

(5.13)  | ker(ψ)| = \frac{|JW_{r}[p]| |JW_{D_{p}}[2]|}{| ker f_{r}^{∗}| | ker f_{r,D_{p}}^{∗}|}

= \begin{cases}2^{2g−1}p^{4g−3} & if α = β = 0 \\ 2^{2g}p^{4g−3+2β} & if α = 0 and β > 0 \\ 2^{2g−1}p^{4g−2+2β} & if α > 0 and β = 0 \\ 2^{2g}p^{4g−2+2β} & if α β > 0 \end{cases}

We can now finish the proof by concluding that

| ker(R)| = | ker(φ_{k})| | ker(ψ)| from Diagram (5.12) and then combining (5.13) and (5.10).

6. The case n is a power of 2

In this section we assume n = 2^{α}.

The case α = 1 corresponds to the Klein group, which was completely described in [Rec–Ro2], as well as the case α = 2. Therefore, we also assume α ≥ 2 in this section.

In this case Ω(n) = {2^{j}; 0 ≤ j ≤ α}, and for each d ∈ Ω(n) with d ≠ n there are two non conjugate subgroups of D_{n} = ⟨r, s⟩ both isomorphic to D_{d}: D_{d} = ⟨r^{\frac{n}{d}} , s⟩ and D̃_{d} = ⟨r^{\frac{n}{d}} , rs⟩.

Furthermore, it follows from Theorem 3.2 that the rational irreducible representations of D_{n} are the following in this case.

χ_{0}, χ_{1}, χ_{2}, χ_{3}, W_{n/4}, W_{n/8}, …, W_{1}

where deg(W_{d}) = φ (\frac{n}{d}) = \frac{n}{2d}.

The isotypical decomposition in this case is the following.

Theorem 6.1.

JW ∼ JT × P (W_{r}/W_{D_{n}}) × P (W_{D_{\frac{n}{2}}} /W_{D_{n}}) × P (WD̃_{\frac{n}{2}} /W_{D_{n}})

× 2P (W_{D_{\frac{n}{4}}} /W_{D_{\frac{n}{2}}} ) × … × 2P (W_{D_{1}}/W_{D_{2}})

with respective actions

χ_{0} × χ_{1} × χ_{2} × χ_{3} × W\frac{n}{4} × … × W_{1}

Furthermore, for each d ∈ Ω(n) with d ≠ n, \frac{n}{2}, we have that

P (W_{D_{d}}/W_{D_{2d}}) is isogenous to P (WD̃_{d}/WD̃_{2d})
2P (W_{D_{d}}/W_{D_{2d}}) is isogenous to P (W_{r^{\frac{n}{d}}} /W_{r^{\frac{n}{2d}}})

and

P (W_{D_{\frac{n}{2}}} /W_{D_{n}}) × P (WD̃_{\frac{n}{2}} /W_{D_{n}}) is isogenous to P (W_{r^{2}}/W_{r})

Proof. For each d ∈ Ω(n) with d ≠ n, \frac{n}{2} we have

ρ_{D_{d}} = χ_{0} + χ_{2} + W\frac{n}{4} + … + W_{d} = ρ_{D_{\frac{n}{2}}} + W\frac{n}{4} + … + W_{d} ;
ρD̃_{d} = χ_{0} + χ_{3} + W\frac{n}{4} + … + W_{d} = ρD̃_{\frac{n}{2}} + W\frac{n}{4} + … + W_{d}

and also

ρ_{⟨r^{\frac{n}{d}} ⟩} = χ_{0} + χ_{1} + χ_{2} + χ_{3} + 2W\frac{n}{4} + … + 2W_{d}
= ρ_{⟨r^{2}⟩} + 2W\frac{n}{4} + … + 2W_{d}

14 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

7. The case n even and not a power of two

In this section we assume n is even and not a power of two. Let n = 2^{α_{0}} · p_{1}^{α_{1}} · . . . · p_{r}^{α_{r}} be the prime decomposition of n.

Then Ω(n) = {2^{k_{0}} · p_{1}^{α_{k_{1}}} · . . . · p_{r}^{α_{kr}}; 0 ≤ k_{j} ≤ α_{j}, 0 ≤ j ≤ r, }.

For each d in Ω(n) such that \frac{n}{d} is even (equivalently, for each d not divisible by 2^{α}) there are two non conjugate subgroups of D_{n} = ⟨r, s⟩ both isomorphic to the dihedral group D_{d}; we choose a representatives D_{d} = ⟨r^{\frac{n}{d}} , s⟩ and D̃_{d} = ⟨r^{\frac{n}{d}} , r^{n−1}s⟩, respectively. For each d in Ω(n) with \frac{n}{d} odd, the conjugacy class of D_{d} in D_{n} is unique; we choose the

representative D_{d} = ⟨r^{\frac{n}{d}} , s⟩.

Furthermore, it follows from Theorem 3.2 that the rational irreducible representations of D_{n} are given by

χ_{0}, χ_{1}, χ_{2}, χ_{3}, \{W_{d}\}\begin{matrix}d∈Ω(n) \\ d≠n,n/2 \end{matrix}

where deg(W_{d}) = φ(\frac{n}{d}).

The isotypical decomposition in this case is the following.

Theorem 7.1. Let n = p_{0}^{α_{0}} · p_{1}^{α_{1}} · . . . · p_{r}^{α_{r}} be the prime decomposition of the even natural number n, not a power of 2. Assume p_{0} = 2.

Let W be a curve with D_{n} action.

Furthermore, for each d ∈ Ω(n) with d ≠ n, \frac{n}{p_{j}} for 0 ≤ j ≤ r, write

d = _{\frac{n}{p_{0}^{i_{0}}·p_{1}^{i_{1}}·...·p_{r}^{ir}}} and let

B_{d} = ⋂_{\begin{matrix}j∈\{0,... ,r\} \\ i_{j}≥1 \end{matrix}} P (W_{D_{d}}/W_{D_{p_{j} ·d}}) .

Then

JW ∼ JW_{D_{n}} × P (W_{r}/W_{D_{n}}) × P (W_{D_{\frac{n}{2}}} /W_{D_{n}}) × P (WD̃_{\frac{n}{2}} /W_{D_{n}})

× ∏_{i=1}^{r} 2P (W_{D_{\frac{n}{p_{i}}}} /W_{D_{n}}) × ∏_{\begin{matrix}d∈Ω(n) \\ d≠n, \frac{n}{p_{j}} \\ 0≤j≤r \end{matrix}} 2B_{d}

DIHEDRAL GROUPS ACTING ON JACOBIANS 15

with respective actions

χ_{0} × χ_{1} × χ_{2} × χ_{3} × ∏_{i=1}^{r} W_{\frac{n}{p_{i}}} × ∏_{\begin{matrix}d∈Ω(n) \\ d≠n, \frac{n}{p_{i}} \end{matrix}} W_{d} .

Furthermore,

B_{d} is isogenous to B̃_{d} = ⋂_{\begin{matrix}j∈\{0,... ,r\} \\ i_{j}≥1 \end{matrix}} P (WD̃_{d}/W{ \atop D_{p_{j} ·d}})

2P (W_{D_{d}}/W_{D_{2d}}) is isogenous to P (W_{r^{\frac{n}{d}}} /W_{r^{\frac{n}{2d}}})

and

P (W_{D_{\frac{n}{2}}} /W_{D_{n}}) × P (WD̃_{\frac{n}{2}} /W_{D_{n}}) is isogenous to P (W_{r^{2}}/W_{r})

Proof. If d = 2^{α}c = \frac{n}{p_{1}^{i_{1}} · . . . · p_{r}^{i_{r}}} is a divisor of n of the form multiple of 2^{α} and different from n then

ρ_{D_{d}} = χ_{0} + ∑_{\begin{matrix}k∈Ω(n),k≠n \\ k·p_{1}^{i_{1}}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{k}

and (if i_{1} ≥ 1)

ρ_{D_{p_{1}·d}} = χ_{0} + ∑_{\begin{matrix}ℓ∈Ω(n),ℓ≠n \\ ℓ·p_{1}^{i_{1}−1}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{ℓ}

For any other d = \frac{n}{2^{i_{0}} · p_{1}^{i_{1}} · . . . · p_{r}^{i_{r}}} divisor of n different from n and from n/2 and not a multiple of 2^{α} we have

ρ_{D_{d}} = χ_{0} + χ_{2} + ∑_{\begin{matrix}k∈Ω(n),k≠n \\ k·2^{i_{0}}·p_{1}^{i_{1}}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{k}

and (if i_{1} ≥ 1)

ρ_{D_{p_{1}·d}} = χ_{0} + χ_{2} + ∑_{\begin{matrix}ℓ∈Ω(n),ℓ≠n \\ ℓ·2^{i_{0}}·p_{1}^{i_{1}−1}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{ℓ} .

16 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

Furthermore, if 2d is not a multiple of 2^{α}

ρ_{D_{2·d}} = χ_{0} + χ_{2} + ∑_{\begin{matrix}ℓ∈Ω(n),ℓ≠n \\ ℓ2^{i_{0}−1}··p_{1}^{i_{1}}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{ℓ}

and if 2d is a multiple of 2^{α}

ρ_{D_{2·d}} = χ_{0} + ∑_{\begin{matrix}ℓ∈Ω(n),ℓ≠n \\ ℓ2^{i_{0}−1}··p_{1}^{i_{1}}·...·p_{r}^{ir} ≡0(n) \end{matrix}} W_{ℓ}

Hence, if d = \frac{n}{p_{1}^{i_{1}} · . . . · p_{r}^{i_{r}}} is a divisor of n and a multiple of 2^{α}, different from n and with 1 ≤ i_{1} < α_{1} then

(7.14)  ρ_{D_{d}} − ρ_{D_{p_{1}·d}} = ∑_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} W_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

and therefore (by Corollary 2.2)

P (W_{D_{d}}/W_{D_{p_{1}·d}}) ∼ ∏_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} B_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

For any other d = \frac{n}{2^{i_{0}} · p_{1}^{i_{1}} · . . . · p_{r}^{i_{r}}} divisor of n, not a multiple of 2^{α} and different from n and from n/2 we have

(7.15)  ρ_{D_{d}} − ρ_{D_{p_{1}·d}} = ∑_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} W_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}} .

Furthermore, if 2d is not a multiple of 2^{α} then

(7.16)  ρ_{D_{d}} − ρ_{D_{2·d}} = ∑_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} W_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

and if 2d is a multiple of 2^{α} then

(7.17)  ρ_{D_{d}} − ρ_{D_{2·d}} = χ_{2} + ∑_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} W_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

and therefore (by Corollary 2.2)

P (W_{D_{d}}/W_{D_{p_{1}·d}}) ∼ ∏_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} B_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

and

P (W_{D_{d}}/W_{D_{2·d}}) ∼ ∏_{\begin{matrix}j∈\{2,... ,r\} \\ s_{j}∈\{0,... ,i_{j}\} \end{matrix}} B_{d·p_{2}^{s_{2}}·...·p_{r}^{sr}}

7.1. The case n = 2p, with p an odd prime. In this case we can be more precise.

There are two conjugacy classes of involutions of the form r^{j}s: one for j even, say j = 0, and the other for j odd, say j = p.

Also, there are two non conjugate dihedral groups of order 2p inside D_{2p}: one generated by r^{2} and s and the other by r^{2} and r^{p} s. They will be denoted by D_{p} and D̃_{p} respectively. Note that we may apply the results of section 5.1 to both of these dihedral groups.

There is one conjugacy class of Klein (or D_{2}) subgroups of D_{2p}, of the form K_{j} = ⟨r^{j}s, r^{p}⟩ = {1, r^{j}s, r^{p}, r^{p+j}s} for j ∈ {0, … p − 1}. The quotient curve W/K_{0} will we denoted by W_{D_{2}}.

Since p is an odd prime, in this case the complete diagram of curves and covers is the following:

[figure: W; f_{r^{2}}; f_{r^{p}s}; f_{r^{p}}; W_{r^{p} s}; f_{r^{p}s,D̃p}; f_{r^{p}s,D_{2}}; W_{r^{2}}; f_{r^{2},D̃p}; f_{r^{2},Dp}; f_{r^{2},r}; WD̃_{p}; f_{D̃p,D_{2p}}; f_{s}; W_{r^{p}}; f_{r^{p},D_{2}}; W_{r}; f_{r,D_{2p}}; W_{s}; f_{s,Dp}; f_{s,D_{2}}; W_{D_{2}}; f_{D_{2},D_{2p}}; W_{D_{p}}; f_{Dp,D_{2p}}; W_{D_{2p}}] (7.18)

The conditions on the numbers of fixed points are the following: 2β_{1} for any involution conjugate to s, 2β_{2} for any involution conjugate to rs, 2κ for r, 4γ for the number of points fixed by r^{2} but not by r, 2pδ for the number of points fixed by r^{p} but not by r, where β_{1} + β_{2},

18 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

β_{1} + δ + κ and β_{2} + δ + κ are all divisible by 2; i.e., β_{1}, β_{2} and δ + κ have the same parity.

If we let g_{T} = g, then we have

genus  order of ramification |B|

g_{W} = 4pg + 1 − 4p + p(β_{1} + β_{2} + δ)+ +(2p − 1)κ + 2(p − 1)γ

g_{W_{s}} = 2pg + 1 − 2p + p \frac{β_{2} + δ}{2} +  |B(W → W_{s})| = 2β_{1}

+ \frac{p − 1}{2} (β_{1} + 2γ) + \frac{2p − 1}{2} κ

g_{W_{r^{p}s}} = 2pg + 1 − 2p + p \frac{β_{1} + δ}{2} +  |B(W → W_{r^{p}s})| = 2β_{2}

+ \frac{p − 1}{2} (β_{2} + 2γ) + \frac{2p − 1}{2} κ

g_{W_{r^{2}}} = 4g − 3 + β_{1} + β_{2} + κ + δ  |B(W → W_{r^{2}})| = 2(p − 1)(2γ + κ)
g_{W_{r^{p}}} = 2pg + 1 − 2p + p \frac{β_{1} + β_{2}}{2} +  |B(W → W_{r^{p}})| = 2pδ + 2κ

+(p − 1)(κ + γ)

g_{W_{Dp}} = 2g − 1 + \frac{β_{2} + κ + δ}{2}  |B(W_{s} → W_{D_{p}})| = (p − 1)(β_{1} + 2γ + κ)

|B(W_{r^{2}} → W_{D_{p}})| = 2β_{1}

g_{WD̃p} = 2g − 1 + \frac{β_{1} + κ + δ}{2}  |B(W_{r^{p}s} → WD̃_{p})| = (p − 1)(β_{2} + 2γ + κ)

|B(W_{r^{2}} → WD̃_{p})| = 2β_{2}

|B(W_{r^{p}s} → W_{0})| = β_{1} + pδ + κ

g_{W_{D_{2}}} = pg + \frac{p − 1}{2} (−2 + κ + γ+  |B(W_{s} → W_{0})| = β_{2} + pδ + κ
+ \frac{β_{1} + β_{2}}{2} )  |B(W_{r^{p}} → W_{0})| = β_{1} + β_{2}
g_{W_{r}} = 2g − 1 + \frac{β_{1} + β_{2}}{2}  |B(W_{r^{2}} → W_{r})| = 2δ + 2κ

|B(W_{r^{p}} → W_{r})| = 2(p − 1)(κ + γ)

|B(W_{D_{p}} → W_{D_{2p}})| = β_{2} + κ + δ |B(W_{r} → W_{D_{2p}})| = β_{1} + β_{2} |B(WD̃_{p} → W_{D_{2p}})| = β_{1} + κ + δ |B(W_{0} → W_{D_{2p}})| = (p − 1) ( \frac{β_{1}+β_{2}}{2} + β + γ)

Remark 7.2. It follows from Theorem 3.2 that there are 6 rational irreducible representations of D_{2p}: χ_{0}, χ_{1}, χ_{2}, χ_{3}, W_{1} = ⊕_{i=1}^{(p−1)/2} V_{2i−1} y W_{2} = ⊕_{i=1}^{(p−1)/2} V_{2i}.

In this case, Theorem 7.1 may be stated as follows.

Theorem 7.3. Let n = 2p with p an odd prime and let W be a curve with D_{2p} action.

Furthermore, let

B_{1} = P (W_{s}/W_{D_{p}}) ∩ P (W_{s}/W_{D_{2}}) .

Then

JW ∼ JW_{D_{n}} × P (W_{r}/W_{D_{2p}}) × P (W_{D_{p}}/W_{D_{2p}}) × P (WD̃_{p}/W_{D_{2p}})

× 2B_{1} × 2P (W_{D_{2}}/W_{D_{2p}})

with respective actions

χ_{0} × χ_{1} × χ_{2} × χ_{3} × W_{1} × W_{2} .

Furthermore,

B_{1}XXX is isogenous to P (W_{r^{p}s}/WD̃_{p}) ∩ P (W_{r^{p}s}/W_{D_{2}})
2P (W_{D_{1}}/W_{D_{2}}) is isogenous to P (W/W_{r^{p}})

and

P (W_{D_{p}}/W_{D_{2p}}) × P (WD̃_{p}/W_{D_{2p}}) is isogenous to P (W_{r^{2}}/W_{r}) .

But in this case we can obtain more information.

Proposition 7.4. By the theory of D_{p} developed in Section 5.1, there are natural isogenies as follows. (XXXX general even?)

2P (W_{s}/W_{D_{p}}) → P (W/W_{r^{2}})
2P (W_{r^{p} s}/WD̃_{p}) → P (W/W_{r^{2}})
2P (W_{D_{2}}/W_{D_{2p}}) → P (W_{r^{p}}/W_{r})

Proposition 7.5. By the theory of the Klein group developed in [Rec–Ro2], there are natural isogenies as follows. (XXXX general even?)

P (W_{D_{p}}/W_{D_{2p}}) × P (WD̃_{p}/W_{D_{2p}}) → P (W_{r^{2}}/W_{r})
P (W_{r}/W_{D_{2p}}) × P (WD̃_{p}/W_{D_{2p}}) → P (W_{r^{2}}/W_{D_{p}})
P (W_{D_{p}}/W_{D_{2p}}) × P (W_{r}/W_{D_{2p}}) → P (W_{r^{2}}/WD̃_{p})

Furthermore, particular to the case D_{2p} is the following result.

20 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

Theorem 7.6. (XXXX general even?)

There is a natural isogeny

h = (Nm f_{rs} ∘ f_{s}^{∗})|_{P (W_{s}/W_{Dp})} : P (W_{s}/W_{D_{p}}) → P (W_{rs}/WD̃_{p}) whose kernel has cardinality given as follows.

| ker(h)| = \begin{cases}p^{2g(4p−1)−6p+3+4pβ_{1}+2pβ_{2}+2pδ+(6p−2)κ+(8p−3)γ} & if γ > 1 and κ ≥ 2; \\ p^{2g(4p−1)−6p+2+4pβ_{1}+2pβ_{2}+2pδ+(6p−2)κ+(8p−2)γ} & if γ ≤ 1 and κ ≥ 2; \\ p^{2g(4p−1)−6p+2+4pβ_{1}+2pβ_{2}+2pδ+(6p−1)κ+(8p−3)γ} & if γ > 1 and κ ≤ 1; \\ p^{2g(4p−1)−2p+2+4pβ_{1}+2pβ_{2}+2pδ} & if γ = 0 and κ = 1; \\ p^{2g(4p−1)−8p+2+4pβ_{1}+2pβ_{2}+2pδ} & if κ = 0 and \\ & γ = 0 or 1. \end{cases} (7.19)

Proof. Denote by

h = (Nm f_{rs} ∘ f_{s}^{∗})|_{P (W_{s}/W_{Dp})} : P (W_{s}/W_{D_{p}}) → P (W_{rs}/WD̃_{p}) and consider the following commutative diagram

\xymatrix{ & A \ar[dr]_{Nm f_{rs}} \ar[rr]^{1+rs} & & B \\ P (W_{s}/W_{D_{p}}) \ar[rr]_{h} \ar[ur]^{f_{s}^{∗}} & & P (W_{rs}/WD̃_{p}) \ar@{-->}[ur]_{f_{rs}^{∗}}} (7.20)

where (see [Rec–Ro2])

A = {z ∈ JW^{⟨s⟩} : (1 + r^{2} + r^{4} + … + r^{2(p−1)})(z) = 0}^{0} and

B = {w ∈ JW^{⟨rs⟩} : (1 + r^{2} + r^{4} + … + r^{2(p−1)})(w) = 0}^{0} . > Note that

(1 + rs)_{|_{A}} = (1 + r)_{|_{A}} .

Consider the complex isotypical decomposition

T_{0}JW = χ_{0}^{m_{0}} ⊕ χ_{1}^{m_{1}} ⊕ χ_{2}^{m_{2}} ⊕ χ_{3}^{m_{3}} ⊕ V_{1} ⊕ … ⊕ V_{p−1} where V_{j} = V_{j}^{n_{j}} for each complex irreducible representation V_{j}.

Comparing it with the respective actions one obtains that

T_{0}A = (df_{s}^{∗})_{0}(T_{0}P (W_{s}/W_{D_{p}})) = ⨁_{i=1}^{p−1} V_{i}^{⟨s⟩}

and

T_{0}B = (df_{rs}^{∗})_{0}(T_{0}P (W_{rs}/WD̃_{p})) = ⨁_{i=1}^{p−1} V_{i}^{⟨rs⟩}

Furthermore, it is clear that we have

(1 + rs)(V_{i}^{⟨s⟩}) = V_{i}^{⟨rs⟩} for each i . (7.21)

and then it follows that

(1 + rs)(V_{1}^{⟨s⟩} ⊕ … ⊕ V_{p−1}^{⟨s⟩}) = V_{1}^{⟨rs⟩} ⊕ … ⊕ V_{p−1}^{⟨rs⟩};

Therefore h is an isogeny as claimed.

To compute the kernel of h we first look at

ker(1 + r)_{|_{A}} = {z ∈ A : z = −r(z)}

Claim 7.7. ker(1 + r)_{|_{A}} ⊆ A[p]

Proof of the claim: If z ∈ ker(1 + r)_{|_{A}} then

0 = (1 + r^{2} + r^{4} + … + r^{2(p−1)})(z) = pz

where the last equality holds since r^{2}(z) = z.

We know that

f_{s}^{∗}^{−1}(A[p]) = P (W_{s}/W_{D_{p}})[p] + ker f_{s}^{∗}

and

ker f_{s}^{∗} ⊆ JW_{s}[2] ;

since p and 2 are relatively prime, it follows that

f_{s}^{∗}^{−1}(A[p]) = P (W_{s}/W_{D_{p}})[p]

and therefore

ker h ⊆ P (W_{s}/W_{D_{p}})[p]

Similarly, ker f_{rs}^{∗} ⊆ JW_{rs}[2] implies that

ker(f_{rs}^{∗} ∘ h) = ker h

Therefore we have proven that

ker h = {x ∈ P (W_{s}/W_{D_{p}})[p] : rf_{s}^{∗}(x) + f_{s}^{∗}(x) = 0} ≈ K/L (7.22)

where

K = {x ∈ P (W_{s}/W_{D_{p}})[p] : r^{2}f_{s}^{∗}(x) = f_{s}^{∗}(x)}

and

L = {x ∈ P (W_{s}/W_{D_{p}})[p] : rf_{s}^{∗}(x) = f_{s}^{∗}(x)} .

We may apply the results of Theorem 5.4 to the subgroup D_{p} (dihedral of order 2p, with p odd) of D_{2p}, with α = κ + 2γ and β = β_{1}, to conclude the following.

K = \begin{cases}f_{s,D_{p}}^{∗}JW_{D_{p}}[p] & if α = 0 or 1; \\ f_{s,D_{p}}^{∗}JW_{D_{p}}[p] ⊕ ⊕_{j=2}^{α}F_{j}ℤ/pℤ & if α ≥ 2. \end{cases}

Therefore,

|K| = \begin{cases}p^{4pg_{W_{Dp}} +4pβ+(4p−2)α+2−4p} & if α = 0 or 1 \\ p^{4pg_{W_{Dp}} +4pβ+(4p−1)α+1−2p} & if α ≥ 2 \end{cases}

We will now compute

L = {x ∈ P (W_{s}/W_{D_{p}})[p] : rf_{s}^{∗}(x) = f_{s}^{∗}(x)}
= {x ∈ P (W_{s}/W_{D_{p}})[p] : f_{s}^{∗}(x) is D_{2p}– invariant}

First observe that

f_{s,D_{p}}^{∗} : JW_{D_{p}} → JW_{s}

is injective, since the p-fold cover f_{s,D_{p}} : W_{s} → W_{D_{p}} is not a Galois cover. Hence ker Nm f_{s,D_{p}} has only one connected component; i.e.,

P (W_{s}/W_{D_{p}}) = ker(Nm f_{s,D_{p}})

On the other hand, r descends to an involution

r̃: W_{D_{p}} → W_{D_{p}}

whose quotient gives the morphism

f_{D_{p},D_{2p}} : W_{D_{p}} → W_{D_{2p}}

and therefore

(f_{s}^{∗} ∘ f_{s,D_{p}}^{∗})JW_{D_{p}}^{⟨r̃⟩} ⊆ JW^{D_{2p}}

On the other hand, since Nm f_{s,D_{p}} ∘ f_{s,D_{p}}^{∗} = p_{JW_{Dp}}, we have that

f_{s,D_{p}}^{∗}(JW_{D_{p}}^{⟨r̃⟩}[p]) ⊆ ker(Nm f_{s,D_{p}}) = P (W_{s}/W_{D_{p}}) .

Therefore

L = f_{s,D_{p}}^{∗}(JW_{D_{p}}^{⟨r̃⟩}[p]) + part coming from the ramification of f_{s,D_{p}}

Since p is an odd prime, it follows from Corollary 3.2 in [Rec–Ro2] that

JW_{D_{p}}^{⟨r̃⟩}[p] = f_{D_{p},D_{2p}}^{∗}(JW_{D_{2p}}[p])

To compute the part coming from the ramification, note that we obtain p–torsion points from the ramification of f_{s,D_{p}} : W_{s} → W_{D_{p}} only from points of type κ and γ.

So let P_{1}, …, P_{κ} denote the κ points in W_{s} (i.e., each P_{i} is a total ramification point of f_{s,D_{p}} such that r̃f_{s,D_{p}}(P_{i}) = f_{s,D_{p}}(P_{i})) and let Q_{1}, …, Q_{2γ} denote the γ points in W_{s} (i.e., Q_{j} is a total ramification point of f_{s,D_{p}} such that r̃f_{s,D_{p}}(Q_{j}) = f_{s,D_{p}}(Q_{j+1}) for each j = 1, 3, …, 2γ − 1).

For each i = 2, 3, …, κ and for each j = 3, 5, …, 2γ − 1 choose m_{i} and n_{j} in JW_{D_{p}}^{⟨r̃⟩} such that

m_{i}^{⊗p} = O_{W_{Dp}}(f_{s,D_{p}}(P_{1}) − f_{s,D_{p}}(P_{i}))

and

n_{j}^{⊗p} = O_{W_{Dp}}(f_{s,D_{p}}(Q_{1}) + f_{s,D_{p}}(Q_{2}) − f_{s,D_{p}}(Q_{j}) − f_{s,D_{p}}(Q_{j+1}))

and define

F_{i} = O_{W_{s}}(P_{i} − P_{1}) ⊗ f_{s,D_{p}}^{∗}(m_{i})  i = 2, 3, …, κ

and

G_{j} = O_{W_{s}}(Q_{j} + Q_{j+1} − Q_{1} − Q_{2}) ⊗ f_{s,D_{p}}^{∗}(n_{j}) j = 3, 5, …, 2γ − 1

Then

L = f_{s,D_{p}}^{∗} ∘ f_{D_{p},D_{2p}}^{∗}(JW_{D_{2p}}[p]) ⊕ ⊕_{i=2}^{κ}F_{i} ℤ/pℤ ⊕ ⊕_{j=1}^{γ−1}G_{2j+1} ℤ/pℤ

Observe that there is no relation among the F_{i}’s and G_{j}’s because we have already taken away some of the elements of P (W_{s}/W_{D_{p}})[p] coming from the ramification.

It follows that, if we let g denote the genus of W_{D_{2p}},

(7.23) |L| = \begin{cases}p^{2g−2+κ+γ} & if γ > 1 and κ ≥ 2; \\ p^{2g−1+κ} & if γ ≤ 1 and κ ≥ 2; \\ p^{2g−1+γ} & if γ > 1 and κ ≤ 1; \\ p^{2g} & if γ ≤ 1 and κ ≤ 1. \end{cases}

Remark 7.8. Note that B_{1} is the orthogonal complement in P (W_{s}/W_{D_{2}}) of f_{s,D_{p}}^{∗}(P (W_{D_{p}}/W_{D_{2p}})) with respect to the polarization induced from the canonical one on JW_{s}.

This subvariety may be described as follows.

f_{s}^{∗}(B_{1}) = {w ∈ JW^{⟨s⟩} : (1 + r)(w) = 0, ∑_{j=0}^{p−1} r^{2j}(w) = 0}^{0}

Even though it is not a Prym variety of some intermediate cover, in the sense of being the orthogonal complement of a Jacobian inside another Jacobian, it is an analog, in the sense of being the orthogonal complement of a Prym inside another Prym.

24 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

Appendix A.

In this section we prove Proposition 5.2.

Proof. Since (see [Rec–Ro2])

P (W/W_{r}) = {w ∈ JW : (∑_{j=0}^{p−1} r^{j}) w = 0}^{0}

and

f_{k}^{∗}(P (W_{k}/W_{D_{p}})) = {z ∈ JW^{⟨k⟩} : (∑_{j=0}^{p−1} r^{j}) z = 0}^{0}

it is clear that both A = f_{k}^{∗}(P (W_{k}/W_{D_{p}})) and r(A) are contained in P (W/W_{r}) and therefore

φ_{k}(P (W_{k}/W_{D_{p}}) × P (W_{k}/W_{D_{p}})) ⊆ P (W/W_{r}) .

Since f_{k}^{∗}|_{P (W_{k}/W_{Dp})} : P (W_{k}/W_{D_{p}}) → A is an isomorphism (by Corollary 4.4), in order to describe the kernel of φ_{k} it is enough to study the kernel of

A × A ⟶ P (W/W_{r})
(x_{1}, x_{2}) −−→ x_{1} + r(x_{2})

and we do this next.

Given (x_{1}, x_{2}) ∈ A × A such that

x_{1} + r(x_{2}) = 0 (1.24)

and using that both x_{1} and x_{2} are fixed under k, we obtain

0 = k(x_{1} + r(x_{2})) = x_{1} + kr(x_{2}) = x_{1} + r^{p−1}(x_{2}) . (1.25)

Combining (1.24) and (1.25) it follows that

r^{2}(x_{1}) = x_{1}

That is, x_{1} is invariant under ⟨r^{2}⟩ = ⟨r⟩. But then x_{1} in A ⊆ P (W/W_{r}) implies that 0 = (1 + r + … + r^{p−1})(x_{1}) = p x_{1}; i.e., x_{1} is in A[p] and it is invariant under D_{p}; furthermore, x_{2} = −r^{p−1}(x_{1}) = −x_{1}. Thus (5.8) and (5.9) are proved.

To prove i) note that f_{k,D_{p}}^{∗} is injective, by Proposition 4.2 i), hence ker Nm f_{k,D_{p}} has only one connected component (see [Rec–Ro2]) and P (W_{k}/W_{D_{p}}) = ker Nm f_{k,D_{p}}. Now (f_{k}^{∗} ∘ f_{k,D_{p}}^{∗})(x) is D_{p}−invariant for every x ∈ JW_{D_{p}} and therefore f_{k,D_{p}}^{∗}(JW_{D_{p}}[p]) is contained in K, since Nm f_{k,D_{p}} ∘ f_{k,D_{p}}^{∗} = p_{JW_{Dp}} is multiplication by p in JW_{D_{p}}.

We consider first the case α = 0; we will show that then K = f_{k,D_{p}}^{∗}(JW_{D_{p}}[p]).

Since the Galois cover W →^{f_{r}} W_{r} is étale, it follows that JW^{⟨r⟩} = f_{r}^{∗}(JW_{r}) or, equivalently,

ker(1 − r) = Im (∑_{j=0}^{p−1} r^{j})

If we observe that

K ≅ f_{k}^{∗}(K) = f_{k}^{∗}(P (W_{k}/W_{D_{p}})) ∩ ker(1 − r)

= f_{k}^{∗}(P (W_{k}/W_{D_{p}})) ∩ Im (∑_{j=0}^{p−1} r^{j})

then we obtain that

for any z ∈ K there exists w ∈ JW such that f_{k}^{∗}(z) = (∑_{j=0}^{p−1} r^{j}) (w) .

Now let x = Nm γ(w), where γ = f_{k,D_{p}} ∘ f_{k}. From Nm γ(w) = Nm γ(g(w)) for any g ∈ D_{p}, we obtain

p x = p Nm γ(w) = Nm γ (∑_{j=0}^{p−1} r^{j}(w)) = Nm γ(f_{k}^{∗}(z))

= Nm f_{k,D_{p}} ∘ Nm f_{k}(f_{k}^{∗}(z)) = Nm f_{k,D_{p}}(2z) = 2 Nm f_{k,D_{p}}(z) = 0

and hence x is in JW_{D_{p}}[p].

Therefore f_{k,D_{p}}^{∗} (\frac{p−1}{2} x) is in K and it follows that

z + f_{k,D_{p}}^{∗} (\frac{p − 1}{2} x) is in K .

But then

f_{k}^{∗} (z + f_{k,D_{p}}^{∗} (\frac{p − 1}{2} x)) = f_{k}^{∗}(z) + \frac{p − 1}{2} f_{k}^{∗} ∘ f_{k,D_{p}}^{∗}(x)

= f_{k}^{∗}(z) + \frac{p − 1}{2} γ^{∗} ∘ Nm γ(w)

= f_{k}^{∗}(z) + \frac{p − 1}{2} ∑_{g∈D_{p}} g(w)

= f_{k}^{∗}(z) + \frac{p − 1}{2} (∑_{j=0}^{p−1} r^{j}(w) + k ∑_{j=0}^{p−1} r^{j}(w))

= f_{k}^{∗}(z) + \frac{p − 1}{2}(f_{k}^{∗}(z) + kf_{k}^{∗}(z))

= f_{k}^{∗}(z) + (p − 1)f_{k}^{∗}(z)

= p f_{k}^{∗}(z) = 0 .

Therefore

z = −f_{k,D_{p}}^{∗} (\frac{p − 1}{2} x) is in f_{k,D_{p}}^{∗}(JW_{D_{p}})[p]

and we have proven that K = f_{k,D_{p}}^{∗}(JW_{D_{p}})[p] for α = 0.

Now assume α ≥ 1 and denote by {P_{1}, …, P_{α}} ⊆ W_{k} the set of total ramification points of f_{k,D_{p}} : W_{k} → W_{D_{p}}. If we denote

f_{k}^{∗}(P_{j}) = Pj^{′} + Pj^{′′}

for j ∈ {1, …, α} with Pj^{′} and Pj^{′′} in W, then r(Pj^{′}) = Pj^{′}, r(Pj^{′′}) = Pj^{′′}

and k(Pj^{′}) = Pj^{′′}.

Let m be such that 2m ≥ g_{W} (the genus of W) and consider the Abel-Jacobi map given as follows.

φ : W^{(2m)} → Pic^{2m} W → Pic^{0} W
D → O_{W} (D) → O_{W} (D − m(P1^{′} + P1^{′′}))

Let z ∈ K. Then f_{k}^{∗}(z) ∈ Pic^{0} W and (since 2m ≥ g_{W}) we have that φ^{−1}(f_{k}^{∗}(z)) = |f_{k}^{∗}(z)| is non empty.

Since r(f_{k}^{∗}(z)) = f_{k}^{∗}(z), r induces a projective transformation on |f_{k}^{∗}(z)| which has a fixed point D in |f_{k}^{∗}(z)|, because r generates a cyclic group (see [M1]).

As D is an r-invariant effective divisor, it must be of the form

D = ∑_{j=1}^{α}(n_{j}Pj^{′} + m_{j}Pj^{′′}) + ∑_{t=1}^{T}(∑_{j=1}^{p−1} r^{j})w_{t}

for some points w_{t} ∈ W and we also have

f_{k}^{∗}(z) = O_{W} (D − m(P1^{′} + P1^{′′})) .

But k(f_{k}^{∗}(z)) = f_{k}^{∗}(z) and hence

f_{k}^{∗}(2z) = O_{W} (∑_{j=1}^{α} s_{j}(Pj^{′} + Pj^{′′}) + ∑_{t=1}^{T}( ∑_{g∈D_{p}} g)w_{t} − 2m(P1^{′} + P1^{′′})) .

Applying Nm f_{k} to the last equality we obtain

4z = \begin{cases}O_{W_{k}} (∑_{j=2}^{α} 2s_{j}(P_{j} − P_{1})+ \\ + ∑_{t=1}^{T} [2f_{k,D_{p}}^{∗}(γ(w_{t})) − 2pP_{1}]) & if α ≥ 2 \\ O_{W_{k}} (∑_{t=1}^{T} [2f_{k,D_{p}}^{∗}(γ(w_{t})) − 2pP_{1}]) & if α = 1 \end{cases}

which we may rewrite as follows

4z = \begin{cases}O_{W_{k}} (∑_{j=2}^{α} b_{j}(P_{j} − P_{1})) ⊗ f_{k,D_{p}}^{∗}(L) & if α ≥ 2 \\ f_{k,D_{p}}^{∗}(L) & if α = 1 \end{cases}

with 0 ≤ b_{j} ≤ p − 1 and L ∈ Pic^{0} W_{D_{p}}.

Since 4z is a point of order p and since f_{k,D_{p}}^{∗} is injective, from the last expression for 4z we obtain that

L^{p} = \begin{cases}O_{W_{Dp}} (∑_{j=2}^{α} b_{j}(f_{k,D_{p}}(P_{1}) − f_{k,D_{p}}(P_{j}))) & if α ≥ 2 \\ O_{W_{Dp}} & if α = 1 \end{cases}

If α = 1 we have just shown that K = f_{k,D_{p}}^{∗}(JW_{D_{p}})[p]. If α ≥ 2 then we have

4z ⊗ F_{2}^{−b_{2}} ⊗ … ⊗ F_{α}^{−b_{α}} = f_{k,D_{p}}^{∗}(L ⊗ m_{2}^{−b_{2}} ⊗ … ⊗ m_{α}^{−b_{α}})

with

(L ⊗ m_{2}^{−b_{2}} ⊗ … ⊗ m_{α}^{−b_{α}})^{p} = O_{W_{Dp}} .

Hence 4z = F_{2}^{b_{2}} ⊗ … ⊗ F_{α}^{b_{α}} ⊗ f_{k,D_{p}}^{∗}(t), with t ∈ Pic^{0} W_{D_{p}} such that t^{p} = O_{W_{Dp}}; also, any multiple of 4z has a similar expression, including z, since the subgroups of K generated by 4z and by z coincide.

We now prove that the b_{j} are unique. Assume there is a non-trivial combination of the F_{j}’s with coefficients

in ℤ/pℤ that is trivial modulo f_{k,D_{p}}^{∗}(JW_{D_{p}})[p].

28 ANGEL CAROCCA, SEVÍN RECILLAS, AND RUBÍ E. RODRÍGUEZ

Then, after reordering the P_{i}’s if needed, we may assume there is t ∈ Pic^{0} W_{D_{p}} such that t^{⊗p} = O_{W_{Dp}} and such that

⊗_{i=1+s}^{s_{1}}F_{i} ⊗_{i=1+s_{1}}^{s_{2}} F_{i}^{⊗2} ⊗ … ⊗_{i=1+s_{p−2}}^{s_{p−1}} F_{i}^{⊗(p−1)} = f_{k,D_{p}}^{∗}(t) It follows from (5.7) that then

O_{W_{k}} ( ∑_{j=s+1}^{s+s_{1}} P_{j} + 2 ∑_{j=s+s_{1}+1}^{s+s_{1}+s_{2}} P_{j} + …

+(p − 1) ∑_{j=s+s_{1}+...+s_{p−2}+1}^{s+s_{1}+...+s_{p−2}+s_{p−1}} P_{j} − (s_{1} + 2s_{2} + … + (p − 1)s_{p−1})P_{1})

= f_{k,D_{p}}^{∗} (t ⊗ ⨂_{j=s+1}^{s+s_{1}}(m_{j})^{−1} ⊗ ⨂_{j=s+s_{1}+1}^{s+s_{1}+s_{2}}(m_{j})^{−⊗2} ⊗ …

⊗ ⨂_{j=s+s_{1}+...+s_{p−2}+1}^{s+s_{1}+...+s_{p−2}+s_{p−1}}(m_{j})^{−⊗(p−1)})

Therefore, if we let s_{1} + 2s_{2} + … + (p − 1)s_{p−1} = pm − ε , with ε ∈ {0, 1, …, p − 1}

we obtain

O_{W_{k}} ( ∑_{j=s+1}^{s+s_{1}} P_{j} + 2 ∑_{j=s+s_{1}+1}^{s+s_{1}+s_{2}} P_{j} + … + (p − 1) ∑_{j=s+s_{1}+...+s_{p−2}+1}^{s+s_{1}+...+s_{p−2}+s_{p−1}} P_{j}+

+εP_{1})

= f_{k,D_{p}}^{∗} (O_{W_{Dp}} (mf_{k,D_{p}}(P_{1})) ⊗ t ⊗ ⨂_{j=s+1}^{s+s_{1}}(m_{j})^{−1}⊗

⨂_{j=s+s_{1}+1}^{s+s_{1}+s_{2}}(m_{j})^{−⊗2} ⊗ … ⊗ ⨂_{j=s+s_{1}+...+s_{p−2}+1}^{s+s_{1}+...+s_{p−2}+s_{p−1}}(m_{j})^{−⊗(p−1)})

Therefore, if we let

L = O_{W_{Dp}} (mf_{k,D_{p}}(P_{1})) ⊗ t ⊗ ⨂_{j=s+1}^{s+s_{1}}(m_{j})^{−1} ⊗ ⨂_{j=s+s_{1}+1}^{s+s_{1}+s_{2}}(m_{j})^{−⊗2} ⊗ …

… ⊗ ⨂_{j=s+s_{1}+...+s_{p−2}+1}^{s+s_{1}+...+s_{p−2}+s_{p−1}}(m_{j})^{−⊗(p−1)}

then L ∈ Pic^{m}(W_{D_{p}}) and we have just seen that

f_{k,D_{p}}^{∗}(L) = O_{W_{k}}( ∑_{j=s+1}^{s+s_{1}} P_{j} + 2 ∑_{j=s+s_{1}+1}^{s+s_{1}+s_{2}} P_{j} + …

… + (p − 1) ∑_{j=s+s_{1}+...+s_{p−2}+1}^{s+s_{1}+...+s_{p−2}+s_{p−1}} P_{j} + εP_{1})

But we also have

L^{⊗p} = O_{W_{Dp}} ( ∑_{j=s+1}^{s+s_{1}} f_{k,D_{p}}(P_{j}) + 2 ∑_{j=s+s_{1}+1}^{s+s_{1}+s_{2}} f_{k,D_{p}}(P_{j}) + …

… + (p − 1) ∑_{j=s+s_{1}+...+s_{p−2}+1}^{s+s_{1}+...+s_{p−2}+s_{p−1}} f_{k,D_{p}}(P_{j}) + εf_{k,D_{p}}(P_{1}))

These conditions on L imply that f_{k,D_{p}} : W_{k} → W_{D_{p}} is a Galois cover with group ℤ/pℤ (and branched over ∑_{j=s+1}^{s+s_{1}} f_{k,D_{p}}(P_{j})+2 ∑_{j=s+s_{1}+1}^{s+s_{1}+s_{2}} f_{k,D_{p}}(P_{j})+ … + εf_{k,D_{p}}(P_{1})), hence a contradiction.

References

[C–R] A. Carocca and R. E. Rodríguez. Little groups. [L–B] H. Lange and C. Birkenhake, Complex Abelian Varieties, Grundlehren der

Mathematischen Wissenschaften 302. Springer-Verlag, Berlin, 1992.

[M1] D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research, Bombay: Oxford University Press, 1970.

[Rec–Ro2] S. Recillas and R. E. Rodríguez, Prym varieties and fourfold covers.

Preprint 2000.

[Ri] J. Ries,

[SA] A. Sánchez-Argáez, Acciones des grupo A_{5} en variedades Jacobianas. Aport. Mat. Com. 25, (1999), 99-108.

[S] J. P. Serre,

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

  E-mail address: acarocca@mat.puc.cl

Instituto de Matemáticas, UNAM, Nicolás Romero 150, Morelia, Mich., 58000, México and CIMAT, Callejón Jalisco s/n, Valenciana, Guanajuato, México

  E-mail address: sevinrec@servidor.dgsca.unam.mx

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

  E-mail address: rubi@mat.puc.cl