Prym varieties of pairs of coverings Herbert Lange and Sevin Recillas*

(de Gruyter 2004

Prym varieties of pairs of coverings

Herbert Lange and Sevin Recillas*

(Communicated by K. Strambach)

Abstract.The Prym variety of a pair of coverings is defined roughly speaking as the comple- ment of the Prym variety of one morphism in the Prym variety of another morphism. We show that this definition is symmetric and give conditions when such a Prym variety is isogenous to an ordinary Prym variety or to another such Prym variety. Moreover in order to show that these varieties actually occur we compute the isogeny decomposition of the Jacobian variety of a curve with an action of the symmetric groupS₅.

Key words.Prym variety, group action.

2000 Mathematics Subject Classification. Primary: 14H40; Secondary: 14K02

1 Introduction

LetXbe a smooth projective curve over an algebraically closed fieldkandGa finite group of automorphisms ofX. This induces an action ofGon the JacobianJX ofX which can be used to decompose JX into a product of smaller dimensional abelian varieties up to isogeny:

JX@B₁^d1 B_r^dr

The abelian subvarieties Bi correspond one-to-one to the irreducible Q- representations of the groupG, which also determine the numbersdi. One would like to understand the decomposition in terms of the curve and its group action itself. In fact, for many small groups theBi’s can be interpreted as Prym varieties of coverings XM !XN, where MHN are subgroups ofGandXM andXN denote the quotients X=M andX=N. This is the case for example for the groupsS₃;S₄;A₄;A₅;D_p;WD₄ andQ₈(see [7], [8], [2] and [5]).

For other groups such asS₅ (see Theorem 4.1 below) or the dihedral groups D_n, (see [1] Remark 8.8) not for everyB_i there is such a Prym variety. Another type of abelian variety turns up: Let M;N₁ and N₂ be subgroups of G with MHN₁ and MHN₂. This gives the following diagram of coverings:

* Supported by CONACYT 40033-F and DFG Contract Ba 423/8-1

X_M f1 . & f2

XN1 XN2 ð1:1Þ

g1 & .g2

X_N

where N¼hN1;N2i, the subgroup generated by N1 and N2. Let PðfiÞdenote the Prym variety of the covering fi. SimilarlyPðgiÞis defined fori¼1;2. Then f₂Pðg2Þ is an abelian subvariety ofPðf1Þ. Since the canonical polarization ofJX induces a polarization of Pðf1Þ, the complementary abelian subvariety of f₂Pðg2ÞinPðf1Þis well defined. Similarly the complementary abelian subvariety of f₁Pðg1ÞinPðf2Þis well defined. It turns out that both complementary abelian subvarieties coincide as subvarieties ofJXM. We denote this subvariety byPðf1;f2ÞorPðXN1 XM !XN2Þ and call it thePrym variety of the pair of coveringsðf1;f2Þ.

In Section 2 we introduce the Prym varietyPðf₁;f₂Þslightly more generally for any pair of coverings of smooth projective curves ðf₁:X!X₁;f₂:X !X₂Þand prove its main properties. In Section 3 we prove some auxiliary results on group actions needed in the last section, where we work out the decomposition ofJX in the case of an action of the symmetric groupS₅of degree 5.

2 Definition ofP(f₁,f₂)

Let f :X !Y be a morphism of degreenof smooth projective curves over an alge- braically closed fieldk. Denote byJ_X:¼Pic⁰ðXÞandJ_Y :¼Pic⁰ðYÞthe Jacobians of X andY. Pulling back line bundles defines a homomorphism

f:J_Y!J_X:

f has finite kernel and is an embedding if and only if f does not factor via a cyclic e´tale cover of degreed2 (see [4], Proposition 11.4.3). The norm map of line bundles (see [3], Section 6.5) defines a homomorphism

N_f :J_X !J_Y:

ThePrym variety PðfÞof the morphism f is defined to be the abelian subvariety PðfÞ:¼kerðNfÞ⁰

ofJ_X where the 0 means the connected component containing 0. Note thatN_f is not necessarily a Prym variety in the classical sense, i.e. the canonical polarization ofJ_X does not necessarily induce a multiple of a principal polarization on PðfÞ. Suppose g:Y !Zis a second morphism of smooth projective curves, say of degreem. The Prym varieties of f;gandgf are related as follows:

Proposition 2.1. PðfÞ and fPðgÞare abelian subvarieties of PðgfÞand the addition map gives an isogeny

PðfÞ fPðgÞ !PðgfÞ:

Proof.The addition map yields an isogeny

PðgfÞ ðgfÞJ_Z!J_X:

Combing the analogous isogenies PðfÞ fJY !JX and fPðgÞ ðgfÞJZ! fJY we obtain that the addition map gives an isogeny

PðfÞ fPðgÞ ðgfÞJZ!JX:

Since PðfÞand fPðgÞare obviously abelian subvarieties of PðgfÞ, this implies the

assertion. r

Now suppose that we are given a commutative diagram of finite morphisms of smooth projective curves:

X

f1 . & f2

X1 X2 ð2:1Þ

g1 & .g2

Y Then we have

Proposition 2.2. Suppose g₁ and g₂ do not both factorize via the same morphism Y⁰!Y of degreed2.Then the Prym variety f₂Pðg2Þis an abelian subvariety of the Prym variety Pðf₁Þ.

Proof.First assume that f1and f2do not both factorize via a morphism f :X !X⁰. The universal property of the fibre product overY yields a diagram

X?y X1YX2

p₁ . &p₂

X1 X2

g1 & .g2

Y

wheren:X !X1YX2 denotes the normalization map and pi:X1YX2 !Xi the projection maps and fi¼ pinfori¼1 and 2. According to [3], Proposition 6.5.8 we have

N_p1p₂ðLÞ ¼g₁N_g2ðLÞ

for any line bundleLonX₂. But the norm map of line bundles is also defined for the mapn(see [3] Section 6.5, condition II is satisfied) and we have

Nf1ðf₂ðLÞÞ ¼Np1Nnðnp₂ðLÞÞ ¼Np1p₂ðLÞ:

Both equations together imply the assertion is this case.

In the general case suppose fi factorizes as fi¼ f_i⁰f with some morphism of smooth projective curves f :X !X⁰and f_i⁰:X⁰!Xifori¼1 and 2. By what we have just shown, f₂⁰Pðg2Þis an abelian subvariety ofPðf₁⁰Þ. So f₂Pðg2Þis an abelian subvariety of fPðf₁⁰Þwhich is an abelian subvariety of Pðf1Þaccording to Proposi-

tion 1.1. r

Remark 2.3. The assumption that g1 and g2 do not factorize via the same mor- phismY⁰!Y is necessary for the validity of Proposition 1.2. To give an example, leth:Y!P₁be a finite covering. Replacegibyhgifori¼1;2. ThenPðhg2Þ ¼JX2

and and it is easy to give an example of a diagram (1.1) where f₂JX₂ is not an abe- lian subvariety ofPðf₁Þ.

The canonical principal polarization induces a polarization on Pðf₁Þ. Hence the complementary abelian subvariety P₁ of the abelian subvariety f₂Pðg2Þ inPðf₁Þis well defined (see [4], Section 5.3). The addition map induces an isogeny of polarized abelian varieties

P1f₂Pðg2Þ !Pðf1Þ:

In the same way the canonical principal polarization ofJX induces a polarization onPðf₂Þ. Hence the complementary abelian subvarietyP₂of f₁Pðg1ÞinPðf₂Þis well defined and the addition map induces an isogeny of polarized abelian varieties

f₁Pðg1Þ P2!Pðf2Þ:

P1andP2are both abelian subvarieties ofJXwith induced polarizations, sayH1and H2. We have:

Proposition 2.4.The polarized abelian subvarieties ðP1;H1Þand ðP2;H2Þof JX coin- cide.

Proof.It su‰ces to show thatP₁ ¼P₂ since the polarizations are induced by the ca- nonical principal polarization ofJX. By definition of the Prym varieties the addition maps induce isogenies

f₁g₁JYf₁Pðg1Þ f₂Pðg2Þ P1! f₁JX1Pðf1Þ !JX

and

f₂g₂J_Yf₂Pðg2Þ f₁Pðg1Þ P₂! f₂J_X2Pðf₂Þ !J_X

where all abelian varieties are subvarieties of J_X. Obviously we have f₁g₁J_Y ¼ f₂g₂J_Y. So ifZdenotes the image of f₁g₁J_Yf₁Pðg1Þ f₂Pðg2ÞinJ_X the addi- tion map gives isogenies

ZP₁!J_X and ZP₂!J_X:

Now the corresponding decompositions of the tangent spaces are orthogonal with respect to the hermitian form associated to the canonical polarization of JX. This implies that on the one handP1 and on the other handP2 is the complement of the abelian subvarietyZinJX. Since the complement is uniquely determined, this implies

the assertion. r

We call the abelian variety P1¼P2 or more precisely the polarized abelian va- rietyðP1;H1Þ ¼ ðP2;H2ÞthePrym variety of the pair of coveringsðf1;f2Þand denote it by Pðf₁;f₂Þ or PðX1 X!X₂Þ. Note that Pðf₁;f₂Þ is defined for any pair ðf₁:X !X₁;f₂ :X!X₂Þof coverings of smooth projective curves. Givenðf₁;f₂Þ, the curve Y in the diagram (2.1) is the smooth projective curve corresponding to the function field kðX1ÞVkðX2Þ. If for example f₁¼ f₂ then we have obviously Pðf₁;f₁Þ ¼0.

Applying the Hurwitz formula, it is easy to compute the dimension of Pðf₁;f₂Þ.

We do this only in the most important case where the function fields satisfy kðX1ÞkðX2Þ ¼kðXÞandkðX1ÞVkðX2Þ ¼kðYÞ, i.e. the hypotheses of Proposition 2.2 are satisfied andX is the normalization ofX1YX2. Then we haved1:¼degðf1Þ ¼ degðg2Þandd2:¼degðf2Þ ¼degðg1Þ. Moreover for any covering f of smooth pro- jective curves letQf denote the degree of the ramification divisor of f. Then we have Proposition 2.5.

dimPðf1;f2Þ ¼ ðd11Þðd21ÞðgðYÞ 1Þ þ1=2½Qf1þ ðd11ÞQg1Qg2: 3 Isogenies between Prym varieties and Prym varieties of pairs

Let againGbe a finite group acting on a smooth projective curveX. IfMHNand M⁰HN⁰ are two pairs of subgroups of G, it may happen that the Prym varieties PðXM=X_NÞandPðXM⁰=X_N⁰Þare isogenous. Similarly this may happen for Prym vari- eties of pairs. In this section we give a criterion for this. Since we need this only in the case of the symmetric groupS₅, we will assume in this section and without further notice that every irreducible Q-representation of the group Gis absolutely irreduc- ible. We will see that then the Prym varieties and Prym varieties of pairs depend only on the induced representations of the trivial representations of the subgroups in question. For a general group we will come to this question in a subsequent paper.

The action ofGon the curveX induces an action on its JacobianJX and thus an algebra homomorphism

r:Q½G !End_QðJXÞ:

Ifedenotes any idempotent of the algebraQ½G, we define ImðeÞ:¼ImðrðmeÞÞJJX

wheremis some positive integer such thatmeAZ½G. ImðeÞis an abelian subvariety ofJX, which certainly does not depend on the chosen integerm.

Let W₁;. . .;W_r denote the irreducible Q-representations of G. We assume in the

sequel thatW1 is the trivial representation and thatdi¼dimWifori¼1;. . .;r. Ifei

denotes the central idempotent of Q½Gassociated to Wi andAi¼ImðeiÞthe corre- sponding abelian subvariety ofJX fori¼1;. . .;r, then the addition map induces an isogeny (see [5], Proposition 2.1)

m:A1 Ar!A: ð3:1Þ If di>1 the abelian variety Ai can be decomposed further: Since Wi is absolutely irreducible, it admits up to a positive constant a uniquely determined G-invariant scalar product (see [9]). Fix one of these for every i and denote it by ð;Þ. Let fwi;1;. . .;w_i;digbe a basis ofW_i, orthogonal with respect toð;Þ, and define

pwi;j :¼ di

jGj kwi;jk² X

gAG

ðwi;j;gwi;jÞg:

Schur’s character relations (see [9], Chapter 2, Corollary 3 of Proposition 4) can be translated into terms of idempotents as follows (see [5], Proposition 3.3): pwi;1;. . .; pwi;di are orthogonal idempotents inQ½Gsatisfying

pwi;1þ þpwi;di ¼ei:

This implies that ifBi;j:¼Imðpwi;jÞ, the addition map induces an isogeny

m_i:Bi;1 Bi;di !Ai: ð3:2Þ Moreover, since the minimal left ideals of Q½G generated by the idempotents p_wi;j are pairwise isomorphic for a fixedi, it follows that the abelian varietiesB_i;1;. . .;B_i;di are pairwise isogenous (see [5]). Combining everything we obtain:

There are abelian subvarietiesB₁;. . .;B_rand an isogeny

JX@B₁^d1 B_r^dr: ð3:3Þ The action ofGonJX induces an action on the tangent spaceT0JX. DenotingVi¼ WinC, we obtain a decomposition

T0JXFV₁ⁿ¹ V_r^nr: ð3:4Þ Comparing this with the decomposition (3.3) implies T0ðB_i^diÞFV_iⁿⁱ. This gives di dimB_i¼n_idimV_i. But dimV_i¼d_iand thus

n_i¼dimB_i:

Let H denote the canonical polarization ofJX. It can be considered as a positive definite hermitian form onT₀JX. Since the groupGof automorphisms ofJX is in- duced by the automorphism groupGof the curveX, it preserves the polarizationH.

This implies that we may change the isomorphism (3.4) in such a way thatH restricts to the scalar productð ; ÞonWiHVifor alli¼1;. . .;r. We fix this isomorphism in the sequel. Using this we can show:

Proposition 3.1.Let MHN be subgroups of the group G.Then PðXM=X_NÞ@B₂^s2 B_r^sr with si¼dimW_i^MdimW_i^N for i¼2;. . .;r.

Note that in the special caseM¼ f1g andN¼GProposition 3.1 gives the well known fact

PðX=YÞ@B₂^d2 B_r^dr since dimW_i^f1gdimW_i^G¼dimWi¼difori¼2;. . .;r.

Proof.Fori¼1;. . .;rchoose an orthogonal basis

fwi;1;. . .;wi;ti;wi;tiþ1;. . .;wi;tiþsi;wi;tiþsiþ1;. . .;wi;dig ofW_iin such a way that

W_i^N¼hwi;1;. . .;wi;tii and W_i^M ¼hwi;1;. . .;wi;tiþsii: Then

W_i^M ¼W_i^Nþhw_i;tiþ1;. . .;w_i;tiþs_ii; the sum being orthogonal.

It is easy to see that p_wi;j is the projection ofW_i onto the 1-dimensional subspace spanned byw_i;j(see [6], Remarque (2), page 53). It follows that

W_i^M ¼X^tiþsi

j¼1

pwi;jðWiÞ and W_i^N ¼X^ti

j¼1

pwi;jðWiÞ:

Since the sums are orthogonal, this implies

W_i^M ¼W_i^Nþ X^tiþsi

j¼tiþ1

pwi;jðWiÞ:

This equation immediately yields, if we again denoteV_i¼W_inC:

V_i^M ¼V_i^Nþ X^tiþsi

j¼tiþ1

p_wi;jðViÞ the sums being orthogonal.

On the other hand the tangent map at the origin to pwi;j :JX !JX is pwi;j :T0JX !T0JX. So the tangent space at the origin of the subvariety Pr

i¼1

Ptiþsi

j¼1 Imp_wi;jHJX isPr i¼1

Ptiþsi

j¼1 p_wi;jðT0JXÞ. But X^r

i¼1

X^tiþsi

j¼1

pwi;jðT0JXÞ ¼ ðT0JXÞ^M ¼T0JXM: It follows that

JX_M@X^r

i¼1

X^tiþsi

j¼1

Imp_wi;j

(which is the image of the sum mapU^r

i¼1U^tiþsi

j¼1 B_i;j!JXÞ. Similarly we have JXN@X^r

i¼1

X^ti

j¼1

Impwi;j:

Hence, sincet1¼d1ð¼1Þand thuss1 ¼0, we obtain the orthogonal decomposition JX_M@JX_NX^r

i¼2

X

tiþs_i j¼tiþ1

Imp_wi;j:

On the other hand, by definition of the Prym variety we have the orthogonal decom- position

JX_M@JX_NPðXM !X_NÞ:

Comparing both, orthogonal cancellation gives

PðXM !X_NÞ@X^r

i¼2

X^tiþsi

j¼tiþ1

Imp_wi;j: ð3:5Þ

This implies the assertion, sinceBiis isogenous to Impwi;j for all j. r For any subgroup M of G let QðG=MÞ denote the induced representation of the trivial representation of M in G. Note that for subgroups MHN of G,

QðG=NÞis a subrepresentation ofQðG=MÞso thatQðG=MÞ QðG=NÞis in fact a Q-representation.

Corollary 3.2.SupposeQðG=MÞ QðG=NÞF 0^r

i¼2W_i^si.Then PðXM !X_NÞ@B₂^s2 B_r^sr:

Proof. For any representationW of G let w_W denote its character. Since any irre- ducible representation ofG is absolutely irreducible, we may apply Frobenius reci- procity, to give for any j¼1;. . .;r

dimW_j^MdimW_j^N ¼ ðw_{QðG=MÞQðG=NÞ};w_WjÞ

¼X^r

i¼1

s_i ðw_Wi;w_WjÞ ¼s_j:

Hence Proposition 3.1 gives the assertion. r

Applying Corollary 3.2 twice we obtain

Corollary 3.3. Let M_iHN_i be subgroups of G such that the representations QðG=MiÞ QðG=NiÞare isomorphic for i¼1and2.Then

PðXM1!X_N1Þ@PðXM2!X_N2Þ:

Now suppose we are given the following diagram of subgroups ofG N₁

% &

M N¼hN1;N2i!G

& % N2

where all the maps are the canonical inclusions. This induces the diagram (1.1) of coverings of curves. The equationN¼hN₁;N₂iimplies thatg₁ andg₂ do not both factorize via a morphism Y⁰!X_N of degreed2. With the notation of above we have

Proposition 3.4.Pðf₁;f₂Þ@B₂^s2 B_r^sr with

si¼dimW_i^MþdimW_i^NdimW_i^N1dimW_i^N2 for i¼2;. . .;r:

Note that MHN₁ and MHN₂ imply W^N1þW^N2HW^M and N¼hN₁;N₂i impliesW^N1VW^N2¼W^N. Hence

dimW^N1þdimW^N2dimW^NcdimW^M:

So s_id0. This in turn implies that QðG=MÞ þQðG=NÞ QðG=N1Þ QðG=N2Þis actually a representation.

One can also state the inequality as:

dimW^N2dimW^NcdimW^MdimW^N1

which is the reason why we can choose the basis the way we do in the proof below.

Proof. For i¼1;. . .;r we choose an orthogonal basis fwi;1;. . .;wi;ti;wi;tiþ1;. . .; w_i;tiþs¹

i;w_i;tiþs¹

iþ1;. . .;w_i;tiþs¹

iþs_i²;w_i;tiþs¹

iþs_i²þ1;. . .;wi;digofWi in such a way that W_i^N¼hw_i;1;. . .;w_i;tii; W_i^N1¼hw_i;1;. . .;w_i;tiþs¹

i

i; W_i^N2 ¼hwi;1;. . .;wi;ti;w_i;tiþs¹

iþ1;. . .;w_i;tiþs¹

iþs_i²i and

W_i^M ¼hwi;1;. . .;w_i;tiþs¹

iþs_i²;. . .;wmi: By (3.5) we have

f_N2PðXN2=X_NÞ@X^r

i¼2 tiþsX_i¹þs₂² j¼tiþs_i¹þ1

Imp_wi;j

all sums being orthogonal with respect to the polarization induced by the canonical polarizationH ofJX. Since f_N2¼ f₂ f_M and f_M is an isogeny, this gives

f₂PðXN2=XNÞ@X^r

i¼2 tiþsX_i¹þs²₂ j¼t_iþs_i¹þ1

Impwi;j:

In the same way we get

PðXM=XN1Þ@X^r

i¼2

X^m

j¼tiþs_i¹þ1

Impwi;j:

Since all sums are orthogonal andPðf1;f2Þis by definition the orthogonal comple- ment of f₂PðXN2=X_NÞinPðXM=X_N1Þ, this implies

Pðf₁;f₂Þ@X^r

i¼2

X^m

j¼tiþs_i¹þs_i²þ1

Imp_wi;j:

Since Impi;jis isogenous toBifor all jand moreover (3.1) and (3.2) are isogenies, we obtain

Pðf₁;f₂Þ@Y^r

i2

Y^m

j¼tiþs_i¹þs_i²þ1

B_i:

Now the assertion follows from mt_is_i¹s²_i ¼dimW_i^MþdimW_i^NdimW_i^N1

dimW_i^N2. r

It is easy to see thatQðG=MÞ þQðG=NÞ QðG=N1Þ QðG=N2Þis actually a rep- resentation. Hence in the same way that Corollary 3.2 follows from Proposition 3.1, Proposition 3.4 implies:

Corollary 3.5. Let ðMHN1;MHN2Þ be a triple of subgroups of G and N ¼ hN1;N2i.If

QðG=MÞ þQðG=NÞ QðG=N1Þ QðG=N2ÞF 0

r i¼2

W_i^si then

Pðf₁;f₂Þ@B₂^s2 B_r^sr Finally Corollaries 3.2 and 3.5 imply

Corollary 3.6.(a) If M⁰HN⁰ is another pair of subgroups of G such that the repre- sentationQðG=MÞ þQðG=NÞ QðG=N1Þ QðG=N2Þis isomorphic to the represen- tationQðG=M⁰Þ QðG=N⁰Þ,then

Pðf₁;f₂Þ@PðXM⁰ !X_N⁰Þ:

(b)If ðM⁰HN₁⁰;M⁰HN₂⁰Þis another triple of subgroups of G and N⁰¼hN₁⁰;N₂⁰i such that the representations QðG=MÞ þQðG=NÞ QðG=N1Þ QðG=N2Þ and QðG=M⁰Þ þQðG=N⁰Þ QðG=N₁⁰Þ QðG=N₂⁰Þ are isomorphic and f₁⁰:XM⁰ !XN₁⁰

and f₂⁰:XM⁰ !XN₂⁰ denote the corresponding coverings,then Pðf1;f2Þ@Pðf₁⁰;f₂⁰Þ:

In somewhat vague terms Corollaries 3.3 and 3.6 can be expressed by saying: The induced representations of the trivial representations determine the isogeny decom- position.

4 Example: The symmetric group of degree 5

LetX be smooth projective curve with an action of the symmetric groupS5of degree 5. The group action induces the decomposition (3.3) of the JacobianJX. Note that if we assume that gðX=S5Þd2, then every abelian subvariety Bi occurring in (3.3) is positive dimensional according to [5] Theorem 4.1. In this section we apply the results of Section 3 in order to express the abelian subvarietiesBiof decomposition (3.3) in terms of Prym varieties of subgroups and pairs of subgroups ofS5.

We consider S5 as the group of permutations of the set of integers f1;. . .;5g. In order to state the result consider the following subgroups ofS₅:

A₅:¼hð1;2;3;4;5Þ;ð3;4;5Þiof order 60, S₄:¼hð2;3Þ;ð2;4Þ;ð2;5Þiof order 24,

A₄:¼hð2;3Þð4;5Þ;ð2;4Þð3;5Þ;ð3;4;5Þiof order 12, D₅:¼hð1;2;3;4;5Þ;ð2;5Þð3;4Þiof order 10, D₄:¼hð2;3Þ;ð2;4;3;5Þiof order 8,

K:¼hð2;3Þ;ð4;5Þiof order 4,

L:¼hð2;3Þ;ð4;5Þ;ð1;2;3Þiof order 12 and

M:¼hð1;2;3;4;5Þ;ð2;5Þð3;4Þ;ð2;4;5;3Þiof order 20.

For any subgroup M of S5 let XM :¼X=M denote the quotient curve of X by the action ofM and denote Y:¼X=S5. IfMHN is any pair of subgroups of S5, we denote by PðXM !XNÞthe Prym variety of the associated coveringXM !XN. Similarly for any triple of subgroups ðMHN1;MHN2Þ let PðXN1 XM !XN2Þ denote the Prym variety of the pair of morphismsðXM !X_N1;X_M !X_N2Þ. With this notation we have:

Theorem 4.1.

JX@JY PðXA5!YÞ PðXS4 !YÞ⁴PðXA5 X_A4!X_S4Þ⁴ PðXM XD5!XA5Þ⁵PðXM !YÞ⁵PðXD4 XK !XLÞ⁶ There is no pair of subgroups MHN of S5 whose associated Prym variety PðXM !XNÞis isogenous to a Prym variety of a pair of morphisms occurring in this decomposition.

Proof.LetU;U⁰;V;V⁰;W;W⁰;5²V denote the irreducibleC-representations ofS₅. They are determined by the following character table:

1 (12) (12)(34) (123) (12)(345) (1234) (12345)

# 1 10 15 20 20 30 24

U 1 1 1 1 1 1 1

U⁰ 1 1 1 1 1 1 1

V 4 2 0 1 1 0 1

V⁰ 4 2 0 1 1 0 1

W 5 1 1 1 1 1 0

W⁰ 5 1 1 1 1 1 0

5²V 6 0 2 0 0 0 1

Observe that all irreducible representations are defined over Q. Hence according to

Equation (3.3) there are abelian subvarietiesB_U⁰;B_V;B_V⁰;B_W;B_W⁰ andB5²V ofJX, uniquely determined up to isogeny, such that

JX@JYB_U⁰B_V⁴ B⁴_V0B_W⁵ B_W⁵ 0B₅⁶2V: We have to identifyB_U⁰;. . .;B5²V in terms of Prym varieties.

(a)BU⁰: Using the above character table one easily checks:

QðS5=A5Þ QðS5=S5ÞFU⁰: So Corollary 3.2 impliesBU⁰@PðXA5!YÞ.

(b) BV: One checks QðS5=S4Þ QðS5=S5ÞFV. So Corollary 3.2 implies BV@ PðXS4!YÞ.

(c) BW⁰: QðS5=MÞ QðS5=S5ÞFW⁰. Hence Corollary 3.2 implies BW⁰@ PðXM !Y).

(d)B_V⁰: Consider the diagram

X_A4 . &

XA5 XS4

& . Y

One checkshS4;A5i¼S5andQðS5=A4Þ þQðS5=S5Þ QðS5=S4Þ QðS5=A5ÞFV⁰. So Corollary 3.5 yieldsBV⁰@PðXA5 XA4!XS4Þ.

(e)BW: Consider the diagram

XD5

. &

XM XA5

& . Y

One checks hM;A₅i¼S₅ and QðS5=D₅Þ þQðS5=S₅Þ QðS5=MÞ QðS5=A₅ÞF W. So Corollary 3.5 impliesB_W@PðXM X_D5 !X_A5Þ.

(f )B5²V: Consider the diagram

XK

. &

X_D4 X_L

& . Y

One checkshD4;Li¼S5andQðS5=KÞ þQðS5=S5Þ QðS5=D4Þ QðS5=LÞF5²V.

So Corollary 3.5 givesB5²V@PðXD4 X_K!X_LÞ.

It remains to show thatB_V⁰;B_W andB5²Vare not isogenous to a Prym variety of a covering associated to a pair of subgroupsMHNofS₅. For this we computed the Prym varieties of all conjugacy classes of pairs of such subgroups. The computations

are a little too long to repeat them here. r

Finally let us give some examples of isogenous Prym varieties as well as Prym vari- eties of pairs which are isogenous to Prym varieties of coverings. For this consider the following subgroups ofS5:

C2:¼hð2;5Þð3;4Þiof order 2, C3:¼hð3;4;5Þiof order 3, C5:¼hð1;2;3;4;5Þiof order 5, N:¼hð3;4;5Þ;ð1;2Þð4;5Þiof order 6, S₃:¼hð4;5Þ;ð3;4;5Þiof order 6 and K₁:¼hð2;3Þð4;5Þ;ð2;4;3;5Þiof order 4.

Examples 4.2.(a)PðXD5 !X_A5Þ@PðXD4!X_S4Þ.

(b)PðXC5 X !X_C2Þ@PðXC2!X_D5Þ.

(c)PðXN X_C3 !X_A4Þ@PðXC5 !X_D5Þ.

(d)PðXA4 XC3 !XS3Þ@PðXK1!XD4Þ.

Note that in (a)XD5 !XA5 is of degree 6 whereasXD4!XS4 is of degree 3.

Proof. We have QðS5=D5Þ CðS5=A5ÞFWlW⁰FQðS5=D4Þ QðS5=S4Þ. So Corollary 3.3 implies (a). As for (b), note first that hC5;C2i¼D5. Then we have QðS5=ð1ÞÞ þQðS5=D5Þ QðS5=C2Þ QðS5=C5ÞFVþV⁰þWþW⁰þ ð5²VÞ²F QðS5=C2Þ QðS5=D5Þ. So Corollary 3.6 (a) implies the assertion. The proof of (c)

and (d) is similar. r

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Received 15 April, 2003

H. Lange, Mathematisches Institut, Universita¨t Erlangen-Nu¨rnberg, Bismarckstraße 1¹₂, 91054 Erlangen, Germany

Email: [email protected]

S. Recillas, Instituto de Matematicas, UNAM Campus Morelia, Morelia, Mich., 58089, Me´xico

Email: [email protected] and

CIMAT Callejo´n de Jalisco s/n, 36000 Valenciana, Gto., Me´xico Email: [email protected]