Prym varieties of \'etale covers of hyperelliptic curves

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by  Herbert Lange

PRYM VARIETIES OF E´TALE COVERS OF HYPERELLIPTIC
CURVES
HERBERT LANGE AND ANGELA ORTEGA
Abstract. ItiswellknownthatthePrymvarietyofan´etalecycliccoveringofahyper-
6
1 ellipticcurveisisogenoustotheproductoftwoJacobians. Moreover,ifthedegreeofthe
0 covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism.
2 We compute the degree of this isogeny in the remaining cases and show that only in the

n case of coverings of degree 4 it is an isomorphism.
a
J

5
1
1. Introduction

]
G Let H denote a hyperelliptic curve of genus g ≥ 2 and f : X → H an ´etale cyclic
A covering of degree n ≥ 2. Let σ denote the automorphism of X defining f. It is well
. known that the hyperelliptic involution of H lifts to an involution τ on X. Then σ and τ
h
t generate the dihedral group D of order 2n. The Prym variety P(f) of f is defined as the
a n
m connected component containing 0 of the kernel of the norm map Nmf : JX → JH of f.
[ For any element α ∈ Dn we denote by Xα the quotient of X by the subgroup generated

by α. The Jacobians JX and JX are abelian subvarieties of the Prym variety P(f) so
1 τ τσ
v the the addition map
2
a : JX ×JX → P(f)
8 τ τσ
0 is well defined. Mumford showed in [3] that for n = 2 the map a is an isomorphism. J.
4
Ries proved the same for any odd prime degree n ([6]). The second author generalized
0
. this statement largely to show that a is an isomorphism for any odd number and, more
1
0 important, for any even n ≡ 2 mod 4 ([4]).
6 It is an obvious question whether this is true for any degree n. In fact, using the ac-
1
tion of the group D on P(f) and a little representation theory, it is not difficult to see
: n
v
that a is an isogeny. For more precise results on the decomposition of P(f) up to isogeny
i
X
see [2]. It isthe aimofthis noteto compute thedegree of theisogeny a. Our mainresult is
r
a
Theorem 4.1 Let f : X → H be as above with n = 2rm, r ≥ 2 and m odd. Then a is
an isogeny of degree
dega = 2[(2r−r−1)m−(r−1)](g−1).
So a is an isomorphism for odd n, for n ≡ 2 mod 4, and for n = 4. The proof proceeds
by induction on the exponent r, the beginning of the induction being Ortega’s theorem
in [4].
1991 Mathematics Subject Classification. 14H40, 14H30.
Key words and phrases. Prym variety, Prym map.
The second author was supported by Deutsche Forschungsgemeinschaft,SFB 647.
1
2 HERBERTLANGEAND ANGELA ORTEGA
2. Preliminaries
Let H be a smooth hyperelliptic curve of genus g with hyperellptic covering π : H → P1
andf : X → H beacyclic´etalecoveringofdegreen ≥ 2. SoX isofgenusg = n(g−1)+1
X
and the Prym variety P := P(f) of f is an abelian variety of dimension
(2.1) dimP = (n−1)(g −1).
The canonical polarization of JX induces a polarization on P of type
( 1,...,1 ,n,...,n).
| {z } | {z }
(n−2)(g−1) g−1
The hyperelliptic involution of H lifts to an involution τ on X which together with the
automorphism σ defined by the covering f generate the dihedral group
D := hσ,τ | σn = τ2 = (στ)2 = 1i.
n
The automorphism σ induces an automorphism of the same order n of P compatible with
the polarization, which we denote by the same letter. Each eigenvalue ζi,i = 1,...,n−1
n
(with ζ a fixed primitive n-th root of unity) of the induced map on the tangent space
n
T P occurs with multiplicity g −1.
0
In the whole paper we write
n = 2rm
with r ≥ 0 and m odd.
In any case the group D admits n involutions, namely τσν for ν = 0,...n − 1. For
n
n
odd n, these are all the involutions. For even n, there is one more, namely σ2. For odd n
all involutions are conjugate to τ and for even n there are 3 conjugacy classes. They are
represented by
τ, τσm and σn/2.
For any subgroup S ⊂ D and for any element α ∈ D we denote by
n n
X := X/S and X := X/hαi
S α
the corresponding quotients.
Consider the following diagram (for odd n only the left hand side of the diagram, since
in this case m = n, so both sides are the same).
X
❋
2}}⑤:⑤1⑤⑤⑤⑤⑤⑤f (cid:15)(cid:15) n:1❋❋❋❋2❋:❋1❋""
X H X
τ τσm
❇
n❇:1❇❇❇❇❇ π (cid:15)(cid:15) 2:||②1②②②②n②:②1②
P1
Let W denote the set of 2g+2 branch points of the hyperelliptic covering π. Then denote
for arbitrary n,
(cid:12) (cid:12)
s := (cid:12){x ∈ W | (πf)−1(x) contains a fixed point of τ}(cid:12)
0
and
(cid:12) (cid:12)
s := (cid:12){x ∈ W | (πf)−1(x) contains a fixed point of τσm}(cid:12).
1
3
According to [4, Proposition 2.4] the Jacobians JX and JX are contained in the
τ τσm
Prym variety P. With these notations the following theorem is proved in [4].
Theorem 2.1. (a) For odd n the map
ψ : (JX )2 → P, (x,y) 7→ x+σ(y)
τ
is an isomorphism.
(b) For n = 2m ≡ 2 mod 4 the map
ψ : JX ×JX → P, (x,y) 7→ x+y
τ τσm
is an isomorphism. Moreover,
s s
g(X ) = m(g −1)+1− 0 and g(X ) = m(g −1)+1− 1.
τ τσm
2 2
In particular s and s are even.
0 1
It is the aim of this paper to study the map ψ in the remaining cases n = 2rm with
r ≥ 2. So in the sequel we assume r ≥ 2. We first need some preliminaries.
There are 2 non-conjugate Kleinian subgroups of D , namely
n
K = {1,σn/2,τ,τσn/2} and K = {1,σn/2,τσm,τσm+n/2}.
τ τσm
Moreover, consider the dihedral subgroups of order 8,
T = hτ,σn/4i and T = hτσm,σn/4i.
τ τσm
Note that for r ≥ 3 the groups T and T are non-conjugate, whereas
τ τσm
(2.2) T = T for r = 2,
τ τσm
sincethen n = mandhτ,σmi = hτσm,σmi. Inanycasewehavethefollowingcommutative
4
diagram
(2.3) X
ayysτsσsnss/2sssssssf1 (cid:15)(cid:15) 2:❑1❑❑❑❑❑❑a❑τ❑σ❑m❑❑%%
X X X
τσn/2 σn/2 τσm
bτσn/2 (cid:15)(cid:15) cyytτtσtntt/t2tttttf2 (cid:15)(cid:15) 2:1❑❑❑❑❑❑cτ❑❑σ❑m❑❑%% (cid:15)(cid:15) bτσm
X X X
dτσn/2 K(cid:15)(cid:15) τ eyytτtσtntt/t2ttttt σn/4❏❏❏❏❏❏e❏τ❏σ❏m❏❏%% K(cid:15)(cid:15)τdστmσm
XTτ✽ f3 n4:1 XTτσm
✽ ✆
✽ ✆
✽ ✆
✽ ✆
✽ ✆
✽ ✆
✽✽ (cid:15)(cid:15) ✆✆
✽✽ X = H ✆✆
n4:1 ✽✽✽ σ ✆✆✆ n4:1
✽ ✆
✽ ✆
✽ π ✆
✽ ✆
✽(cid:28)(cid:28) (cid:15)(cid:15) (cid:2)(cid:2)✆✆
P1
4 HERBERTLANGEAND ANGELA ORTEGA
In the sequel we use the following notation: if an involution of the group D induces
n
an involution on a curve of the diagram, we denote the induced involution by the same
letter. In order to compute the genera of the curves in the diagram, we need the following
lemma.
Lemma 2.2. Suppose that the dihedral group D = hσ,τi of order 2n with n ≥ 3 acts on
n
a finite set S of n elements such that the subgroup hσi acts transitively on S.
(a) If n is odd, τ admits exactly one fixed point,
(b) for even n, either τ acts fixed-point free or admits exactly 2 fixed points.
(c) for even n, exactly one of the involutions τ and τσ admits a fixed point.
Proof. Let S = {x ,...x }. We may enumerate the x in such a way that σ(x ) = x
1 n i i i+1
for i = 1,...,n where x = x . If n is odd, then clearly τ admits a fixed point. So
n+1 1
in any case we may assume that x is a fixed point of τ. Then we have inductively for
1
i = 1,...⌊n+3⌋,
2
(2.4) τ(x ) = x .
i n+2−i
In fact, the induction step is τ(x ) = τσ(x ) = σ−1τ(x ) = σ−1(x ) = x .
i i−1 i−1 n−i+3 n−i+2
Hence for odd n the involution τ admits no further fixed point and for even n τ admits
exactly one additional fixed point, namely x . This gives (a) and (b).
n+2
2
(c): Suppose n is even and τ admits a fixed point, say x . Hence we have (2.4) for all
1
i. This implies
τσ(x ) = τ(x ) = x .
i i+1 n+1−i
and τσ acts fixed point free. Conversely, suppose τ acts fixed point free. Suppose that
τ(x ) = x for some i ≥ 2. Then σ1−iτ(x ) = x . So σ1−iτ admits a fixed point and thus
1 i 1 1
cannot be equivalent to τ. Hence τσ is equivalent to σ1−iτ and admits a fixed point. (cid:3)
Lemma 2.3. Suppose n = 2rm with m odd and r ≥ 2. Then
(i)
s +s = 2g +2 with s ,s ≥ 2 even;
0 1 0 1
(ii) for r = 2, X → X and X → X are ramified exactly at 2g +2 points.
σn/4 Tτ σn/4 Tτσm
(iii) X → X and X → X as well as X → X , if r ≥ 3, are ramified exactly
τ σn/2 Kτ σn/4 Tτ
at 2s points. X → X and X → X as well as X → X , if r ≥ 3, are
0 τσm σn/2 Kτσm σn/4 Tτσm
ramified exactly at 2s points.
1
Proof. Thefixedpointsofτ andτσm lieover the2g+2Weierstrass pointsofH. Moreover,
accordingtoLemma2.2. overeachWeierstrasspointofH exactlyoneofτ andτσm admits
a fixed point. This gives the first assertion of (i). The evenness of s and s follows from
0 1
the Hurwitz formula. Now s = 0 means that τ acts fixed-point free. Since also σ acts
0
fixed-point free, so does τσm which means s = 0. But this contradicts the equation
1
s +s = 2g +2. Hence s ,s ≥ 2.
0 1 0 1
If x is a Weierstrass point of H and τ admits a fixed point over x, then D acts on the
n
fibre f−1(x). Similarly, the group D = hσn/2,τi acts on the fibre (f ◦f )−1(x) and the
n/2 3 2
group D = hσn/4,τi acts of the fibre f−1(x). Hence Lemma 2.2 implies (ii), since in
n/4 3
these cases the order of the fibre is even, and (iii), since in this case the order of the fibre
(cid:3)
is odd.
5
By checking the ramification of the maps in diagram (2.3) we immediately get from
Lemma 2.3 the following corollaries.
Corollary 2.4. All vertical left and right hand maps are ramified.
Corollary 2.5. If n = 2rm with m odd and r ≥ 2, then
n n
g(X) = n(g −1)+1 g(X ) = (g −1)+1 g(X ) = (g −1)+1;
σn/2 2 σn/4 4
n s n s
0 1
g(X ) = (g −1)+1− g(X ) = (g −1)+1− ;
τσn/2 2 2 τσm 2 2
n s n s
0 1
g(X ) = (g −1)+1− g(X ) = (g −1)+1− ;
Kτ 4 2 Kτσm 4 2
and for r ≥ 3,
n s n s
0 1
g(X ) = (g −1)+1− g(X ) = (g −1)+1− .
Tτ 8 2 τσm 8 2
For r = 2,
1
g(X ) = g(X ) = (m−1)(g −1).
Tτ τσm 2
Proof. All assertions follow from the Hurwitz formula. For the first line of assertions we
use the fact that f is ´etale. For the other formulas we use Lemma 2.3(ii) and (iii). (cid:3)
The following lemma is well known. In fact, it is an easy consequence of [1, Proposition
11.4.3] and [1, Corollary 12.1.4].
Lemma 2.6. Let g : Y → Z be a covering of smooth projective curves of degree d ≥ 2.
The addition map
g∗JZ ×P(Y/Z) → JY
is an isogeny of degree
|JZ[d]|
|g∗JZ ∩P(Y/Z)| = .
|kerg∗|2
We need a result on curves with an action of the Klein group. Let Y be a curve with
an action of the group
V = hr,s | r2 = s2 = (rs)2 = 1i.
4
Then we have the following diagram
(2.5) Y
~~⑥a⑥s⑥⑥⑥⑥⑥⑥ (cid:15)(cid:15) a❇r❇❇❇❇❇a❇r❇s!!
Y Y Y
s r rs
❆❆❆❆❆❆❆❆ (cid:15)(cid:15) }}⑤⑤⑤⑤⑤⑤⑤⑤
Z
with Y := Y/hvi for any v ∈ V and Z = Y/V . The following theorem is a special case
v 4 4
of [5, Theorem 3.2].
6 HERBERTLANGEAND ANGELA ORTEGA
Proposition 2.7. Suppose a is ´etale, that a respectively a are ramified at 2α > 0,
r s rs s
respectively 2α > 0 points and Z is of genus g(Z). Then P(Y /Z) and P(Y /Z) are
rs s rs
subvarieties of P(Y/Y ) and the addition map
r
φ : P(Y /Z)×P(Y /Z) → P(Y/Y )
r s rs r
is an isogeny of degree 22g(Z).
3. A degree computation
As above, let n = 2rm with m odd and r ≥ 2. Again we consider a curve X with
action of the dihedral group D := hσ,τ | σn = τ2 = (στ)2 = 1i. With the notation as in
n
Section 2 we have diagram (2.3) and s ,s ≥ 2. Then, apart from f ,f and f , all the
0 1 1 2 3
maps in diagram (2.3) are ramified. So the pullbacks of the corresponding Jacobians are
embeddings. Recall that P(f) denotes the Prym variety of the covering f.
We consider the isogenies
h := Nma ◦a∗ : P(b ) −→ P(b ).
τσm τσn/2 τσn/2 τσm
and
h′ := Nma ◦a∗ : P(b ) −→ P(b ).
τσn/2 τσm τσm τσn/2
Let
A := a∗ (P(b )) and B := a∗ (P(b ))
τσn/2 τσn/2 τσm τσm
be subvarieties of JX. Now τ (respectively σn/2) induces an involution on X (respec-
τσn/2
tively X ), which we denote by the same letter. Thus the Prym variety P(b ) is
τσm τσn/2
Ker(1+τ)0 and P(b ) = Ker(1+σn/2)0. Hence we have (for example by [5, Corollary
τσm
2.7]),
A = {z ∈ JXhτσn/2i | z +τz = 0}0, B = {w ∈ JXhτσmi | w+σn/2w = 0}0.
Moreover, as in [5], there is a commutative diagram:
(3.1) A 1+τσm // B 1+τσn/2 // A
a∗τ③σ③n③/③2③③③③③③③③==Nm❇a❇τ❇σ❇m❇❇❇❇❇❇❇❇!! a⑤∗τ⑤σ⑤m⑤⑤⑤⑤⑤⑤⑤⑤⑤N== ma❉τ❉σ❉n❉❉/❉2❉❉❉❉❉❉!! a∗τ③σ③n③/③2③③③③③③③③==
P(b ) // P(b ) // P(b )
τσn/2 h τσm h′ τσn/2
Lemma 3.1. For any n = 2rm with m odd and r ≥ 2 we have
|Kerh| = |Ker(1+τσm) |
|A
and
Ker(1+τσm) = (JX[2])hτ,σmi.
A
Proof. The first assertion follows from diagram (3.1), since a∗ : P(b ) → A and
τσn/2 τσn/2
a∗ : P(b ) → B areisomorphisms. Forthelast assertionnotethat z ∈ Ker(1+τσm)
τσm τσm |A
if and only if
τσn/2z = z, τz = −z τσm(z) = −z,
which implies that σmz = z. So
z = τσn/2z = τσ2r−1mz = τ(σm)2r−1(z) = τz = −z,
7
then z ∈ A[2].
Therefore z ∈ Ker(1+τσm) if and only if z ∈ JX[2] such that τz = z and σmz = z
|A
(cid:3)
which was to be shown.
The following proposition is a generalization of a special case of [5, Theorem 4.1,(ii)].
Proposition 3.2. For every n = 2rm with m odd and r ≥ 2, we have
degh = 2(m−1)(g−1)+s1−2.
Proof. The proof is by induction on the exponent r ≥ 2. Suppose first r = 2, i.e. n = 4m.
Consider the curve X with the action of the dihedral subgroup
D := hσm,τi ⊂ D .
4 n
It has 2non-conjugateKleinian subgroups, namely K = hσ2m,τi and K = hσ2m,τσmi
τ τσm
Note that by (2.2), T = T = hσm,τi. Then according to [5, Theorem 4.1,(ii)]
τ τσm
|Kerh| = 22g(XTτ)−2+s1.
So Corollary 2.5 give the proposition in this case.
Suppose now r ≥ 3 and the proposition holds for r − 1. Let X be a curve with an
action of D with X/hσi = H, so that we have the diagram (2.3). Then the subgroup
n
D = hσ2,τi of index 2 acts of the curve X , so that we can apply the inductive
n n/2
2
hypothesis. This gives that the map
h := Nmc ◦c∗ : P(d ) −→ P(d )
n/2 τσm τσn/2 τσn/2 τσm
is an isogeny of degree 2(m−1)(g−1)−2+s1.
Hence it suffices to show that
Kerh = b∗ (Kerh ).
τσn/2 n/2
This implies the proposition, since the map b∗ is injective.
τσn/2
Now Lemma 3.1 applied to the induction hypothesis, i.e. to h gives
n
2
|Kerh | = |(JX [2])hτ,σmi| = |a∗ (JX [2])hτ,σmi|.
n2 τσn/2 τσn/2 τσn/2
But
a∗ (JX [2])hτ,σmi = {z ∈ JX[2] | τσn/2z = z,τz = z,σmz = z}
τσn/2 τσn/2
= {z ∈ JX[2] | τz = z,σmz = z},
since the equation τσn/2z = z is a consequence of the last 2 equations. This gives
|Kerh| = |Kerh |
n
2
(cid:3)
which completes the proof of Proposition 3.2.
8 HERBERTLANGEAND ANGELA ORTEGA
4. Decomposition for n = 2rm, r ≥ 2 with m odd
NowletthenotationbeasinSection1withn = 2rm, r ≥ 2andmodd. Letf : X → H
be a cyclic ´etale covering of degree n of a hyperellitic curve H. The main result of the
paper is the following theorem.
Theorem 4.1. Let n and f : X → H be as above. Then JX and JX are abelian
τ τσm
subvarieties of the Prym variety P(f) and the addition map
a : JX ×JX → P(f)
τ τσm
is an isogeny of degree 2[(2r−r−1)m−(r−1)](g−1).
The proof is by induction on r. Since the proofs for r = 2 and for the inductive step in
case r ≥ 3 are almost the same, we give them simultaneously. The difference is only that
for r = 2 we use Theorem 2.1 instead of the induction hypothesis.
So in the whole of this section we assume that for r ≥ 3, Theorem 4.1 is true for r−1,
i.e. for covering of degree 2r−1m for all m. Let r ≥ 2 and f : X → H be an ´etale covering
of degree n = 2rm with odd m ≥ 1. We use the notation of diagram (2.3). In addition
let b : X → X denote the canonical projection.
τ τ Kτ
Proposition 4.2. The varieties JX ,JX ,P(b ) and P(b ) are abelian subvari-
Kτ Kτσm τ τσn/2
eties of JX and the addition map
φen : f∗JH ×JXKτ ×JXKτσm ×P(bτ)×P(bτσn/2) → JX
is an isogeny of degree
degφe = m2g−2 ·2[(2r+1−r)m+r](g−1)+2−s0.
n
Proof. All maps in diagram (2.3) are ramified apart from f ,f and f , which gives the
1 2 3
first assertion. The dihedral group D = hσ2,τi acts on the curve X .
n/2 σn/2
If r = 2, we can apply Theorem 2.1 to get that the canonical map
α : JX ×JX → P(f ◦f )
Kτ Kτσm 3 2
is an isomorphism. For r ≥ 3 we can apply the induction hypothesis, which gives that α
is an isogeny of degree 2[(2r−1−r)m−(r−2)](g−1). Since this number is equal to 1 for r = 2,
this is valid for all r ≥ 2.
Now the addition map α : (f ◦f )∗JH ×JX ×JX → JX factorizes as
1 3 2 Kτ Kτσm σn/2
(f ◦f )∗JH ×JX ×JX α1 // JX
3 2 Kτ ❲❲❲❲K❲❲τ❲σ❲mi❲d❲×❲α❲❲❲❲❲❲❲❲❲❲++ ❦❦❦❦❦❦❦❦❦❦ψ❦❦❦❦❦55 σn/2
(f ◦f )∗JH ×P(f ◦f )
3 2 3 2
where ψ is the addition map. So Lemma 2.6 implies that
degα = degα·degψ = 2[(2r−1−r)m−(r−2)](g−1) ·(2r−1m)2g−2 = m2g−2 ·2[(2r−1−r)m+r)](g−1).
1
9
Clearly α and its pullback via f∗ are of the same degree. Moreover, considering X with
1 1
the action of the Klein group hσn/2,τi, we have the diagram
X
{{①①①a①τ①①①①① (cid:15)(cid:15) f1❏❏❏❏❏❏a❏τ❏σ❏n❏/$$2
X X X
τ ❊❊b❊τ❊❊❊❊❊❊"" σ(cid:15)(cid:15)nc/τ2σzz✉n✉✉/✉2✉b✉τ✉σ✉n✉/2τσn/2
X
Kτ
Then Proposition 2.7 gives that the addition map
α : P(b )×P(b ) → P(f )
2 τ τσn/2 1
is an isogeny of degree 22g(XKτ) = 22r−1m(g−1)+2−s0.
Now note that the map φe factorizes as
n
e
[f∗JH ×JXKτ ×f1∗Jα1X×Kα2τσ(cid:15)(cid:15)m]×[✐P✐✐(✐b✐τ✐)✐✐×✐✐P✐✐(✐b✐τα✐σ3✐n✐/✐2✐)✐]✐✐✐✐✐✐φ✐n✐✐✐✐✐44// JX
f∗JX ×P(f )
1 σn/2 1
By Lemma 2.6 the addition map α3 is an isogeny of degree 22g(Xσn/2)−2 = 22rm(g−1),
therefore the map φe is an isogeny of degree
n
degφe = degf∗α ·degα ·degα
n 1 1 2 3
= m2g−2 ·2[(2r−1−r)m+r)](g−1) ·22r−1m(g−1)+2−s0 ·22rm(g−1)
= m2g−2 ·2[(2r+1−r)m+r](g−1)+2−s0.
(cid:3)
Corollary 4.3. The canonical map
φ : f∗JH ×JX ×JX ×P(b )×P(b ) → JX
n Kτ Kτσm τ τσm
is an isogeny of degree
degφ = m2g−22[(2r+1−r−1)m+r−1](g−1).
n
Proof. According to Proposition 3.2 the canonical map
h : P(b ) → P(b )
τσn/2 τσm
is an isogeny of degree 2(m−1)(g−1)−2+s1.
Now with the definition of the map h one checks that the following diagram commutes
e
[f∗JH ×JXKτ ×JiXd×Khτ(cid:15)(cid:15)σm ×P(❣b❣τ❣)❣]❣×❣❣❣P❣❣(❣b❣τ❣σ❣nφ❣/n❣2❣)❣❣❣❣❣φ❣n❣❣❣❣//33 JX
[f∗JH ×JX ×JX ×P(b )]×P(b )
Kτ Kτσm τ τσm
10 HERBERTLANGEAND ANGELA ORTEGA
So Propositions 4.2 and 3.2 imply that φ is an isogeny of degree
n
degφe
n
degφ =
n
degh
m2g−2 ·2[(2r+1−r)m+r](g−1)+2−s0
= = m2g−2 ·2[(2r+1−r−1)m+r−1](g−1)
2(m−1)(g−1)−2+s1
where we used again that s +s = 2g +2. (cid:3)
0 1
Corollary 4.4. The canonical map
ψ : JX ×JX ×P(b )×P(b ) → P(f)
n Kτ Kτσm τ τσm
is an isogeny of degree 2[(2r+1−r−1)m−(r+1)](g−1).
Proof. Clearly the addition maps the source of ψ into P(f) and the following diagram is
n
commutative
f∗JH ×[JXKτ ×iJd×XψnKτ(cid:15)(cid:15) σm ×P❣(❣b❣τ❣)❣×❣❣❣P❣❣(❣ϕb❣τ❣σ❣m❣)❣]❣❣❣❣❣φ❣n❣❣❣❣33// JX
f∗JH ×P(f)
where ϕ denotes the addition map. According to Lemma 2.6, ϕ is an isogeny of degree
(16m)2g−2. Hence ψ is an isogeny of degree
n
degφ m2g−2 ·2[(2r+1−r−1)m+r−1](g−1)
degψ = n = = 2[(2r+1−r−1)m−(r+1)](g−1)
n
degϕ (2rm)2g−2
(cid:3)
Proof of Theorem 4.1. The following diagram is commutative
JX ×JX ×P(b )×P(b )
Kτ Kτσm≃ (cid:15)(cid:15) τ❯❯❯❯❯❯❯❯❯τ❯σ❯mψ❯n❯❯❯❯❯❯❯❯**
[JX ×P(b )]×[JX ×P(b )] P(f)
Kτ τϕ1×ϕ2 (cid:15)(cid:15) Kτσ❤m❤❤❤❤❤❤❤❤a❤τ❤σ❤m❤❤❤❤❤❤❤❤❤44
JX ×JX
τ τσm
where ϕ and ϕ denote the addition maps. According to Lemma 2.6, ϕ and ϕ are
1 2 1 2
isogenies of degrees 28m(g−1)+2−s0 and 28m(g−1)+2−s1 respectively. This implies that a is an
isogeny of degree
degψ 2[(2r+1−r−1)m−(r+1)](g−1)
dega = n = = 2[(2r−r−1)m−(r−1)](g−1).
degϕ ·degϕ 2(2rm−2)(g−1)
1 2
(cid:3)
which completes the proof of the theorem.