On Jacobians with group action and coverings
aa r X i v : . [ m a t h . AG ] N ov ON JACOBIANS WITH GROUP ACTION AND COVERINGS
SEBASTI ´AN REYES-CAROCCA AND RUB´I E. RODR´IGUEZ
Abstract.
Let S be a compact Riemann surface and let H be a finite group.It is known that if H acts on S then there is a H -equivariant isogeny decompo-sition of the Jacobian variety JS of S, called the group algebra decompositionof JS with respect to H. If S → S is a regular covering map, then it is also known that the groupalgebra decomposition of JS induces an isogeny decomposition of JS . In this article we deal with the converse situation. More precisely, we provethat the group algebra decomposition can be lifted under regular coveringmaps, under appropriate conditions. Introduction
Let H be a finite group acting on a compact Riemann surface S . It is a knownfact that this action induces an action of H on the Jacobian variety JS of S and this,in turn, gives rise to a H -equivariant isogeny decomposition of JS as a product ofabelian subvarieties. This decomposition is called the group algebra decomposition of JS with respect to H ; see [5] and [15].The decomposition of Jacobians with group actions has been extensively studiedin different settings, with applications to theta functions, to the theory of integrablesystems and to the moduli spaces of principal bundles of curves, among others. Thesimplest case of such a decomposition is when H is a group of order two; this factwas already noticed in 1895 by Wirtinger [26] and used by Schottky-Jung in [23].For decompositions of Jacobians with respect to other special groups, we referto the articles [1], [4], [6], [10], [11], [12], [16], [17], [18], [20] and [22].Let C be a compact Riemann surface admitting the action of a finite group G. For every subgroup N of G consider the associated regular covering map C → S = C N given by the action of N on C. It was proved in [5] that the group algebra decomposi-tion of JC with respect to G induces an isogeny decomposition of JS.
Furthermore,the factors arising in the induced decomposition of JS are exactly the same as theones arising in the decomposition of JC, possibly with lower multiplicity.This article is mainly devoted to generalize the aforementioned result, by in-vestigating the unstudied converse situation. More precisely, let S be a compactRiemann surface admitting the action of a finite group H, and consider any Rie-mann surface C satisfying the next two conditions: Mathematics Subject Classification.
Key words and phrases.
Riemann surfaces, Group actions, Jacobian varieties.Partially supported by Postdoctoral Fondecyt Grant 3160002, Fondecyt Grant 1141099 andAnillo ACT 1415 PIA-CONICYT Grant. (a) there exists a regular covering map π : C → S , and(b) C admits the action of a supergroup G of the deck group K of π, in sucha way that K is normal in G and G/K ∼ = H. The main result of this paper states that the group algebra decomposition of JS with respect to H lifts to obtain an isogeny decomposition of JC, which is closelyrelated to the group algebra decomposition of JC with respect to G. More precisely,the factors arising in the decomposition of JC can be of two types, namely:(a) the factors arising in the group algebra decomposition of JS , with exactlythe same multiplicities, or(b) factors whose product is isogenous to the Prym variety P ( C/S ) associatedto the covering map π .As a matter of fact, the main result tells us about the compatibility between thegroup algebra decomposition and the classically known decomposition JC ∼ JS × P ( C/S )associated to the covering map π : C → S .We anticipate that in order to determine the group algebra decomposition of JC with respect to G it is necessary to have a complete knowledge of the rationalirreducible representations of G. By contrast, as we shall see later, the application ofthe main result of this paper significantly simplifies those computations, by reducingthe problem to study the representations of G that are not trivial in K. As a further application of the main result, we derive a result related to decom-position of Jacobians JC as products of Jacobians of quotients of C. This article is organized as follows. In Section 2 we shall briefly review the basicbackground; namely, group actions on Riemann surfaces, representation of groups,abelian varieties and the group algebra decomposition theorem for Jacobians. InSection 3 we shall prove two basic purely algebraic lemmata. The results of thispaper will be stated and proved in Sections 4 and 5. Finally, in Section 6 we shallexhibit two explicit examples in order to show how our results can be applied.
Acknowledgments.
The authors are grateful to their colleague Angel Caroccafor his helpful suggestions throughout the preparation of this manuscript.2.
Preliminaries
Group actions on Riemann surfaces.
Let S be a compact Riemann surfaceand let Aut( S ) be its automorphism group. We recall that a finite group H actson S if there is a monomorphism ǫ : H → Aut( S ) . The space of orbits S H of theaction of H ∼ = ǫ ( H ) on S is endowed with a Riemann surface structure, such thatthe natural projection π H : S → S H is holomorphic. The degree of π H is the order | H | of H and the multiplicity of π H at p ∈ S is | H p | , where H p denotes the stabilizerof p in H . If | H p | 6 = 1 then p is called a branch point of π H ; its image by π H is a branch value of π H .Let { p , . . . , p l } be a maximal collection of non- H -equivalent branch points of π H . The signature of the action of H on S is the tuple ( γ ; m , . . . , m l ) where γ isthe genus of the quotient S H and m i = | H p i | . The branch value π H ( p i ) is said tobe marked with m i . The Riemann-Hurwitz formula relates these numbers with the
N JACOBIANS WITH GROUP ACTION AND COVERINGS 3 order of H and the genus g of S. Namely,2 g − | H | [2 γ − li =1 (1 − m i )]A tuple ( a , . . . , a γ , b , . . . , b γ , c , . . . , c l ) of elements of H is called a generatingvector of H of type ( γ ; m , . . . , m l ) if the following conditions are satisfied:(a) H is generated by a , . . . , a γ , b , . . . , b γ , c , . . . , c l ;(b) the order of c i is m i for all i, and(c) Π γi =1 ( a i b i a − i b − i )Π li =1 c i = 1 . Riemann’s existence theorem ensures that H acts on a Riemann surface of genus g with signature ( γ ; m , . . . , m l ) if and only if the Riemann-Hurwitz formula issatisfied and H has a generating vector of type ( γ ; m , . . . , m l ); see [3].We refer to [9] for further material related to Riemann surfaces and group actions.2.2. Representations of groups.
Let H be a finite group and let ρ : H → GL( V )be a complex representation of H. Abusing notation, we shall also write V to referto the representation ρ. The degree d V of V is the dimension of V as a complexvector space, and the character χ V of V is the map obtained by associating to each h ∈ H the trace of the matrix ρ ( h ) . Two representations V and V are equivalent ifand only if their characters agree; we write V ∼ = V . The character field K V of V isthe field obtained by extending the rational numbers by the values of the characterof V . The Schur index s V of V is the smallest positive integer such that there existsa degree s V field extension L V of K V over which V can be defined.It is classically known that for each rational irreducible representation W of H there is a complex irreducible representation V of H such that W ⊗ Q C ∼ = ( ⊕ σ V σ ) ⊕ s V · · · ⊕ ( ⊕ σ V σ ) = s V ( ⊕ σ V σ ) (2.1)where the sum ⊕ σ V σ is taken over the Galois group associated to Q ≤ K V . We shallsay that V is associated to W. Two complex irreducible representations associatedto the same rational irreducible representation are termed
Galois associated .Let N be a subgroup of H and consider V N := { v ∈ V : ρ ( h )( v ) = v for all h ∈ N } , the vector subspace of V consisting of those elements fixed under N ; let d NV denoteits dimension. By Frobenius Reciprocity theorem, d NV = h ρ N , V i H where ρ N stands for the representation of H induced by the trivial one of N, andthe brackets for the usual inner product of representations.We refer to [24] for further basic facts related to representations of groups.2.3. Complex tori and abelian varieties. A g -dimensional complex torus X = V /
Λ is the quotient of a g -dimensional complex vector space V by a maximal rankdiscrete subgroup Λ. Each complex torus is an abelian group and a g -dimensionalcompact connected complex analytic manifold. Homomorphisms between complextori are holomorphic maps which are also group homomorphisms; we shall denoteby End( X ) the ring of endomorphisms of X, that is, homomorphisms of X intoitself. An isogeny is a surjective homomorphism with finite kernel; isogenous toriare denoted by X ∼ X . The isogenies of a complex torus X into itself are theinvertible elements of the algebra of rational endomorphismsEnd Q ( X ) := End(X) ⊗ Z Q . SEBASTI´AN REYES-CAROCCA AND RUB´I E. RODR´IGUEZ An abelian variety is a complex torus which is also a complex projective algebraicvariety. The Jacobian variety JS of a compact Riemann surface S is an (irreducibleprincipally polarized) abelian variety; its dimension is the genus of S. By the well-known Torelli’s theorem, two compact Riemann surfaces are isomorphic if and onlyif their Jacobians are isomorphic (as principally polarized) abelian varieties.Associated to every covering map π : C → S between compact Riemann surfacesthere are two homomorphism; namely, the pull-back and the norm of π : π ∗ : JS → JC and N π : JC → JS. As π ∗ ( JS ) is an abelian subvariety of JC isogenous to JS,
Poincar´e’s Reducibilitytheorem implies that there exists an abelian subvariety P ( C/S ) of JC such that JC ∼ JS × P ( C/S ) . The factor P ( C/S ) (which is termed the
Prym variety associated to π ) agreeswith the connected component of zero of the kernel of the norm N π . We refer to [2] and [8] for basic material on this topic.2.4.
Group algebra decomposition theorem.
Let us suppose that H is a finitegroup and that W , . . . , W r are its rational irreducible representations. It is clas-sically known that each action ǫ H,S : H → Aut( S ) of H on S induces a Q -algebrahomomorphism Φ H,S : Q [ H ] → End Q ( JS )where Q [ H ] stands for the rational group algebra of H (see, for example [2, p. 431]).For every α ∈ Q [ H ] we define the abelian subvariety A α := Im( α ) = Φ H,S ( lα )( JS ) ⊂ JS where l is some positive integer chosen such that Φ H,S ( lα ) ∈ End( JS ).Let e i be the idempotent of Q [ H ] given by e i = d V i | H | X h ∈ H tr K Vi | Q ( χ V i ( h − )) h, where V i is a complex irreducible representation of H associated to W i , of degree d V i ,and tr K Vi | Q is the trace of the extension Q ≤ K V i . Then the equality 1 = e + · · · + e r yields an isogeny JS ∼ A e × · · · × A e r which is H -equivariant; the factors A i := A e i above are called the isotypical factors of JS with respect to H . See [15].Additionally, there are n i = d V i /s V i idempotents f i , . . . , f in i in Q [ H ] such that e i = f i + · · · + f in i , which provide n i pairwise isogenous abelian subvarieties of JS.
Let B i be one of them, for every i. Thus A i ∼ B n i i and therefore the isogeny JS ∼ H B n × · · · × B n r r (2.2)is obtained. This last isogeny is called the group algebra decomposition of JS withrespect to H ; see [5].If the representations are labeled in such a way that W (= V ) denotes the trivialone (as we will do in this paper) then n = 1 and B ∼ JS H . N JACOBIANS WITH GROUP ACTION AND COVERINGS 5
Let N be a subgroup of H and consider the associated regular covering map S → S N . It was proved in [5] that the group algebra decomposition of JS withrespect to H induces the following isogeny decomposition of JS N : JS N ∼ B n N × · · · × B n Nr r where n Ni = d NV i /s V i . The isogeny above provides a criterion to identify if a factor in the group algebradecomposition of JS with respect to H is isogenous to the Jacobian variety of aquotient of S or isogenous to the Prym variety of an intermediate covering of π H .More precisely, if two subgroups N ≤ N ′ of H satisfy d NV i − d N ′ V i = s V i for some fixed 2 ≤ i ≤ r and d NV l − d N ′ V l = 0for all l = i such that dim( B l ) = 0 , then B i ∼ P ( S N /S N ′ ) . Furthermore if, in addition, the genus of S N ′ is zero then B i ∼ JS N . See also [13].Assume that ( γ ; m , . . . , m l ) is the signature for the action of H on S and thatthe tuple ( a , . . . , a γ , b , . . . , b γ , c , . . . , c l ) is a generating vector representing thisaction. Following [21, Theorem 5.12], the dimension of B i in (2.2) isdim( B i ) = k V i h d V i ( γ −
1) + 12 Σ lk =1 ( d V i − d h c k i V i ) i (2.3)for 2 ≤ i ≤ r, where k V i is the degree of the extension Q ≤ L V i . Two basic algebraic lemmata
Let G be a finite group, let K be a normal subgroup of G and let Φ : G → H bea surjective homomorphism of groups whose kernel is K. If ρ : H → GL( V ) is a complex representation of H then by precomposing ρ byΦ we obtain a complex representation˜ ρ := ρ ◦ Φ : G → GL( V )of G whose kernel contains K. Conversely, given a complex representation ˜ ρ : G → GL( V ) of G which is trivial in K, for h ∈ H we define ρ ( h ) := ˜ ρ ( g ) , where g is chosen in such a way that Φ( g ) = h. It is easy to see that ρ is well-defined. Remark 1. (a)
We shall write V and ˜ V instead of ρ and ˜ ρ respectively. (b) Throughout the paper we shall use repeatedly the following obvious remark.After fixing a basis of the vector space V , the sets of matrices { ρ ( h ) : h ∈ H } and { ˜ ρ ( g ) : g ∈ G } agree, showing that the character fields, the degrees and the Schur indicesof the representations V and ˜ V agree. The next lemma is a particular case of [7, Theorem 11.25]; we include a substan-tially simpler proof that fits this context better.
SEBASTI´AN REYES-CAROCCA AND RUB´I E. RODR´IGUEZ
Lemma 1.
Let G be a finite group and let K be a normal subgroup of G. If H = G/K then the correspondence V ˜ V defines a bijection between (a) the set of complex irreducible representations of H , and (b) the set of complex irreducible representations of G that are trivial in K. Proof.
We only have to prove that the rule V ˜ V restricts to a bijection at thelevel of complex representations that are irreducible. To do this, we recall the well-known fact that the irreducibility of a representation U of a group G is equivalentto h U, U i G = 1 . Now, the proof is clear after noticing that h V, V i H = h ˜ V , ˜ V i G for every complex representation V of H. (cid:3) Lemma 2.
Let G be a finite group and let K be a normal subgroup of G. If H = G/K then the correspondence V ˜ V defines a bijection between (a) the set of rational irreducible representations of H , and (b) the set of rational irreducible representations of G that are trivial in K. Proof. If V , V are representations of H, then it is straightforward to check that ^ V ⊕ V = ˜ V ⊕ ˜ V and ] ( V σ ) = ( ˜ V ) σ for every σ in the Galois group associated to the extension Q ≤ K V = K ˜ V . Theresult follows directly from Lemma 1, from Remark 1 and from the way in whichthe rational irreducible representations of a group are constructed (see (2.1)). (cid:3) Stamement and proof of the results
Let S be a compact Riemann surface and let H be a finite group acting on S. Inthis section we assume the existence of a compact Riemann surface C such that:(a) there exists a regular covering map π : C → S , and that(b) C admits the action of a supergroup G of the deck group K of π, in sucha way that K is normal in G and G/K ∼ = H. Note that S H ∼ = ( C/K ) / ( G/K ) ∼ = C G as Riemann surfaces, and the following diagram commutes. C SC G S H π ∼ = π G π H Proposition 1.
Let
Φ : G → H be a surjective homomorphism of groups whosekernel is K. If σ = ( a , . . . , a γ , b , . . . , b γ , c , . . . , c l ) is a generating vector of type ( γ ; m , . . . , m l ) representing the action of G on C, then the tuple Φ( σ ) = (Φ( a ) , . . . , Φ( a γ ) , Φ( b ) , . . . , Φ( b γ ) , Φ( c ) , . . . , Φ( c l )) is a generating vector of type ( γ ; m d , . . . , m l d l ) and represents the action of H on S, where d j = |h c j i ∩ K | . N JACOBIANS WITH GROUP ACTION AND COVERINGS 7
Proof.
Let p j denote the branch value of π G marked with m j . Then, over p j by π G there are exactly t j := | G | /m j different points; say π − G ( p j ) = { q j, , . . . , q j,t j } . Following [21, Section 3.1], the multiplicity d j,i of π at q j,i agrees with the order ofthe intersection group g i h c j i g − i ∩ K for a suitable representative g i of the left cosets of the normalizer of h c j i in G. The normality of K in G implies that d j,i := | g i ( h c j i ∩ K ) g − i | = |h c j i ∩ K | is independent of i , so we can set d j = d j,i ; note that d j divides m j .It follows that π − G ( p j ) is sent by π to( | G | /m j ) / ( | K | /d j ) = | H | / ( m j /d j )different points in S. This, in turn, says that π H has a branch value marked with m j /d j and, in the end, that the signature of the action of H on S is( γ ; m d , . . . , m l d l ) . Now, the proof is done after noticing that the elements of the tuple Φ( σ ) generate H, that the order of Φ( c j ) is m j /d j , and that Φ(1) = 1 . (cid:3) Let us employ the following notation:End Q ,G ( JC ) := Φ G,C ( Q [ G ]) ⊂ End Q ( JC )and End Q ,H ( JS ) := Φ H,S ( Q [ H ]) ⊂ End Q ( JS ) . Proposition 2.
Let
Φ : G → H be a surjective homomorphism of groups whosekernel is K. Then Φ induces surjective Q -algebra homomorphisms ˆΦ : Q [ G ] → Q [ H ] and ˇΦ : End Q ,G ( JC ) → End Q ,H ( JS ) , that make the following diagram commutative: Q [ G ] End Q ,G ( JC ) Q [ H ] End Q ,H ( JS )Φ G,C Φ H,S ˆΦ ˇΦ
Proof.
The map ˆΦ is the natural extension of Φ by linearity; namely Q [ G ] ∋ X g ∈ G λ g g X g ∈ G λ g Φ( g ) = X h ∈ H µ h h ∈ Q [ H ]where µ h = X g ∈ G, Φ( g )= h λ g . (4.1)It is not a difficult task to check that this is, in fact, a surjective homomorphismbetween Q -algebras. SEBASTI´AN REYES-CAROCCA AND RUB´I E. RODR´IGUEZ
Since a typical element of End Q ,G ( JC ) is of the form Σ g ∈ G λ g Φ G,C ( g ) , in a similarway as before, we define ˇΦ by the ruleEnd Q ,G ( JC ) ∋ X g ∈ G λ g Φ G,C ( g ) X g ∈ G λ g Φ H,S (Φ( g )) , and the last expresion can be rewritten as X h ∈ H µ h Φ H,S ( h ) ∈ End Q ,H ( JS )with µ h as defined in (4.1). Again, it is not difficult to check that ˇΦ is a surjectivehomomorphism between Q -algebras, and the diagram commutes by construction. (cid:3) We recall that associated to the covering map π : C → S there is a surjectivehomomorphism between the corresponding Jacobians: N π : JC → JS, called the norm of π. This map corresponds to the push-forward with respect to π, when we regard the Jacobian variety as the group of (equivalence classes of) degreezero divisors (or line bundles) on the Riemann surface; see e.g. [2, Chapter 11]. Proposition 3.
For every ϕ ∈ End Q ,G ( JC ) the following diagram commutes: JC JCJS JSϕ ˇΦ( ϕ ) N π N π Proof.
Let p , . . . , p t be different points of C and consider the degree zero divisor D = t X i =1 n i p i as a point of JC. If ϕ = P g ∈ G λ g Φ G,C ( g ) ∈ End Q ,G ( JC ) then ϕ ( D ) is defined as ϕ ( D ) = X g ∈ G λ g t X i =1 n i ǫ G,C ( g )( p i ) , where ǫ G,C : G → Aut( C ) is the monomorphism defining the action of G on C. Thus ( N π ◦ ϕ )( D ) = X g ∈ G λ g t X i =1 n i π ◦ ǫ G,C ( g )( p i ) . (4.2)Now, as K is a normal subgroup of G, we have that ǫ H,S ( h ) ◦ π = π ◦ ǫ G,C ( g )where ǫ H,S : H → Aut( S ) is the monomorphism defining the action of H on S, and h = Φ( g ) . It follows that the equality (4.2) can be rewritten as( N π ◦ ϕ )( D ) = X h ∈ H µ h t X i =1 n i ǫ H,S ( h ) ◦ π ( p i ) N JACOBIANS WITH GROUP ACTION AND COVERINGS 9 where µ h is as in (4.1). The last expression equals X h ∈ H µ h ǫ H,S ( h ) π t X i =1 n i p i = ( ˇΦ( ϕ ) ◦ N π )( D )and the proof follows. (cid:3) We are now in position to state and prove the main result of this paper:
Theorem 1.
Let S be a compact Riemann surface with action of a finite group H, and let JS ∼ H JS H × B n × · · · × B n r r (4.3) be the group algebra decomposition of JS with respect to H. Assume the existence of a compact Riemann surface C such that: (a) there exists a regular covering map π : C → S , and (b) C admits the action of a supergroup G of the deck group K of π, in such away that K is normal in G and G/K ∼ = H. Then the group algebra decomposition of JC with respect to G is JC ∼ G ( JS H × B n × · · · × B n r r ) × ( ˜ B ˜ n r +1 r +1 × · · · × ˜ B ˜ n s s ) where the factors ˜ B i , for r + 1 ≤ i ≤ s, are associated to the rational irreduciblerepresentations of G that are not trivial in K. Proof.
Let W . . . , W r be the rational irreducible representations of H . By Lemma2, the representations ˜ W , . . . , ˜ W r of G are precisely the rational irreducible repre-sentations of G that are trivial in K . If we denote by ˜ W r +1 , . . . , ˜ W s the rationalirreducible representations of G that are not trivial in K, then the group algebradecomposition of JC with respect to G is JC ∼ G ( JC G × ˜ B ˜ n × · · · × ˜ B ˜ n r r ) × ( ˜ B ˜ n r +1 r +1 × . . . × ˜ B ˜ n s s ) , (4.4)where the factor ˜ B i is associated to ˜ W i and ˜ n i = d ˜ V i /s ˜ V i . As S H ∼ = C G and as ˜ n i = n i for 2 ≤ i ≤ r (see Remark 1), the isogeny (4.4) canbe rewritten as JC ∼ G ( JS H × ˜ B n × · · · × ˜ B n r r ) × ( ˜ B ˜ n r +1 r +1 × . . . × ˜ B ˜ n s s ) , (4.5)and therefore to prove the theorem we have to show that ˜ B i and B i are isogenous,for each 2 ≤ i ≤ r. To accomplish this task, we proceed in three steps.
Claim 1.
The dimensions of B i and ˜ B i agree for every 2 ≤ i ≤ r. Let us suppose the signature of the action of G on C to be ( γ ; m , . . . , m l ) and σ = ( a , . . . , a γ , b , . . . , b γ , c , . . . , c l )to be a generating vector representing this action. Note that for every 2 ≤ i ≤ r and 1 ≤ k ≤ l we have ˜ ρ i ( c k ) = ( ρ i ◦ Φ)( c k ) = ρ i (Φ( c k ))and, consequently, the subspace of V i fixed under the subgroup h c k i of G ˜ V h c k i i = { v ∈ V i : ˜ ρ i ( c k )( v ) = v } agrees with the subspace of V i fixed under the subgroup h Φ( c k ) i of HV h Φ( c k ) i i = { v ∈ V i : ρ i (Φ( c k ))( v ) = v } . By Proposition 1, the signature of the action of H on S is( γ ; m d , . . . , m l d l )and the tuple Φ( σ ) is a generating vector representing this action. Now, we employthe aforementioned generating vectors in order to apply the formula (2.3) to (4.3)and (4.4). It follows thatdim( B i ) = k V i h d V i ( γ −
1) + 12 l X k =1 ( d V i − d h Φ( c k )) V i ) i = k ˜ V i h d ˜ V i ( γ −
1) + 12 l X k =1 ( d ˜ V i − d h c k )˜ V i ) i = dim( ˜ B i )and the proof of Claim 1 is done.Let us now consider the isotypical factors ˜ A i ∼ ˜ B n i i and A i ∼ B n i i of JC and JS with respect to G and H respectively, for each 2 ≤ i ≤ r. We recall that ˜ A i and A i agree with the image of the endomorphisms Φ G,C (˜ e i )and Φ H,S ( e i ) respectively, where˜ e i = d ˜ V i | G | X g ∈ G tr K ˜ Vi | Q ( χ ˜ V i ( g − )) g and e i = d V i | H | X h ∈ H tr K Vi | Q ( χ V i ( h − )) h. Claim 2. ˆΦ(˜ e i ) = e i for every 2 ≤ i ≤ r. The next equality follows directly from the definition of ˆΦ :ˆΦ(˜ e i ) = d ˜ V i | G | X g ∈ G tr K ˜ Vi | Q ( χ ˜ V i ( g − ))Φ( g ) . Furthermore, the fact that ˜ V i is trivial in K together with Remark 1 imply X g ∈ G, Φ( g )= h tr K ˜ Vi | Q ( χ ˜ V i ( g − )) = | K | tr K Vi | Q ( χ V i ( h − )for each h ∈ H. TherebyˆΦ(˜ e i ) = d V i | G | X h ∈ H | K | tr K Vi | Q ( χ V i ( h − )) h = e i . Claim 3.
The norm N π restricts to a homomorphism ˜ A i → A i for each 2 ≤ i ≤ r. Propositions 2 and 3 together with Claim 2 imply that the equalityΦ
H,S ( e i ) ◦ N π = N π ◦ Φ G,C (˜ e i )holds, for each 2 ≤ i ≤ r. By considering images, we obtain that A i = N π ( ˜ A i )proving Claim 3. N JACOBIANS WITH GROUP ACTION AND COVERINGS 11
Finally, from Claim 1 follows the fact that the factors ˜ A i and A i have the samedimension, showing that the homomorphism N π : ˜ A i → A i is, in fact, an isogeny;consequently ˜ B i and B i are isogenous for every 2 ≤ i ≤ r. This brings the proof to an end. (cid:3)
Remark 2. As S is isomorphic to the quotient C K , the isogeny decomposition (4.5) yields the following isogeny: JS ∼ ( JS H × ˜ B n K × · · · × ˜ B n Kr r ) × ( ˜ B ˜ n Kr +1 r +1 × · · · × ˜ B ˜ n Ks s ) (4.6)= ( JS H × ˜ B n × · · · × ˜ B n r r ) × ( ˜ B ˜ n Kr +1 r +1 × · · · × ˜ B ˜ n Ks s ) . (4.7) Note that from the comparison of dimensions in the isogenies (4.3) and (4.7) itfollows that the product ˜ B ˜ n Kr +1 r +1 × · · · × ˜ B ˜ n Ks s must be zero.The previous observation says that, if ˜ V is a representation of G which is nottrivial in K then ˜ V is “totally non-trivial”. More precisely: n K ˜ V = n ˜ V = ⇒ n K ˜ V = 0 . (4.8) The implication above can be understood in purely algebraic terms, as follows.Let ˜ V be a complex irreducible representation to G. Following [25, p. 264] , as K is a normal subgroup of G, the restriction ˜ V | K of ˜ V to K decomposes in terms ofirreducible representations of K as m ( ρ ⊕ · · · ⊕ ρ ν ) , (4.9) where m is some non-negative integer and ρ , . . . , ρ ν are conjugate representationsof ρ . If we assume that n K ˜ V = 0 then the trivial representation χ ,K of K must appearin the decomposition (4.9) . Without loss of generality, we can suppose ρ = χ ,K and therefore, as ρ , . . . ρ s are conjugate representations of ρ , we obtain that ˜ V | K ∼ = ( mν ) χ ,K . Note that mν equals the degree d ˜ V of ˜ V and, by Frobenius Reciprocity theorem,we obtain that d K ˜ V = h ˜ V , ρ K i G = h ˜ V | K , χ ,K i K = h d ˜ V χ ,K , χ ,K i K = d ˜ V showing that n K ˜ V = n ˜ V , as desired.In terms of Q -algebras, the implication (4.8) reveals the existence of a surjective Q -algebra homomorphism a : Q [ G ] ∼ = Q [ G ] K ⊕ R → Q [ H ∼ = G/K ] such that a ( R ) = 0 , and a | Q [ G ] K is an isomorphism onto Q [ H ] . More precisely, if G = g K ∪ . . . ∪ g d K is a decomposition of G into left cosets of K , with d = | H | , then a is defined as: e ˜ W | G | X g ∈ G ˜ α g Φ( g ) = 1 | G | d X i =1 ˜ α g i Φ( g i ) X k ∈ K ˜ α k , with ˜ α g = d ˜ V tr K ˜ V | Q ( χ ˜ V ( g − )) . Note that R is generated by those elements e ˜ W ∈ Q [ G ] with ˜ W being not trivialin K. In this case the result (4.8) ensures that n K ˜ V = 1 | K | X k ∈ K χ ˜ V ( k ) = 0 . In particular X k ∈ K ˜ α k = d ˜ V X k ∈ K tr K ˜ V | Q ( χ ˜ V ( k − )) = 0 showing that a ( R ) = 0 . By contrast, as Q [ G ] K is generated by those elements e ˜ W ∈ Q [ G ] with ˜ W beingtrivial in K, it is easy to see that the restriction of a to Q [ G ] K is defined by e ˜ W → e W showing that a restricts to an isomorphism from Q [ G ] K onto Q [ H ] . According to the notation used in the proof of Theorem 1, if B i is the abeliansubvariety of JS associated to the representation W i of H then ˜ B i denotes thecorresponding abelian subvariety of JC associated to the representation ˜ W i of G. We have proved that B i and ˜ B i are isogenous. Moreover: Theorem 2.
Consider subgroups N and N ′ of H such that N ≤ N ′ and i ∈{ , . . . , r } . Then: (1) B i ∼ JS N ⇐⇒ ˜ B i ∼ JC Φ − ( N ) . (2) B i ∼ P ( S N /S N ′ ) ⇐⇒ ˜ B i ∼ P ( C Φ − ( N ) /C Φ − ( N ′ ) ) . Proof.
By using the same arguments employed in the proof of Proposition 1, it isnot difficult to see that if σ is a generating vector representing the action of Φ − ( N )on C then Φ( σ ) is a generating vector representing the action of N on S. Thus, ina similar way as done in the proof of Theorem 1, it can be seen that d Φ − ( N )˜ V j = d NV j (4.10)for every 2 ≤ j ≤ r. Let us assume that ˜ B i ∼ JC Φ − ( N ) . This is equivalent to thegenus of C G being zero and d Φ − ( N )˜ V j = (cid:26) j = is ˜ V i if j = i for all 2 ≤ j ≤ r such that ˜ B j = 0 . By (4.10) it is clear that B i ∼ JS N . The converse is similar. Let us assume that B i ∼ JS N . This is equivalent to thegenus of S H being zero and d NV j = (cid:26) j = is V i if j = i for all 2 ≤ j ≤ r such that B j = 0 . Now, (4.10) implies that JC Φ − ( N ) ∼ ˜ B i × P for some abelian subvariety P of JC.
N JACOBIANS WITH GROUP ACTION AND COVERINGS 13
As the dimensions of B i and ˜ B i agree, to prove part (1) it is enough to checkthat the genera of S N and of C Φ − ( N ) agree. This fact follows from S N ∼ = ( C/K )(Φ − ( N ) /K ) ∼ = C Φ − ( N ) . To prove part (2) we proceed analogously. The equality d Φ − ( N )˜ V j − d Φ − ( N ′ )˜ V j = d NV j − d N ′ V j holds for every 2 ≤ j ≤ r and the sufficient condition is clearly satisfied. Conversely,if B i ∼ P ( S N /S N ′ ) then P ( C Φ − ( N ) /C Φ − ( N ′ ) ) ∼ ˜ B i × P for some abelian subvariety P of JC.
Now, as B i and ˜ B i have the same dimension,the result follows directly after noticing that the dimensions of P ( S N /S N ′ ) and of P ( C Φ − ( N ) /C Φ − ( N ′ ) ) agree. (cid:3) Jacobian isogenous to product of Jacobians
By arguing as in the proof of Theorem 1, we are able to derive conditions underwhich the Jacobian of a Riemann surface C with group action can be decomposedas a product of Jacobians of quotients of C by subgroups. Lemma 3.
Let ˜ W be a rational irreducible representation of G and let ˜ B ˜ W be thefactor associated to it in the group algebra decomposition of C with respect to G. Assume the existence of two subgroups K and K of G such that: (1) ˜ W is simultaneously trivial in K and K and (2) the genus of the quotient C h K ,K i is zero.Then ˜ B ˜ W = 0 . Proof.
Let us assume that ˜ B ˜ W has positive dimension. It is clear that there existsan abelian subvariety P of JC such that JC ∼ ˜ B ˜ n ˜ W × P, where ˜ n = d ˜ V /s ˜ V and ˜ V is a complex irreducible representation of G associated to˜ W .
Now, as the representation ˜ W is simultaneously trivial in K and K then it isalso trivial in the group h K , K i ; or, equivalently,˜ n h K ,K i = ˜ n. It follows that the induced isogeny decomposition of JC h K ,K i is JC h K ,K i ∼ ˜ B ˜ n ˜ W × Q for a suitable abelian subvariety Q of JC.
The result is now clear because the genus of C h K ,K i is assumed to be zero. (cid:3) Theorem 3.
Let us consider a compact Riemann surface C with action of a group G. If there exist normal subgroups K , . . . , K t of G such that the genus of C h K i ,K j i is zero, for every ≤ i = j ≤ t, then JC ∼ JC K × · · · × JC K t × P for some abelian subvariety P of JC.
Moreover if, in addition, g C = Σ ti =1 g C Ki then JC ∼ JC K × · · · × JC K t . Proof.
The set of rational irreducible representations R of G can be written as adisjoint union R = t [ j =1 R j ∪ t [ i,j =1 ,i = j R ij ∪ C where R j is the subset consisting of those representations which are trivial in K j butnot in K i for every i = j, while R ij is the subsets consisting of those representationswhich are simultaneously trivial in K i and K j , and C = R − ( t [ j =1 R j ∪ t [ i,j =1 ,i = j R ij ) . By the previous lemma, if ˜ W ∈ R ij then ˜ B ˜ W = 0 and therefore JC ∼ Q × t Y j =1 Y ˜ W ∈ R j ˜ B ˜ n ˜ W ˜ W (5.1)for some abelian subvariety Q of JC.
Now, we note that in (5.1)˜ n ˜ W = ˜ n K l ˜ W if and only if ˜ W ∈ R l . Thereby, the induced isogeny decomposition of JC K l is JC K l ∼ Q l × Y ˜ W ∈ R l ˜ B ˜ n ˜ W ˜ W for some abelian subvariety Q l of JC, for each 1 ≤ l ≤ t. This shows that JC ∼ JC K × · · · × JC K t × P where P := Q × Q × · · · × Q t , proving the first statement.Clearly P = 0 provided that g C = Σ ti =1 g C Ki and the proof is done. (cid:3) Remark 3. In [19] the authors have considered the same kind of decompositions,but on Riemann surfaces admitting actions of non necessarily normal subgroups;see also [14] . Examples
Example 1.
Let q ≥ S of genus g S = 1 + 2 q admitting a group ofautomorphism H isomorphic to the dihedral group D q = h a, b : a q = b = ( ab ) = 1 i of order 4 q, acting on S with signature (0; 2 , . . ., . The tuple˜ σ := ( ba, ba, b, b, b, b )can be chosen as a generating vector of H of the desired type.The complex irreducible representations of the dihedral groups are well-known.Namely, D q has four complex irreducible representation of degree one, given by V : a , b V : a , b
7→ − V : a
7→ − , b V : a
7→ − , b
7→ − , N JACOBIANS WITH GROUP ACTION AND COVERINGS 15 all of them being rationals; set W j = V j . Additionally, it has q − V k +4 : a diag( ξ k , ¯ ξ k ) , b ( )where ξ = exp(2 πi/ q ) and 1 ≤ k ≤ q − . We also recall that the dihedral groups only possess representations with Schurindex one. Hence, as the character field of each V l , for 5 ≤ l ≤ q + 3 , has degree( q − / Q , they give rise to exactly two rational irreducible representationsof H, say W and W (associated to V and V respectively).Thus, the group algebra decomposition of JS with respect to H is JS ∼ H JS H × B × B × B × B × B . The dimension of the factors is summarized in the following table:Factor JS H B B B B B Dimension 0 2 0 1 ( q − / q − / q :(1) the dihedral group D q = h r, s : r q = s = ( sr ) = 1 i . (2) the semidirect product D q ⋊ Z = h r, s, x : r q = s = ( sr ) = x = ( xr ) = ( xs ) = 1 i . These groups act as groups of automorphisms -say G and G respectively- offamilies of compact Riemann surfaces -say C and C respectively- of genus g = 1 + 8 q and g = 1 + 4 q, and signatures s = (0; 2 , . . .,
2) and (0; 2 , . . ., , respectively. The tuples σ = ( sr, sr, s, s, s, s, r q , r q ) and σ := ( sr, sr, s, s, x, x )can be chosen as generating vectors of the desired type, in each case.The correspondencesΦ : r a, s b and Φ : r a, s b, x b define epimorphisms of groups Φ j : G j → H for j = 1 ,
2; let K j denote the kernelof Φ j . It is a straightforward task to check the following facts:(1) Φ j ( σ j ) = σ. (2) K = h r q i ∼ = Z acts with 8 q fixed points on C . (3) K = h xs i ∼ = Z acts freely on C . (4) H ∼ = G j /K j acts on S ∼ = ( C j ) K j . We summarize the facts above in the next commutative diagram
S C C P π H π π π G π G Hence, by applying Theorem 1 to these two cases, we can assert that JC ∼ B × B × B × B × P for some abelian subvariety P = P ( C /S ) of JC of dimension 6 q , and JC ∼ B × B × B × B × P for some abelian subvariety P = P ( C /S ) of JC of dimension 2 q. In order to further decompose P j to obtain the group algebra decomposition of JC j with respect to G j we only need to care about of those representations of G j which are not trivial in K j . Note that, by contrast, to have a complete knowledgeof the irreducible representations of G is not a trivial task.We proceed to study both cases separately.(1) For j = 1 . Let us consider the two-dimensional complex irreducible repre-sentations of G given by r diag( i, − i ) , s ( ) and r diag( ω, ¯ ω ) , s ( )where ω = exp(2 πi/ q ) . It is easy to see that both of them are not trivial in K and that thecharacter fields are Q and Q ( ω + ¯ ω ) respectively. Clearly, they are notGalois associated and, in consequence, yield two rational irreducible rep-resentations of G –say ˜ W and ˜ W – and hence two factors in the groupalgebra decomposition of P with respect to G . Thus P ∼ ˜ B × ˜ B × Q for some abelian subvariety Q of P . Applying the formula (2.3), we seethat dim( ˜ B ) = 3 and dim( ˜ B ) = 3( q − Q = 0 . Thereby, the following isogeny is obtained JC ∼ G ( B × B × B × B ) × ( ˜ B × ˜ B ) . (2) For j = 2 . Let us consider the one-dimensional complex irreducible repre-sentations of G given by r
7→ − , s , x
7→ − r , s
7→ − , x , and the two-dimensional ones given by r diag( µ, ¯ µ ) s (cid:0) − − (cid:1) x ( )and r diag( µ , ¯ µ ) s ( ) x (cid:0) − − (cid:1) where µ = exp(2 πi/q ) . It is easy to see that each one of them is not trivial in K and thatthe character fields are Q and Q ( µ + ¯ µ ) respectively. Clearly, they arenot Galois associated and, in consequence, yield four rational irreduciblerepresentations of G –say ˆ W , ˆ W , ˆ W , ˆ W – and hence four factors in thegroup algebra decomposition of P with respect to G . Thus P ∼ ˆ B × ˆ B × ˆ B × ˆ B × Q
2N JACOBIANS WITH GROUP ACTION AND COVERINGS 17 for some abelian subvariety Q of P . Applying the formula (2.3), we seethatdim( ˆ B ) = dim( ˆ B ) = 1 and dim( ˆ B ) = dim( ˆ B ) = ( q − / Q = 0 . Thereby, the following isogeny is obtained JC ∼ G ( B × B × B × B ) × ˆ B × ˆ B × ˆ B × ˆ B . Finally, let us consider the subgroup N = h a q , b i ∼ = Z of H. Note that d NV j = (cid:26) j = 2 , ,
51 if j = 6( B = B = 0) and therefore B ∼ JS N . By Theorem 2, it follows that ˜ B := π ∗ ( B ) ∼ JC N where N = Φ − ( N ) = h r q , s i ∼ = D , while ˆ B := π ∗ ( B ) ∼ JC N where N = Φ − ( N ) = h r q , s, x i ∼ = Z . Example 2.
Let q ≥ C be a compact Riemannsurface with action of G ∼ = Z q such that the signature of the quotient is (0; q, q, q ) . If we denote by a and a two generators of G acting with fixed points, then σ = ( a , a , ( a a ) − )can be chosen as a generating vector of G of the desired type. The subgroups K i = h a a i i ∼ = Z q ≤ i ≤ q − C and h K i , K j i = G for 2 ≤ i = j ≤ q − . Now, by Theorem 3, the isogeny JC ∼ G JC K × · · · × JC K q − × P is obtained, for a suitable abelian subvariety P of JC.
Furthermore, as the genusof C K i equals ( q − / ≤ i ≤ q − g K + · · · + g K q − = ( q − q − / g C and, consequently, P = 0 . We obtain the isogeny JC ∼ JC K × · · · × JC K q − . References [1]
P. Barraza and A. M. Rojas , The group algebra decomposition of Fermat curves of primedegree,
Arch. Math. (Basel) (2015), no. 2, 145–155.[2]
Ch. Birkenhake and H. Lange , Complex Abelian Varieties, nd edition, Grundl. Math.Wiss. , Springer, 2004.[3] S. A. Broughton , Classifying finite group actions on surfaces of low genus.
J. Pure Appl.Algebra (1991), no. 3, 233–270.[4] A. Carocca, S. Recillas and R. E. Rodr´ıguez , Dihedral groups acting on Jacobians,
Contemp. Math. (2011), 41–77.[5]
A. Carocca and R. E. Rodr´ıguez,
Jacobians with group actions and rational idempotents.
J. Algebra (2006), no. 2, 322–343.[6]
M. Carvacho, R. A. Hidalgo and S. Quispe,
Jacobian variety of generalized Fermatcurves,
Q. J. Math. (2016), no. 2, 261–284.[7] C. W. Curtis and I. Reiner,
Methods of representation theory with applications to finitegroups and orders,
Vol 1, John Wiley & Sons (1981). [8]
O. Debarre,
Tores et vari´et´es ab´eliennes complexes,
Cours Sp´ecialis´es, . Soci´et´eMath´ematique de France, Paris; EDP Sciences, Les Ulis (1999).[9] H. Farkas and I. Kra,
Riemann surfaces , Grad. Texts in Maths. , Springer-Verlag(1980).[10] R. A. Hidalgo, L. Jim´enez, S. Quispe and S. Reyes-Carocca,
Quasiplatonic curves withsymmetry group Z ⋊ Z m are definable over Q , Bull. London Math. Soc. (2017) 165–183.[11] R. A. Hidalgo and R. E. Rodr´ıguez,
A remark on the decomposition of the Jacobianvariety of Fermat curves of prime degree,
Arch. Math. (Basel) (2015), no. 4, 333–341.[12]
M. Izquierdo, L. Jim´enez, and A. M. Rojas
Decomposition of Jacobian varieties of curveswith dihedral actions via equisymmetric stratification, arXiv:1609.01562[13]
L. Jim´enez,
On the kernel of the group algebra decomposition of a Jacobian variety . Rev.R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM (2016), no. 1, 185–199.[14]
R. Kani and M. Rosen,
Idempotent relations and factors of Jacobians , Math. Ann. (1989) 307-327.[15]
H. Lange and S. Recillas,
Abelian varieties with group actions . J. Reine Angew. Math. (2004) 135–155.[16]
J. Paulhus,
Decomposing Jacobians of curves with extra automorphisms , Acta Arith. (2008), no. 3, 231–244.[17]
J. Paulhus and A. M. Rojas,
Completely decomposable Jacobian varieties in new genera ,Experimental Mathematics (2017), no. 4, 430–445.[18] S. Recillas and R. E. Rodr´ıguez,
Jacobians and representations of S , Aportaciones Mat.Investig. , Soc. Mat. Mexicana, M´exico, 1998.[19] S. Reyes-Carocca and R. E. Rodr´ıguez,
A generalisation of Kani-Rosen decompositiontheorem for Jacobian varieties , arXiv:1702.00484, To appear in Ann. Sc. Norm. Super. PisaCl. Sci. DOI: 10.2422/2036-2145.201706-003[20]
J. Ries , The Prym variety for a cyclic unramified cover of a hyperelliptic curves , J. ReineAngew. Math. (1983) 59–69.[21]
A. M. Rojas , Group actions on Jacobian varieties , Rev. Mat. Iber. (2007), 397–420.[22] A. S´anchez-Arg´aez , Actions of the group A in Jacobian varieties , Aportaciones Mat.Comun. , Soc. Mat. Mexicana, M´exico (1999), 99–108.[23] F. Schottky and H. Jung , Neue S¨atze ¨uber Symmetralfunctionen und due Abel’schenFunctionen der Riemann’schen Theorie,
S.B. Akad. Wiss. (Berlin) Phys. Math. Kl. (1909),282–297.[24] J. P. Serre , Linear Representations of Finite Groups,
Graduate Texts in Mathematics Springer-Verlag, New York-Heidelberg (1977).[25]
M. Suzuki , Group Theory 2 , Grundlehren der Mathematischen Wissenschaften ,Springer-Verlag, New York (1986).[26]
W. Wirtinger , Untersuchungen ¨uber Theta Funktionen,
Teubner, Berlin, 1895.
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