Jacobians with complex multiplication
aa r X i v : . [ m a t h . AG ] J un JACOBIANS WITH COMPLEX MULTIPLICATION
ANGEL CAROCCA, HERBERT LANGE AND RUB´I E. RODR´IGUEZ
Abstract.
We construct and study two series of curves whose Jacobians admit complexmultiplication. The curves arise as quotients of Galois coverings of the projective linewith Galois group metacyclic groups G q, of order 3 q with q ≡ G m of order 2 m +1 . The complex multiplications arise as quotients of double cosetalgebras of the Galois groups of these coverings. We work out the CM-types and showthat the Jacobians are simple abelian varieties. Introduction
An abelian variety A over an algebraically closed field is said to have or to admit complexmultiplication if there is a CM-field K of degree 2 dim( A ) over Q such that K ⊂ End Q ( A ).A smooth projective curve C is said to admit complex multiplication , if its Jacobian vari-ety does. In these cases one says that A (respectively C ) has complex multiplication by K .In this paper we use Galois coverings of the projective line with metacyclic Galois groups,in order to construct and investigate two series of curves with complex multiplication.For the first series let q be an odd prime and n a positive integer such that n | q − G q,n := h a, b | a q = b n = 1 , b − ab = a k i where 1 < k < q is such that k n ≡ q and k m q for all 1 ≤ m < n (that is, the order of k mod q is n ). Denote the subgroup generated by b by H = h b i . In [4] Ellenberg used Galois coverings Y of the projective line with Galois group G q,n ,such that the Jacobian of the quotient curve X = Y /H admits a totally real field asan endomorphism algebra. In this note we show that his method can also be applied toconstruct smooth projective curves admitting complex multiplication. In fact, we showthat for every q as above with n = 3 there is exactly one Galois covering Y such that X = Y /H has complex multiplication. To be more precise, our first result is the followingtheorem. To state it, let ζ = ζ q denote a primitive q -th root of unity and let Q ( ζ ( n ) )denote the unique subfield of index n of the cyclotomic field Q ( ζ ). Clearly Q ( ζ ( n ) ) is aCM-field if and only if n is odd, which we assume in the sequel. Mathematics Subject Classification.
Key words and phrases.
Complex Multiplications, Jacobians, Abelian Varieties.The first and third author were supported by Fondecyt grants 1095165 and 1060742 respectively.
Theorem 1.
Suppose Y is a Galois covering of P with group G q,n with n odd and branchpoints p i ∈ P of ramification index n i for i = 1 , . . . , r over an algebraically closed field K of characteristic 0. (1) The curve X = Y /H admits complex multiplication by Q ( ζ ( n ) ) if and only if n = r = 3 and { n , n , n } = { q, , } . (2) For every odd prime q ≡ and n = 3 there is, up to isomorphism, exactlyone such curve Y .Furthermore, in this case the following results hold. (3) The Jacobian
J Y is isogenous to
J X . (4) The Jacobian
J X is a simple abelian variety, of dimension q − . (5) The function field of Y over K is K ( Y ) = K ( z, y ) where K ( P ) = K ( x ) and z and y satisfy the equations z = xx − and y q = ( z − z − ω ) k ( z − ω ) k , with ω a primitive third root of unity. For the second series of curves let m ≥ G m = h a, b | a m = b = 1 , bab = a d i where d = 2 m − − d ≡ m ).Let ξ = ξ m denote a primitive 2 m -th-root of unity, and observe that ξ + ξ d = ξ − ξ isnot real.Consider the complex irreducible representation V of G m given by V ( a ) = (cid:18) ξ ξ d (cid:19) , V ( b ) = (cid:18) (cid:19) Its character field K V = Q [ ξ + ξ d ] is a cyclic CM-field of degree [ K V : Q ] = 2 m − . Theorem 2 . Let m ≥ . Then (1)
There exists a Galois covering Y → P with Galois group G m , branched at points in P with monodromy a, b and ab . (2) The curve X = Y / h b i and the Prym variety P of the covering Y → X havecomplex multiplication by K V . (3) J Y is isogenous to
J X . (4) J X and P are isogenous simple principally polarized abelian varieties, of dimen-sion m − . (5) Y and X are hyperelliptic curves. An equation for Y is y = x ( x m − − . In the second section we fix the notation and collect some preliminaries on the repre-sentations of the groups G q,n and G m . Section 3 contains the proof of Theorem 1. To be more precise, in 3.1 we prove parts(1) and (2) of the theorem. In 3.2 we see how the Jacobians of X and Y are related.In fact, J Y is isogenous to the third power of
J X (part (3)). In particular
J Y alsoadmits complex multiplication. In 3.3 we apply the theorem of Chevalley-Weil in orderto compute the CM-type of
J Y . This implies part (4) of the theorem. Finally in 3.4 wework out the function field of the curve Y .In Section 4 we prove Theorem 2. To be more precise, in 4.1 we prove (1), (2) and (3)of Theorem 2. Section 4.2 contains the proof of (4). Moreover, in this section we computethe CM-types of J Y and
J X . Finally, in 4.3 we give the equations of the curves Y and X . We did not include the equation of X in Theorem 2, because it requires some morenotation.We would like to thank Eduardo Friedman and Wulf-Dieter Geyer for some valuableconversations. 2. Preliminaries
Some notation.
For any finite group G we denote by χ the trivial representationof G . If H is any subgroup of G , χ H will denote the character of the representation of G induced by the trivial representation of H . If H is cyclic generated by g ∈ G , we alsowrite χ g for χ h g i . In particular, if H = { } , then χ = χ { } is the character of the regularrepresentation of G .If V is a representation of G , then V H denotes the subspace of V fixed by H .All curves will be smooth, projective and irreducible; for simplicity we assume thecurves to be defined over the field of complex numbers. As in [4], the results remain validover any algebraically closed field of characteristic not dividing the group orders by using l ’adic cohomology and Grothendieck’s algebraic fundamental group instead of singularcohomology and usual monodromy.2.2. Representations of G q,n . Let G q,n denote the group defined in the introduction.As a semidirect product of the subgroup N = h a i by the subgroup H = h b i , it is ametacyclic group of order qn .Let ω = ω n be a primitive n -th root of unity. The non-isomorphic one-dimensionalrepresentations of G q,n are the following: χ i ( a ) = 1 ; χ i ( b ) = ω i for i = 0 , . . . n − . There are exactly s := q − n complex irreducible representations V i of dimension n , defined as follows. If a i , . . . , a i s with s = q − n is a set of representatives for the action of H on N defined by the relation b − ab = a k , the corresponding orbits are { a i j , a ki j , . . . , a k n − i j } for j = 1 , . . . , s . If ζ = ζ q denotes a primitive q -th root of unity, then for j = 1 , . . . , s the representation V j is given ANGEL CAROCCA, HERBERT LANGE AND RUB´I E. RODR´IGUEZ by V j ( a ) = ζ i j · · · ζ ki j · · · · · · ...0 0 0 · · · ζ k n − i j ; V j ( b ) = · · · · · · · · · · · · ... ...0 0 0 · · · . For any ℓ | n consider the subgroup S ℓ = h b nℓ i of H of order ℓ . Also, S ℓ ⊂ ker( χ i ) if andonly if n | nℓ i , and there are exactly nℓ such i ’s. We havedim V Hj = 1 and dim V S ℓ j = nℓ , and it is easy to check that χ N = χ + χ + · · · + χ n − ,χ H = χ + χ V + χ V + · · · + χ V s ,χ S ℓ = X ≤ i Proposition 2.1. Suppose the metacyclic group G q,n acts on the compact Riemann sur-face Y with monodromy g , . . . , g r , g r +1 . . . , g r + t , where g j has order n j (dividing n ) for j = 1 , . . . , r and order q for j = r + 1 . . . , u = r + t and assume g ( Y /G q,n ) = 0 . Then χ Y = ( r − n − X i =1 χ i − r X j =1 X
First observe that if g j has order n j dividing n , then h g j i is conjugate to S n j , andthat if g j has order q then h g j i = N .Then, from (2.1) and using the formulas above for χ N , χ H and χ S ℓ , we obtain χ Y = 2 χ + ( r + t − 2) ( χ + · · · + χ n − + n ( χ V + · · · + χ V s )) − t ( χ + · · · + χ n − ) + ( r X j =1 nn j )( χ V + · · · + χ V s ) + r X j =1 X ≤ i Representations of the group G m . For m ≥ G m denote the group definedin the introduction. As a semidirect product of the subgroup N = h a i by the subgroup H = h b i , it is a metacyclic group of order 2 m +1 .The group G m has 3 nontrivial representations of degree 1, namely χ : a , b 7→ − , χ : a 7→ − , b , χ : a 7→ − , b 7→ − . Let ξ = ξ m denote a primitive 2 m -th root of unity. For i = 1 , . . . , m − V i defined by V i ( a ) = (cid:18) ξ i − ξ i − d (cid:19) , V i ( b ) = (cid:18) (cid:19) . Note that V is the representation V of the introduction. V i is a complex irreduciblerepresentation with Galois character field K i = Q [ ξ i − + ξ i − d ]with [ K i : Q ] = 2 m − − i . Hence, V i has 2 m − − i non-equivalent complex irreducibleGalois-conjugate representations V i , . . . , V m − − i i . Since these representations are obvi-ously pairwise non-equivalent, we get in this way2 m − + 2 m − + ... + 1 = 2 m − − · + (2 m − − · = 2 m +1 = | G m | , which implies that these are all the complex irreducible representations of G m .The non-equivalent rational irreducible representations are, apart from χ , . . . , χ , therepresentations W i , i = 1 , . . . , m − 1, whose complexifications are W i ⊗ Q C ≃ ⊕ m − − i j =1 V ji . Note that W i is of degree 2 m − i .One checks that χ N = χ + χ ,χ H = χ + χ + m − X i =1 χ W i ,χ h ab i = χ + χ + m − X i =2 χ W i . Using this, we immediately obtain from (2.1), Proposition 2.2. Let Y → P denote a Galois covering with Galois group G m , m ≥ ,ramified over points of P with monodromy a , b and ( ab ) − . Then χ Y = χ W . ANGEL CAROCCA, HERBERT LANGE AND RUB´I E. RODR´IGUEZ Curves with Galois group G q,n Curves with complex multiplication by Q ( ζ ( n ) q ) . Let Y → P be a Galois cov-ering with group G q,n over an algebraically closed field K of characteristic 0. Considerthe curve X := Y /H where H denotes the subgroup generated by b . In this section we use Proposition 2.1in order to determine those Galois coverings Y for which the curve X admits complexmultiplication.Recall that a CM-field K of degree 2 g is a totally complex quadratic extension of atotally real field of rank g over Q .The main result of this section is the following Proposition. As in the introductionlet Q ( ζ ( n ) q ) denote the unique subfield of index n of the cyclotomic field Q ( ζ q ). It is aCM-field if and only if n is odd, which we assume in the sequel. Proposition 3.1. Suppose Y is a Galois covering of P with group G q,n and branchpoints p i ∈ P of ramification index n j dividing n for i = 1 , . . . , r and equal to q for j = r + 1 , . . . , r + t .Then the curve X = Y /H has complex multiplication by Q ( ζ ( n ) q ) if and only if n = r + t = 3 and { n , n , n } = { , , q } .Proof. According to the Hurwitz formula, the genus of X is g ( X ) = q − r + t − − r X j =1 n j ! . If the curve X has complex multiplication by Q ( ζ ( n ) q ) then [ Q ( ζ ( n ) q ) : Q ] = 2 dim( J X ).This is the case if and only if q − n = 2 q − r + t − − r X j =1 n j ! which is equivalent to(3.1) r X j =1 nn j = n ( r + t − − . This implies t ≤ t = 2, then (3.1) says P rj =1 nn j = nr − 1, a contradiction, since nn j is either 1 or ≥ t = 0, then (3.1) says P rj =1 nn j = n ( r − − 1. Since n is odd, n j ≥ 3, which implies nr ≥ nr − n − r ≤ n . This gives r = 2 or 3. In both cases (3.1) cannotbe satisfied. For r = 2 the right hand side would be negative and for r = 3 the evennumber n − n . Hence t = 1, in which case (3.1) says P rj =1 nn j = n ( r − − 1. By the same argumentas above, this implies nr ≥ nr − n − 1, which gives 23 r ≤ n . Since n ≥ 3, thisimplies r ≤ 32 (1 + 13 ) = 2 and thus r = 2. But then (3.1) says nn + nn = n − n = n = n = 3.Hence we have shown that the curve X = Y /H has complex multiplication by Q ( ζ ( n ) q )only if n = r + t = 3 and { n , n , n } = { , , q } .Certainly there exists a Galois covering of this type, with branch points p , p , p in P and stabilizers G p = h a i , G p = h b i and G p = h ab i , since ab ( ab ) − = 1.We will now finish the proof by showing that under the assumptions Q ( ζ (3) q ) is containedin End Q ( J X ).Let W be the rational irreducible representation of G q, whose complexification is thedirect sum of all the irreducible complex representations V j of dimension 3.Note that Proposition 2.1 in our case says χ Y = χ W . This, together with [5, p. 202,Corollaire], implies that there is an isomorphism of Q [ G ]-modules H ( Y, Q ) ≃ W and an isomorphism of Q [ H \ G/H ]-modules(3.2) H ( X, Q ) ≃ H ( Y, Q ) H ≃ ( W H ) ⊕ m with m = n ( r + t − − r X j =1 nn j = 3(2 + 1 − − X j =1 Q [ H \ G/H ] → End Q ( J X )induces a homomorphism ρ : Q [ H \ G/H ] → End( H ( X, Q )) = End( W H ), and the imageof ρ is isomorphic to the image of Q [ H \ G/H ] in End Q ( J X ).Now the image of Q [ G ] in End( W ) is isomorphic to the 3 × Q ( ζ (3) q ), and the fact that dim V H = 1 implies that Q [ H \ G/H ] ∼ = Q ⊕ Q ( ζ (3) q ) (see [2,Theorem 4.4] and [4]).Hence the image of Q [ H \ G/H ] in End( W H ) is isomorphic to Q ( ζ (3) q ) and therefore Q ( ζ (3) q ) ⊂ End Q ( J X ) . (cid:3) We have thus proven the following result. Corollary 3.2. Let Y be a Galois covering of P with group G := G q, . Then X = Y /H has complex multiplication by Q ( ζ (3) q ) if and only if Y → P is isomorphic to the G -covering with branch points p , p , p in P and stabilizers G p = h a i , G p = h b i and G p = h ab i . ANGEL CAROCCA, HERBERT LANGE AND RUB´I E. RODR´IGUEZ Remark 3.3. For each q there exists a unique curve Y satisfying the conclusions in theCorollary; hence for every q there is only one such curve X , as was observed already byLefschetz in [6, p. 463].3.2. The Jacobians J Y and J X . Let q be an odd prime with q ≡ G := G q, = < a, b | a q = b = 1 , b − ab = a k > where k ≡ q, < k < q . Let Y and X denote the curves of Corollary 3.2. Inthis section we want to see how the Jacobians J Y and J X are related.First, the Hurwitz formula gives(3.3) g ( Y ) = q − 12 and g ( X ) = q − . In particular g ( Y ) = 3 g ( X ). The following proposition is more precise. Proposition 3.4. The Jacobian of Y is isogenous to the third power of the Jacobian ofX: J Y ∼ J X . In particular J Y admits complex multiplication by Q ( ζ q ) .Proof. There are exactly 2 nontrivial rational irreducible representations of G , namely W , whose complexification is χ ⊕ χ , and W , whose complexification is V ⊕ · · · ⊕ V s ,with χ i and V j as defined in Section 2. Correspondingly, according to [2, Proposition 5.2],there are abelian subvarieties B and B of J Y , uniquely determined up to isogeny, suchthat J Y ∼ B dim χ m × B dim V m and J X ∼ B dim χH m × B dim V H m , where m i is the Schur index of the corresponding representation. Hence m = m = 1.Since dim V = 3 and dim V H = 1, it suffices to show for the first assertion that dim B =0. This is a consequence of [7, Theorem 5.12], which in our case saysdim B = [ K χ : Q ]( − dim χ + 12 X j =1 (dim χ − dim χ G j )) , where K χ denotes the field generated by the values of the character χ over Q , G = h a i , G = h b i , and G = h ab i . Hence we getdim B = 2( − − − − , which completes the proof of the first assertion. According to Corollary 3.2, J X admitscomplex multiplication by Q ( ζ (3) q ). The last assertion follows from the first and the factthat Q ( ζ q ) is a degree-three CM-extension of Q ( ζ (3) q ). (cid:3) Remark 3.5. The Prym variety P of the threefold covering Y → X is defined as theconnected component containing 0 of the kernel of the norm map J Y → J X . Since J Y is isogenous to the product J X × P , we deduce from Corollary 3.2 and Proposition 3.4that P ∼ J X . In particular, P admits complex multiplication by a CM-extension of degree 2 of Q ( ζ (3) q ).3.3. The CM-types of J Y and J X . Recall that a CM-field K of degree 2 g admits g pairs of complex conjugate embeddings into the field of complex numbers. A CM-type of K is by definition the choice of a set of representatives of these pairs; that is, a set of g pairwise non-isomorphic embeddings K ֒ → C .The Jacobian J C of a curve C admits complex multiplication by a CM-field K , if andonly if H ( C, Q ) is a K -vector space of dimension 1. In this case the Hodge decomposition H ( C, C ) = H ( C, ω C ) ⊕ H ( C, ω C )induces a CM-type on the field K . It is called the CM-type of the Jacobian J C . In thissection we compute the CM-types of the Jacobians J Y and J X of Section 4.We need some elementary preliminaries. The congruence k ≡ q admits exactly 2 integer solutions with 2 ≤ k ≤ q − 2. If k is one such solution, q − − k is the other one.For any integer n let [ n ] denote the uniquely determined integer with0 ≤ [ n ] ≤ q − n ] ≡ n mod q and for any integer ℓ, ≤ ℓ ≤ q − O ℓ := { ℓ, [ kℓ ] , [ k ℓ ] } . Lemma 3.6. Let k be an integer with ≤ k ≤ q − and k ≡ q . Then (1) 1 + k + [ k ] = q ;(2) For any ℓ, ≤ ℓ ≤ q − , ℓ + [ kℓ ] + [ k ℓ ] = (cid:26) q if of the numbers in O ℓ are smaller than q − , q otherwise . Proof. (1): We have 1 + k + [ k ] ≡ q . On the other hand, 1 < k + [ k ] ≤ q − + q − q − . This implies the assertion.(2): We have ℓ + kℓ + k ℓ = ℓ (1 + k + k ) ≡ q . Hence q divides ℓ + [ kℓ ] + [ k ℓ ].On the other hand, 1 ≤ ℓ + [ kℓ ] + [ k ℓ ] < q . This implies ℓ + [ kℓ ] + [ k ℓ ] = q or 2 q. Suppose 2 of the 3 numbers are less than q − . Then ℓ + [ kℓ ] + [ k ℓ ] < q − + q − < q and hence ℓ + [ kℓ ] + [ k ℓ ] = q .In the remaining case at least one of the numbers is > q − , implying ℓ + [ kℓ ] + [ k ℓ ] > q − = q which completes the proof. (cid:3) The following lemma gives a criterion for the set O ℓ to contain two elements less than q − . Lemma 3.7. For any integer ℓ ; 1 ≤ ℓ ≤ q − consider the real number β ℓ := sin (cid:18) πℓq (cid:19) + sin (cid:18) πkℓq (cid:19) + sin (cid:18) πk ℓq (cid:19) . Then the set O ℓ contains two elements less than q − if and only if β ℓ > .Proof. Suppose O ℓ contains two elements less than q − . Without loss of generality wemay assume that they are ℓ and [ kℓ ]. Since 1 + k + k ≡ q , we havesin (cid:18) π [ k ℓ ] q (cid:19) = sin (cid:18) π [ − (1 + k ) ℓ ] q (cid:19) = − sin (cid:18) π [(1 + k ) ℓ ] q (cid:19) = − (cid:20) sin (cid:18) πℓq (cid:19) cos (cid:18) π [ kℓ ] q (cid:19) + cos (cid:18) πℓq (cid:19) sin (cid:18) π [ kℓ ] q (cid:19)(cid:21) . Hence β ℓ = sin (cid:18) πℓq (cid:19) + sin (cid:18) π [ kℓ ] q (cid:19) + sin (cid:18) π [ k ℓ ] q (cid:19) = sin (cid:18) πℓq (cid:19) (cid:20) − cos (cid:18) π [ kℓ ] q (cid:19)(cid:21) + sin (cid:18) π [ kℓ ] q (cid:19) (cid:20) − cos (cid:18) πℓq (cid:19)(cid:21) is positive, because both ℓ and [ kℓ ] are positive and smaller than q − .The “only if” part of the assertion follows from the fact that the elements of O q − ℓ arethe negatives mod q of the numbers of O ℓ , and therefore β q − ℓ = − β ℓ . This completes theproof. (cid:3) Let the notation be as in Section 2 with n = 3. In particular { a i , . . . , a i s } ( s = q − )denotes a set of representatives for the action of the group H = h b i on the group N = h a i .The corresponding orbits are { a i j , a ki j , a k i j } for j = 1 , . . . , s . Now with { a i j , a ki j , a k i j } also { a q − i j , a k ( q − i j ) , a k ( q − i j ) } is an orbit, disjoint from it. Hence we can enumerate theorbits in the following way: The orbits of N under the adjoint action of the group H areexactly(3.4) { a i ν , a ki ν , a k i ν } s ν =1 and { a q − i ν , a k ( q − i ν ) , a k ( q − i ν ) } s ν =1 , where 2 of the numbers i ν , [ ki ν ] , [ k i ν ] are less than q − (and thus 2 of the numbers q − i ν , [ k ( q − i ν )] , [ k ( q − i ν )] are ≥ q − ).If V j denotes the complex irreducible representations as in Section 2 for j = 1 , . . . , s ,then we have Proposition 3.8. H ( Y, ω Y ) = s M ν =1 V ν . Proof. Since 3 s q − 12 = g ( Y ), it suffices to show that every V ν with 1 ≤ ν ≤ s occursexactly once in the representation H ( Y, ω Y ) of G .According to the Theorem of Chevalley-Weil (see [3]), the representation V ν occursexactly(3.5) N = − deg V ν + X µ =1 n µ − X α =0 N µ,α (cid:28) − αn µ (cid:29) times in the representation H ( Y, ω Y ), where • µ runs through the branch points of the covering Y → P , • n µ is the order of the µ -th branch point, • N µ,α denotes the multiplicity of the eigenvalue e πiαnµ in the matrix V ν ( g µ ), where g µ is any nontrivial element of G stabilizing a point in the fiber of µ , and • h r i := r − ⌊ r ⌋ denotes the fractional part of the real number r .Hence we have n = q, n = n = 3 and thus N = − (cid:28) − i ν q (cid:29) + (cid:28) − [ ki ν ] q (cid:29) + (cid:28) − [ k i ν ] q (cid:29) + 2 (cid:28) − (cid:29) + 2 (cid:28) − (cid:29) = − q − i ν q + q − [ ki ν ] q + q − [ k i ν ] q = 1 , where the last equation follows from Lemma 3.6 (2), since by assumption, 2 of the numbers q − i ν , q − [ ki ν ] , q − [ k i ν ] are ≥ q − . (cid:3) Corollary 3.9. Let Y and X denote the curves of Section 3, and denote ζ = ζ q = e πiq .Let { a i ν , a ki ν , a k i ν } s ν =1 be the first half of the orbits in (3.4) . Then (1) The CM-type of J Y is given by the following g ( Y ) = 3 s embeddings ϕ j of Q ( ζ ) into C : ϕ j ( ζ ) = ζ j for j in { i , k i , k i , . . . , i s , k i s , k i s } . (2) Denoting α ν := ζ i ν + ζ ki ν + ζ k i ν for ν = 1 , . . . , s , the CM-type of J X is given by thefollowing g ( X ) = s embeddings ψ ν of Q ( α ) into C : ψ ν ( α ) = α ν .Proof. (1) is a direct consequence of Propositions 3.4 and 3.8, and the definition of therepresentations V ν in Section 2.According to Corollary 3.2 the Jacobian J X has complex multiplication by Q ( ζ (3) ).Hence (2) follows from Theorem 3.8 and the fact that Q ( ζ (3) ) = Q ( α ) = Q ( ζ + ζ k + ζ k )is the only subfield of Q ( ζ ) of index 3. (cid:3) Proposition 3.10. The Jacobian J X is a simple abelian variety, of dimension q − . This proves part (4) of Theorem 1 in the Introduction. Proof. Let Φ denote the CM-type of J X , i.e. of the field Q ( ζ (3) ) as given in Corollary 3.9(2). With α = ζ + ζ k + ζ k as above and µ := α − α we have • K = Q ( α + α ) is totally real; • η := − µ = 4 β is a totally positive element of K (by Lemma 3.7); • The elements of Φ are exactly the embeddings ϕ : Q ( µ ) ֒ → C for which theimaginary part of ϕ ( µ ) is positive (according to Lemma 3.7).We have to show that Φ is a primitive CM-type. For this we apply the criterion [8,Prop.27] of Shimura-Taniyama which says the following: Φ is primitive if and only if thefollowing two conditions are satisfied:(i) K ( µ ) = Q ( µ );(ii) for any conjugate α ′ of α over Q , other than α itself, α ′ α is not totally positive.The first condition holds trivially, since both fields are equal to Q ( ζ (3) ) = Q ( α ). As forthe second condition: for any conjugate µ ′ of µ over Q different from µ , µ ′ µ is not totallypositive, because µ ′ µ runs over β ℓ β . (cid:3) The function field K ( Y ) . In this section we use Kummer theory in order to provepart (5) of Theorem 1 in the Introduction.Let Y be the Galois covering of P of Corollary 3.2, with Galois group G q, ramifiedover the points p , p and p of P in affine coordinates. The subgroup N = h a i is normalof index 3 in G q, , and gives a factorization of the covering Y → P into cyclic coverings Y → Z := Y /N of degree q and Z → P of degree 3. The last covering is ramified over p and p . Hence, according to the Hurwitz formula, g ( Z ) = 0 . We choose an affine coordinate x of P in such a way that p = 1 , p = 0 and p = 2.Then the covering Z → P is given by the equation z = xx − Z = P is K ( z ). Proposition 3.11. Let ω denote a primitive third root of unity, and choose < k < q such that k ≡ q . A (singular) model of the curve Y is given by the equation y q = ( z − z − ω ) k ( z − ω ) k . With these notations, automorphisms σ and τ of the curve Y of corresponding orders q and , are given by σ : z z, y ζ q y , and τ : z ω z, y ω m ′ ( z − ω ) m y k where m and m ′ are given by k = mq + 1 and k + k + 1 = m ′ q . An immediate consequence of the proposition is statement (5) of Theorem 1 in theIntroduction. Proof. The covering Y → Z is ramified exactly over the points 1 , ω and ω of Z = P .According to Kummer theory the covering Y → Z is given by the affine equation(3.6) y q = ( z − z − ω ) k ( z − ω ) k , with k as in the statement of the proposition.The automorphism τ : Z → Z given by z ω z extends to the automorphism of Y asdefined in the proposition since, denoting the right hand side of (3.6) by F = F ( z ), wehave τ ( F ) = ω k + k ( z − ω )( z − k ( z − ω ) k = ω k + k ( z − ω ) mq F k = (cid:18) ω m ′ ( z − ω ) m y k (cid:19) q . If σ denotes the automorphism of Y given above, then it is clear that Y → P is a Galoiscovering with Galois group < σ, τ > = G q, . (cid:3) Curves with Galois group G m Complex multiplication of the curves. Let Y → P denote a Galois coveringwith Galois group G m = h a, b | a m = b = 1 , bab = a d i for m ≥ d = 2 m − − 1, branched over 3 points in P with monodromy a, b and ( ab ) − . Notice that such a covering exists, since ab ( ab ) − = 1. In fact, for any m ≥ Y , g ( Y ) = 2 m − . Consider the curve X := Y /H where H denotes the subgroup generated by b . We want to show that X admits complexmultiplication.As in Section 2.3 let ξ = ξ m denote a primitive 2 m -th root of unity. The complexrepresentation V = V has character field K V = Q ( ξ + ξ d ), and its Schur index is equalto 1. The field K V is of CM-type and degree [ K V : Q ] = 2 m − . Proposition 4.1. The curve X has complex multiplication by K V . In particular, g ( X ) = 2 m − . Proof. As in Section 2.3, let W denote the rational irreducible representation whosecomplexification is ⊕ m − j =1 V j . According to [5, p.202, Corollaire] and Proposition 2.2above, there is an isomorphism of Q [ H \ G/H ]-modules(4.1) H ( X, Q ) ≃ W H . Since dim V H = 1, this implies dim W H = 2 m − and thus g ( X ) = 2 m − . Moreover (4.1)implies that the canonical map Q [ H \ G/H ] → End Q ( J X ) induces a homomorphism p : Q [ H \ G/H ] → End( W H ) . whose image is isomorphic to the image of Q [ H \ G/H ] in End Q ( J X ). Now the image of Q [ G ] in End( W ) is isomorphic to the 2 × K V , and Q [ H \ G/H ] isisomorphic to Q ⊕ K V (since dim V H = 1).Hence the image of Q [ H \ G/H ] in End( W H ) is isomorphic to K V , which means thatthe curve X admits complex multiplication by K V . (cid:3) This proves (2) of Theorem 2 of the Introduction. Part (4) of Theorem 2 is proven bythe following proposition. Proposition 4.2. The Jacobian of Y is isogenous to the second power of the Jacobian ofX: J Y ∼ J X . In particular J Y admits complex multiplication by the cyclotomic field Q ( ξ m ) .Proof. We already know from Proposition 2.2 that there is only one nontrivial isogenyfactor of J Y : it is associated to the representation W . Using [2, Proposition 5.2] thismeans that there is an abelian subvariety B of J Y such that J Y ∼ B , using that dim V = 2 and the Schur index of V is 1. Since dim V H = 1 we get moreoverfrom [2] that J X ∼ B . The two isogenies together imply the first assertion. Since J X admits complex multipli-cation by a CM-subfield of index 2 in Q ( ξ m ), J Y ∼ J X admits complex multiplicationby Q ( ξ m ). (cid:3) The projection map π b : Y → X is a double covering ramified at two points. Hence itsPrym variety P = ker( J Y → J X ) is a principally polarized abelian variety, of dimensionequal to g ( X ) = 2 m − . Corollary 4.3. The Prym variety of the covering π b : Y → X has complex multiplicationby K V .Proof. This follows immediately from Propositions 4.1 and 4.2, and the fact that J Y isisogenous to the product J X × P . (cid:3) The CM-types of J Y and J X . In order to determine the CM-types of J Y and J X , recall that, according to Proposition 2.2, the representation H ( Y, Q ) of G m is justthe rational irreducible representation W . Moreover, the complexification of W is thedirect sum of the [ K V : Q ] = 2 m − Galois conjugate representations V j of the complexirreducible representation V . For every positive integer i consider the complex irreduciblerepresentation U i defined by U i ( a ) = (cid:18) ξ i − ξ (2 i − d (cid:19) , U i ( b ) = (cid:18) (cid:19) . and denote U ′ i := U m − + i . Lemma 4.4. The complex irreducible representations V j , j = 1 , . . . , m − are given bythe representations U , . . . , U m − and U ′ , . . . , U ′ m − . The representation U ′ i is the complexconjugate of U i for i = 1 , . . . , m − . Proof. First note that U coincides with the representation V and every U i is Galoisconjugate to U . Moreover clearly U , . . . , U m − are pairwise non-isomorphic. Henceit suffices to show that U ′ i is the complex conjugate of U i . But this follows from thecongruences2 m − + 2 i − ≡ − (2 i − d mod 2 m and (2 m − + 2 i − d ≡ − (2 i − 1) mod 2 m . (cid:3) Proposition 4.5. H ( Y, ω Y ) = ⊕ m − i =1 U i . Proof. Since 2 · m − = g ( Y ), it follows from Lemma 4.4 that it suffices to show that every U i with 1 ≤ i ≤ m − occurs exactly once in the representation H ( Y, ω Y ) of G m . Againthis is a consequence of the Theorem of Chevalley-Weil; that is, equation (3.5).Here Y → P is branched over 3 points in P with n = 2 m , n = 2 and n = 4, and wehave for the representation U i , ≤ i ≤ m − , N = − (cid:28) − (2 i − m (cid:29) + (cid:28) − (2 i − d m (cid:29) + (cid:28) − (cid:29) + (cid:28) − (cid:29) + (cid:28) − (cid:29) = − m − i + 12 m + 2 m − + 2 i − m + 12 + 34 + 14= 1 . (cid:3) As a consequence we get Corollary 4.6. Let ξ = e πi m . Then we have (1): The CM-type of J Y is given by { ξ i − , | i = 1 , . . . , m − } ; (2): The CM-type of J X is given by { ξ i − + ξ (2 i − d | i = 1 , . . . , m − } .Proof. According to Proposition 4.5 the CM-type of J Y is { ξ i − , ξ (2 i − d | i = 1 , . . . , m − } .This implies (1), since ξ (2 i − d ≡ m − − i + 1 mod 2 m for i = 1 , . . . , m − . (2) is animmediate consequence of (1), since the CM-field of X is the fixed field of the involution ξ ξ d . (cid:3) Proposition 4.7. The Jacobian J X and the Prym variety P of the covering Y → X aresimple abelian varieties of dimension m − . This proves part (4) of Theorem 2 of the introduction. Proof. It suffices to show that the field K V = Q ( ξ + ξ d ) does not admit a proper CM-subfield, since then every CM-type of it is primitive, and in particular J X is a simpleabelian variety.It is well known that the Galois group of the extension Q ( ξ ) | Q is h σ i × h τ i , with h σ i is cyclic of order 2 m − generated by σ : ξ ξ , and τ is the involution τ : ξ ξ d .Therefore the Galois group of K V | Q , which is the fixed field of τ , is h σ i ≃ Z / m − Z . Itsonly element of order 2 is σ m − , which must be complex conjugation in K V . Since everynontrivial subgroup of h σ i contains σ m − , this implies that every proper subfield of K V is real. (cid:3) Equations. Let the notation be as in the previous subsections. Here we want togive equations for the curves Y and X .First note that the center Z = Z ( G m ) = h a m − i of the group G m is of order two;furthermore, | Z \ G m / h a i| = 2 , | Z \ G m / h b i| = 2 m − , | Z \ G m / h ab i| = 2 m − . According to [7], for any subgroup H of G acting on a curve Y with monodromy g , . . . , g t ,the genus of the quotient Y /H is given by g Y/H = [ G : H ]( g Y/G − 1) + 1 + 12 t X j =1 ([ G : H ] − | H \ G/ h g j i| ) . We obtain that in our case g ( Y /Z ) = 0, and therefore Y is hyperelliptic.We may choose coordinates in such a way that an affine equation for Y is y = x ( x m − − , and if ξ = ξ m , then the automorphisms a and b of the curve Y are given by a ( x, y ) = ( ξ x, ξ y ) ; b ( x, y ) = (cid:18) ξ x , − i ξ d yx ( m − +1 (cid:19) . Note that a m − ( x, y ) = ( x, − y ) =: j ( x, y ) , where j denotes the hyperelliptic involution of Y ,Also, b and j generate a Klein group, with the following associated diagram of covers. Y π b (cid:15) (cid:15) π bj & & NNNNNNNNNNNN π j w w nnnnnnnnnnnnn Y / h j i = P ' ' PPPPPPPPPPPPP Y / h b i = X (cid:15) (cid:15) Y / h b j i x x qqqqqqqqqqqq P π j : Y → P is the hyperelliptic covering ramified over { , ∞ , ξ i m − : 0 ≤ i ≤ m − − } , π b ramifies at the two fixed points P i = ( ξ d , y ) with y = − ξ d of b , and π bj ramifies atthe two fixed points ( − ξ d , y ) with y = 2 ξ d of bj .Since X is then hyperelliptic and ramifies over π b ( P ), π b ( P ), and the images under π b of the Weierstrass points of Y , we may consider f : P → P of degree two and invariantunder x → ξ x such as f ( x ) = ξ ( ξ x + 1) ξ x + 1and adequate g ( x, y ) so that π b ( x, y ) = ( f ( x ) , g ( x, y ))(= ( u, v )), and we obtain X = Y / h b i : v = u ( u − ξ ) m − − Y k =0 ( u − f ( ξ k )) . Remark 4.8. In the case m = 4 we obtain the curve of genus two X = Y / h b i : v = u − ξ u + 2 ξ u + 14 ξ u − ξ u − ξ u whose Jacobian has complex multiplication by Q ( ξ + ξ ) = Q ( p − √ y = − x + 3 x + 2 x − x − x + 1The curves are isomorphic, because they have the same Igusa invariants i := I /I = 1836660096 = 2 · , i := I · I /I = 28343520 = 2 · · 5, and i := I · I /I = 9762768 = 2 · · References [1] S. Broughton: The homology and higher representations of the automorphism group of a Riem-man surface . Trans. AMS, 300 (1987), 153-158.[2] A. Carocca, R. E. Rodr´ıguez: Jacobians with group actions and rational idempotents . J. Alg.306 (2006), 322-343.[3] C. Chevalley, A. Weil: ¨Uber das Verhalten der Integrale erster Gattung bei Automorphismandes Funktionenk¨orpers . Hamb. Abh. 10 (1943), 358-361.[4] J.S. Ellenberg: Endomorphism Algebras of Jacobians . Adv. in Math. 162 (2001), 243-271.[5] A. Grothendieck: Sur quelques points d’alg`ebre homologique . Tohoku Math. J. 9 (1957), 119-221.[6] S. Leschetz: On certain numerical invariants of algebraic varieties with application to abelianvarieties (cont.) Trans. Amer. Math. Soc. 22 (1921), no. 4, 407–482.[7] A.M. Rojas: Group actions on Jacobian varieties . Rev. Mat. Iber. 23 (2007), 397-420.[8] G. Shimura, Y. Taniyama: Complex Multiplication of Abelian Varieties . Math. Soc. Japan(1961).[9] P van Wamelen: Examples of genus two CM curves defined over the rationals . Math. Comp. 68(1999), 307-320. A. Carocca, Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Casilla306-22, Santiago, Chile E-mail address : [email protected] H. Lange, Mathematisches Institut, Universit¨at Erlangen-N¨urnberg, Germany E-mail address : [email protected] R. E. Rodr´ıguez, Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile,Casilla 306-22, Santiago, Chile E-mail address ::