On a property of Fermi curves of 2-dimensional periodic Schrödinger operators
aa r X i v : . [ m a t h - ph ] J un ON A PROPERTY OF FERMI CURVES OF -DIMENSIONAL PERIODIC SCHR ¨ODINGER OPERATORS EVA L ¨UBCKE
Abstract.
We consider a compact Riemann surface with a holomor-phic involution, two marked fixed points of the involution and a divisorobeying an equation up to linear equivalence of divisors involving all thisdata. Examples of such data are Fermi curves of 2-dimensional periodicSchr¨odinger operators. We show that the equation has a solution if andonly if the two marked points are the only fixed points of the involution. Introduction
Let X be a compact Riemann surface of genus g < ∞ and σ : X → X a holomorphic involution with two fixed points P and P . Then the linearequivalence D + σ ( D ) ≃ K + P + P is solvable by a divisor D of degree g if and only if P and P are the only fixed points of σ . It is known that onFermi curves of 2-dimensional periodic Schr¨odinger operators, there exists aholomorphic involution σ with two fixed points such that the pole divisor ofthe corresponding normalized eigenfunctions obeys this linear equivalence,see [N-V]. It was remarked in [N-V] without proof that I.R. Shafarevich andV. V. Shokurov pointed out that D + σ ( D ) ≃ K + P + P can hold if andonly if P and P are the only fixed points of σ . To prove this assertion,we basically use the results reflecting the connection between the Jabobianvariety and the Prym variety which was shown in [Mu]. Since we are mainlyusing the ideas shown there and not the whole concept, we will explain laterhow this connection involves here.2. A two-sheeted covering
Let X be a compact Riemann surface, σ a holomorphic involution on X and P , P ∈ X fixed points of σ , i.e. σ ( P i ) = P i for i = 1 ,
2. For p, q ∈ X let p ∼ q : ⇔ ( p = q ∨ p = σ ( q )) and define X σ := X/ ∼ . Let π : X → X σ be the canonical two-sheeted covering map. Since the subgroup of Aut( X )generated by σ is Z , X σ is a compact Riemann surface and π : X → X σ is holomorphic, compare [Mir, Theorem III.3.4.]. Due to the constructionof X σ , the fixed points of σ coincide with the ramification points of π . Theset of ramification points of π we denote by r π . Then the map π is locally Date : June 8, 2016.1991
Mathematics Subject Classification.
Primary 14H81; Secondary 14H40.
Key words and phrases.
Fermi curves, divisors, Jacobian variety. biholomorphic on X \ r π , see [Fo, Corollary I.2.5]. We define the ramificationdivisor of π on X as R π := P p ∈ r π p . In general, the ramification divisor isdefined as R π := P p ∈ X (mult p ( π ) − · p , where the multiplicitiy mult p ( π )of π in p denotes the number of sheets which meet in p , compare [Mir,Definition II.4.2]. Since mult p ( π ) = 1 for p ∈ X \ r π and mult p ( π ) = 2 for p ∈ r π , this coincides with the above definition. Furthermore, b π := π [ r π ] isthe set of branchpoints of π on X σ . The involution σ extends to an involutionon the divisors on X by σ (cid:0) P p ∈ X a ( p ) p (cid:1) := P p ∈ X a ( p ) σ ( p ) which we alsodenote as σ . So the degree of a divisor is conserved under σ . We define thepullback of a point p σ ∈ X σ as π ∗ p σ := X p ∈{ π − [ { p σ } ] } mult p ( π ) p. With this definition, the pullback of a divisor D := P p σ ∈ X σ a ( p σ ) p σ on X σ is defined as π ∗ D := P p σ ∈ X σ a ( p σ ) π ∗ p σ . Since π is a non-constantholomorphic map between two Riemann surfaces, every meromorphic 1-form on X σ can be pulled back to a meromorphic 1-form ω := π ∗ ω σ on X ,compare for example [Mir, Section IV.2.]. Lemma 2.1.
Let
X, X σ and π be given as above and let ω σ be a non-constantmeromorphic -form on X σ .(a) The divisor of π ∗ ω σ on X is given by ( π ∗ ω σ ) = π ∗ ( ω σ ) + R π . (b) Let g σ be the genus of X σ . Then there exists a divisor e K on X with deg( e K ) = 2 g σ − such that ( π ∗ ω σ ) = e K + σ ( e K ) + R π .Proof. (a) Due to [Mir, Lemma IV.2.6] one has for p ∈ X thatord p ( π ∗ ω σ ) = (1 + ord π ( p ) ( ω σ ))mult p ( π ) − p ( π ∗ ω σ ) as defined in [Mir, Section IV.1.9]. Inserting this intothe definition of ( π ∗ ω σ ) = P p ∈ X (ord p ( π ∗ ω )) yields the assertion.(b) One has deg K σ = deg( ω σ ) = 2 g σ − K σ is the canonical divisoron X σ . Let p σ ∈ X σ be a point in the support of ( ω σ ) as defined in [Mir,Section V.1]. For p σ b π one has π ∗ p σ = p + σ ( p ) with p = σ ( p ) ∈ X and for p σ ∈ b π it is π ∗ p σ = 2 p with p ∈ r π . For p σ b π , let oneof the pulled back points in π ∗ p σ be the contribution to e K and for p σ ∈ b π the pulled back point is counted with multiplicity one in e K .Then π ∗ ( K ) = e K + σ ( e K ) and the claim follows from (a). (cid:3) Now we are going to construct a symplectic cycle basis of H ( X, Z ) froma symplectic cycle basis of H ( X σ , Z ). The holomorphic map σ : X → X induces a homomorpism of H ( X, Z ) which we denote as σ ♯ : H ( X, Z ) → H ( X, Z ) , γ σ ♯ γ. N A PROPERTY OF FERMI CURVES 3
Let g σ be the genus of X σ and A σ, , . . . , A σ,g σ , B σ, , . . . , B σ,g σ be represan-tatives of a symplectic basis of H ( X σ , Z ), i.e. A σ,i ⋆ A σ,ℓ = B σ,i ⋆ B σ,ℓ = 0 and A σ,i ⋆ B σ,ℓ = δ iℓ , where ⋆ is the intersection product between two cycles. From Riemannsurface theory it is known that such a basis exists, compare e.g. [Mir,Section VIII.4]. Due to Hurwitz’s Formula, e.g. [Mir, Theorem II.4.16], oneknows that that ♯b π = 2 n is even for a two sheeted covering π : X → X σ andthat the genus g of X is given by g = 2 g σ + n −
1. Hence a basis of H ( X, Z )consists of 4 g σ + 2 n − H ( X, Z ) which we denote as A i , σ ♯ A i , B i , σ ♯ B i and C j , D j and whichhas the following two properties: First of all, the only non-trivial pairwiseintersections between elements of the basis of H ( X, Z ) must be given by A i ⋆ B i = σ ♯ A i ⋆ σ ♯ B i = C j ⋆ D j = 1 . (1)Secondly, the involution σ ♯ has to map A i to σ ♯ A i and vice versa, B i to σ ♯ B i and vice versa and has to act on C j and D j as σ ♯ C j = − C j and σ ♯ D j = − D j . Here, and from now on, we consider i, ℓ ∈ { , . . . , g σ } and j, k ∈ { , . . . , n − } as long as not pointed out differently. The difference inthe notation of the cycles indicates the origin of these basis elements: the A - and B -cycles on X will be constructed via lifting a certain symplecticcycle basis of H ( X σ , Z ) via π and the C - and D -cycles originate from thebranchpoints of π .We will start by constructing the C - and D -cycles. A sketch of the idea how todo this is shown for n = 3 and g σ = 0in figure 1 . We connect the points in b π pairwise by paths s j for j = 1 , . . . , n .The set of points corresponding to a path s j : [0 , → X σ we denote by [ s j ] := { s j ( t ) | t ∈ [0 , } and use the same no-tation for any other path considered as aset of points in X or X σ . Let [ s j ] ◦ bethe corresponding set with t ∈ (0 , s j are constructed in such a waythat every branchpoint is connected withexactly one other branchpoint and suchthat s k ∩ s j = ∅ for k = j . This is possi-ble since the branchpoints lie discrete on X σ : suppose the first two branchpointsare connected by s such that s containsno other branchpoint. Then one can finda small open tubular neighborhood N ( s )of s in X σ with boundary ∂N ( s ) in X σ isomorphic to S . To see that X σ \ [ s ] is Figure 1.
E. L ¨UBCKE path connected, let γ be a path in X σ which intersects ∂N ( s ) in the twopoints p , p ∈ X σ . Then there is a path ˜ γ such that ˜ γ | X σ \ N ( s ) = γ | X σ \ N ( s ) and such that the points p and p are connected via a part of ∂N ( s ). Hence X σ \ [ s ] is path connected. Like that one can gradually choose s , . . . s n .To find a path s j not intersecting s , . . . s j − , consider X σ \ ([ s ] ∪ · · · ∪ [ s j − ]) which is path connected and repeat the above procedure until allbranchpoints are sorted in pairs. The preimage of s j under π yields two pathsin X which both connect the preimage of the connected two branchpoints.These preimages are ramification points of π and we denote them as b j and b j . A suitable linear combination of the two paths on X then defines a cycle C j for j = 1 , . . . , n . Since π is unbranched on X \ r π , i.e. a homeomorphism,and since π [ r π ] = b π ⊂ [ s ] ∪ · · · ∪ [ s n ], π − [ X σ \ ([ s ] ∪ · · · ∪ [ s n ])] consistsof two disjoint connected manifolds whose boundaries both are equal to π − [ s ] ∪ · · · ∪ π − [ s n ] and σ interchanges those manifolds. We will call them M and σ [ M ]. Since the n C -cycles are the boundary of M respectively σ [ M ],they are homologous to another, i.e. C n = − P n − i =1 C i , so this constructionyields maximal n − C -cycles which are not homologous to each other. These n cycles we orientate as the boundary of the Riemann surface M . We willsee later on that, due to the intersection numbers, the cycles C , . . . , C n − are not homologous to each other. By construction, each cycle C j containsthe two ramification points b j and b j of π and no other ramification points.The next step is to construct n − D -cycles such that one has C j ⋆ D k = δ jk .We will see that it is possible to connect π ( b j ) with π ( b j +1 ) by a path t j for j = 1 , . . . , n − t j ∩ t k = ∅ for j = k . Since X σ \ ([ s ] ∪ · · · ∪ [ s n ]) ispath connected, also X σ \ ([ s ] ◦ ∪ [ s ] ◦ ∪ [ s ] ∪ · · · ∪ [ s n ]) is path connected. Soone can connect b with b with a path t in X σ not intersecting s , . . . , s n and the path s + t + s in X σ contains no loop. As above, one can chosea small open neighborhood N ([ s ] ∪ [ t ] ∪ [ s ]) with boundary isomorphicto S . Therefore, X σ \ ([ s ] ∪ · · · ∪ [ s n ] ∪ [ t ]) is path connected. Repeatingthis procedure shows that X σ \ ([ s ] ∪ · · · ∪ [ s n ] ∪ [ t ] ∪ · · · ∪ [ t j ]) remainspath connected and that P jm =1 ( s m + t m ) + s j +1 contains no loop for j =1 , . . . , n −
1. This yields the desired n − t j in X σ . Lifting these pathsvia π yields each n − M and n − σ [ M ]. The pathson M and σ [ M ] which result from the lift of t j both start at b j and endin b j +1 . Hence identifying these end points with each other yields a cycleon X which we denote as e D j . We orientate e D j such that C j ⋆ e D j = 1 and C j +1 ⋆ e D j = − j ∈ { , . . . , n − } . Due to the construction of e D j one N A PROPERTY OF FERMI CURVES 5 has C i ⋆ e D j = 0 for i
6∈ { j, j + 1 } . Defining D j := P n − i = j e D i yields for k < jC j ⋆ D j = C j ⋆ n − X l = j e D l = C j ⋆ e D j = 1 , C k ⋆ D j = C k ⋆ n − X l = j e D l = 0 C j ⋆ D k = C j ⋆ n − X l = k e D l = C j ⋆ ( e D j + e D j − ) = 1 − n − C k ⋆ D j = δ kj . Two cycles can not behomologeous to each other if the intersection number of each one of thosecycles with a third cycle is not equal, hence C k ⋆ D j = δ kj implies that theabove construction yields 2 n − C j and D j which are not homologousto each other. To construct the missing 4 g σ cycles, we choose a symplecticcycle basis A σ,i , B σ,i of H ( X σ , Z ) such that they intersect none of the paths s , . . . , s n and t , . . . , t n − . This is possible since all of these paths in X σ are connected and hence can be contracted to a point. On the preimage of X σ \ { S n − j =1 ([ s j ] ∪ [ t j ]) ∪ [ s n ] } , the map π is a homeomorphism. So each ofthe cycles in H ( X σ , Z ) is lifted to one cycle in M and one cycle in σ [ M ] via π and those two cycles are interchanged by σ . Thus lifting the whole basisyields 4 g σ cycles on X where we denote the 2 g σ cycles lifted to M as A i and B i and the corresponding cycles lifted to σ [ M ] as σ ♯ A i and σ ♯ B i . Then thesecycles obey the desired transformation behaviour under σ ♯ . Since M and σ [ M ] are disjoint, the intersection number of the lifted cycles on X stays thesame as the intersection number of the corresponding cycles on X σ if twocycles are lifted to the same sheet M respectively σ [ M ] or equals zero if theyare lifted to different sheets. Furthermore, the construction of these cyclesensured that the lifted A - and B -cycles do not intersect any of the C - and D -cycles on X . Hence A i , σ ♯ A i , B i , σ ♯ B i , C j and D j are in total 4 g σ + 2 n − H ( X, Z ) and the A - and C -cycles are disjoint. Thatthe C - and D -cycles constructed like this have the desired transformationbehavior under σ ♯ is shown in the next lemma. Lemma 2.2.
For C j , D j ∈ H ( X, Z ) as defined above one has σ ♯ C j = − C j and σ ♯ D j = − D j .Proof. Every cycle C j is the preimage of a path in X σ and X σ is invariantunder σ . So σ [ C j ] = [ C j ] and the two points b j and b j stay fixed. Therefore σ ♯ C j = ± C j . Since σ commutes the two lifts of the path s j in X σ , i.e. b j and b j are the only fixed points of σ on C j , one has σ ♯ C j = − C j . By the samemeans, since D j also consists of the two lifts of t j which are interchangedby σ , one also has σ ♯ D j = − D j . (cid:3) Decomposition of H ( X, Z )With help of the Abel map Ab one can identify the elements of H ( X, Z )with a lattice in C g such that Jac( X ) ≃ C g / Λ, compare [Mir, Section VIII.2].
E. L ¨UBCKE
To do so, let ω , . . . , ω g ∈ H ( X, Ω) be a basis of the g = 2 g σ + n − X which are normalized with respect tothe A -, σ ♯ A - and C -cycles, i.e. I A i ω ℓ = δ iℓ , I σ ♯ A i ω g σ + ℓ = δ iℓ , I C j ω g σ + k = δ jk (2)and all other integrals over one of the A - and C -cycles with another elementof the basis of H ( X, Ω) are equal to zero. Furthermore, note that theconstruction of the A -cycles yields σ ∗ ω i = ω g σ + i for i = 1 , . . . , g σ and thatLemma 2.2 implies σ ∗ ω g σ + j = − ω g σ + j . We define ω ± i := 12 ( ω i ± ω g σ + i ) and ω − g σ + j := ω g σ + j . (3)Direct calculation shows that these differential forms also yield a basis of H ( X, Ω). For a path γ in X , we define the vectorsΩ γ := (cid:18) Z γ ω k (cid:19) gk =1 , Ω + γ := (cid:18) Z γ ω + k (cid:19) g σ k =1 , Ω − γ := (cid:18) Z γ ω − k (cid:19) g σ + n − k =1 . and the following lattices generated over Z asΛ := h Ω A i , Ω σ ♯ A i , Ω C j , Ω B i , Ω σ ♯ B i , Ω D j i i =1 ,...,g σ j =1 ,...,n − Λ + := h Ω + A i + σ ♯ A i , Ω + B i + σ ♯ B i i i =1 ,...,g σ Λ − := h Ω − A i − σ ♯ A i , Ω − B i − σ ♯ B i , Ω − C j , Ω − D j i i =1 ,...,g σ j =1 ,...,n − . (4)Furthermore, the mappingΦ : C g → C g σ ⊕ C g σ + n − , v ... v g ( v + v g σ +1 )... ( v g σ + v g σ ) ⊕ ( v − v g σ +1 )... ( v g σ − v g σ ) v g σ +1 ... v g σ + n − is obviously linear and bijective. Hence Φ is a vector space isomorphism. Lemma 3.1.
For every path γ on X one has Φ(Ω γ ) = Ω + γ ⊕ Ω − γ = Ω + ( γ + σ ♯ γ ) ⊕ Ω − ( γ − σ ♯ γ ) . N A PROPERTY OF FERMI CURVES 7
Proof.
The first equality follows immediately from the definition of Φ andthe differential forms in (3):Φ(Ω γ ) = Φ R γ ω ... R γ ω g = ( R γ ω + ω g σ +1 )... ( R γ ω g σ + ω g σ ) ⊕ ( R γ ω − ω g σ +1 )... ( R γ ω g σ − ω g σ ) R γ ω g σ +1 R γ ω g σ + n − = R γ ω +1 ... R γ ω + g σ ⊕ R γ ω − ... R γ ω − g σ R γ ω − g σ +1 R γ ω − g σ + n − . Since ω + k = σ ∗ ω + k for k = 1 , . . . , g σ and ω − k = − σ ∗ ω − k for k = 1 , . . . , g σ + n − Z γ ω + k = 12 (cid:18) Z γ ω + k + σ ∗ ω + k (cid:19) = Z ( γ + σ ♯ γ ) ω + k as well as Z γ ω − k = 12 (cid:18) Z γ ω − k − σ ∗ ω − k (cid:19) = Z ( γ − σ ♯ γ ) ω − k which implies the second equality. (cid:3) Corollary 3.2.
The generators of Φ − (Λ + ⊕ Λ − ) span a basis of C g over R , the generators of Λ + span a basis of C g σ over R and the generators of Λ − span a basis of C g σ + n − over R .Proof. Since Jac( X ) = C g / Λ is a complex torus, the generators of Λ given in(4) are a basis of C g over R , compare for example [L-B, Section II.2]. Basistransformation yields that Ω A i + σ ♯ A i , Ω A i − σ ♯ A i ,Ω B i + σ ♯ B i , Ω B i − σ ♯ B i , Ω C j andΩ D j are also a basis of C g over R . Since Φ is a vector space ismorphism withΦ(Ω A i + σ ♯ A i ) = Ω + A i + σ ♯ A i ⊕ , Φ(Ω A i − σ ♯ A i ) = 0 ⊕ Ω − A i − σ ♯ A i , Φ(Ω B i + σ ♯ B i ) = Ω + B i + σ ♯ B i ⊕ , Φ(Ω B i − σ ♯ B i ) = 0 ⊕ Ω − B i − σ ♯ B i , Φ(Ω C j ) = 0 ⊕ Ω − C j , Φ(Ω D j ) = 0 ⊕ Ω − D j , the generators of Φ − (Λ + ⊕ Λ − ) yield a basis of C g over R . Since Φ is anisomorphism, the generators of Λ + are a basis of C g σ and the generators ofΛ − of C g σ + n − over R . (cid:3) In the sequel, we will apply Φ and Φ − to lattices. Note that, for shortageof notation, we abuse the notation in the sense that Φ(Λ) denotes the latticein C g σ ⊕ C g σ + n − spanned by the image of the generators of Λ under Φ andanalogously for Φ − applied to lattices. E. L ¨UBCKE
Lemma 3.3. (a) Λ + ⊕ ∩ ( C g σ ⊕ , ⊕ Λ − = Φ(Λ) ∩ (0 ⊕ C g σ + n − ) .(b) Φ(Λ) = (Λ + ⊕ Λ − ) + M (5) with M := n g σ X i =1 (cid:16) a i + A i + σ ♯ A i + b i + B i + σ ♯ B i (cid:17) ⊕⊕ g σ X i =1 (cid:16) a i − A i − σ ♯ A i + b i − B i − σ ♯ B i (cid:17) (cid:12)(cid:12)(cid:12) a i , b i ∈ { , } o . (c) M ∩ (Λ + ⊕ Λ − ) = { } Proof.
Obviously, Λ + ⊕ ∩ ( C g σ ⊕ ∩ ( C g σ ⊕
0) is also a subset of Λ + ⊕
0, note that for every γ ∈ Λ thereexists coefficients a i , a σ,i , b i , b σ i , c j , d j ∈ Z such that γ = g σ X i =1 a i Ω A i + a σ,i Ω σ ♯ A i + b i Ω B i + b σ,i Ω σ ♯ B i + c j Ω C j + d j Ω D j . The generators of Λ + and Λ − are linearly independent, compare Corollary3.2. So the second equality in Lemma 3.1 shows that Φ( γ ) ∈ C g σ ⊕ c j = d j = 0, a i = a σ i and b i = b σ i . Then for such γ it isΦ( γ ) = 2 a i Ω + ( A i + σ ♯ A i ) + 2 b i Ω + ( B i + σ ♯ B i ) ⊕ a i Ω + A i + σ ♯ A i + b i Ω + B i + σ ♯ B i ⊕ ∈ Λ + ⊕ . The equality 0 ⊕ Λ − = Φ(Λ) ∩ (0 ⊕ C g σ + n − ) follows in the same manner.So the first part holds.To get insight into the second part, we will show that for the set of cosetsone has Φ(Λ) / Λ + ⊕ Λ − = { λ + (Λ + ⊕ Λ − ) | Φ( λ ) ∈ Λ } = { λ + (Λ + ⊕ Λ − ) | Φ( λ ) ∈ M } . The lattice Λ is a finitely generated abelian group, so also Φ(Λ), Λ + andΛ − are finitely generated abelian groups and Φ(2Λ) ⊂ Λ + ⊕ Λ − ⊂ Φ(Λ),where the second inclusion is obvious and the first inclusion holds sinceany element 2 γ of 2Λ be decomposed as 2 γ = 2 (cid:0) ( γ + σ ♯ γ ) + ( γ − σ ♯ γ ) (cid:1) .Therefore, Φ(Λ) / (Λ + ⊕ Λ − ) ⊂ Φ(Λ) / Φ(2Λ) and the set of the (Λ : 2Λ) =2 g elements contained in Φ(Λ) / Φ(2Λ) is the maximal set of points whichare not contained in Λ + ⊕ Λ − but in Φ(Λ). One has Φ(Ω C j ) , Φ(Ω D j ) ∈ Λ + ⊕ Λ − ⊂ Φ(Λ). Therefore, all points in M are linear combinations ofΦ(Ω A i ) , Φ(Ω σ ♯ A i ) , Φ(Ω B i ) and Φ(Ω σ ♯ B i ) with coefficients in { , } . SinceΩ A i = Ω A i + σ ♯ A i − Ω σ ♯ A i , one has that [Φ(Ω A i )] = [Φ(Ω σ ♯ A i )] and [Φ(Ω B i )] = N A PROPERTY OF FERMI CURVES 9 [Φ(Ω σ ♯ B i )] in Φ(Λ) / (Λ + ⊕ Λ − ) and thus M ⊆ n g σ X i =1 a i Φ(Ω A i ) + b i Φ(Ω B i ) (cid:12)(cid:12) a i , b i ∈ { , } o . (6)Furthermore, Φ(Ω A i ) = (Φ(Ω A i + σ ♯ A i ) + Φ(Ω A i − σ ♯ A i )). Due to Corollary3.2, these representations of Φ(Ω A i ) as a vector in C g in the basis given bythe generators of Λ + ⊕ Λ − is unique, i.e. Φ(Ω A i ) Λ + ⊕ Λ − and by the samemeans Φ(Ω σ ♯ A i ) , Φ(Ω B i ) , Φ(Ω σB i ) Λ + ⊕ Λ − . The linear independence ofthe generators of Λ then yields equality in (6). Hence (Φ(Λ) : Λ + ⊕ Λ − ) =2 g σ and so Λ can be seen as finitely many copies of Λ + ⊕ Λ − translated bythe points in M . The linear independence of the generators of Λ + and Λ − and the definition of M imply (c). (cid:3) Remark . In [Mu] it was shown that Jac( X σ ) ≃ C g σ / Λ + and that thePrym variety P ( X, σ ) can be identified with C g σ + n − / Λ − . Furthermore, itwas also shown that the direct sum Jac( X σ ) ⊕ P ( X, σ ) is only isogenous toJac( X ), but that the quotient of this direct sum divided by a finite set ofpoints is isomorphic to Jac( X ). The explicit calculations in Lemmata 3.1and 3.3 are mirroring this connection and the finite set of points which aredivided out of the direct sum in [Mu, Section 2, Data II] are exactly thepoints in M .4. The fixed points of σ and the linear equivalence Theorem 4.1.
Let X be a Riemann surface of genus g , K a canonicaldivisor on X , σ : X → X a holomorpic involution and P , P ∈ X fixedpoints of σ . Then there exists a divisor D of degree g on X which solves D + σ ( D ) ≃ K + P + P (7) if and only if σ has exactly the two fixed points P and P .Proof. Assume that σ has more fixed points then P and P , i.e. n > e K of degree2 g σ − X such that K = e K + σ ( e K ) + R π and hence equation (7)yields D − e K + σ ( D − e K ) ≃ R π + P + P . We sort the 2 n ramificationpoints in r π into pairs as it was done in the construction of the C -cyclesand denote the two fixed points on C n as P and P . Then equation (7)reads as D − e K + σ ( D − e K ) ≃ P n − j =1 ( b j + b j ) + 2 P + 2 P . With e D := D − e K + P n − j =1 b j − P − P this is equivalent to e D + σ ( e D ) + n − X j =1 ( b j − b j ) ≃ . (8)Furthermore, deg( e D + σ ( e D ) + P n − j =1 ( b j − b j )) = 0 and deg( P n − j =1 ( b j − b j )) =0. Since deg acts linear on divisors and is invariant under σ , this yieldsdeg( e D ) = 0. So counted without multiplicity, there are as many points with positive sign as with negative sign in e D , i.e. e D = P ℓk =1 ( p k − p k ).Let γ k : [0 , → X be a path with γ k (0) = p k and γ k (1) = p k . Then σ ♯ γ k : [0 , → X is a path with σ ( γ k (0)) = σ ( p k ) and σ ( γ k (1)) = σ ( p k ).Then define γ e D := P ℓk =1 γ k and σ ♯ γ e D := P ℓk =1 σ ♯ γ k . Analogously, let γ R,j be defined as the paths γ R,j : [0 , → X such that γ R,j (0) = b j and γ R,j (1) = b j for j = 1 , . . . , n −
1. Then, due to the construction of the C -cycles, one has γ R,j − σ ♯ γ R,j = C j . We define γ R := P n − j =1 γ R,j . Set γ := γ e D + σ ♯ γ e D + γ R andlet ω , . . . , ω g be the canonical basis of H ( X, Ω) normalized with respect to A - and C -cycles as in (2). Again we use the identification Jac( X ) = C g / Λvia the Abel map Ab with the basis of holomorphic 1-forms on X normalizedas in (2). Due to (8), the linear equivalence can also be expressed asAb (cid:16) e D + σ ( e D ) + n − X i =1 ( b i − b i ) (cid:17) = 0 mod Λ . This equation can only hold if Ω γ ∈ Λ. Due to Lemma 3.1 we can split Ω γ ∈ C g uniquely by considering Φ(Ω γ ) = Ω + γ ⊕ Ω − γ Due to the decomposition ofΛ in (5), Ω γ ∈ Λ is equivalent to Ω + γ ⊕ Ω − γ ∈ (Λ + ⊕ Λ − ) + M as defined inLemma 3.3. So we want to show that Ω + γ ⊕ Ω − γ is not contained in any ofthe translated copies of Λ + ⊕ Λ − if n >
1. Since it will turn out that it isΩ − γ which leads to this assertion, we determine the explicit form of Ω − γ . Forevery ω − ∈ H ( X, Ω) such that σ ∗ ω − = − ω − one has Z γ e D + σ ♯ γ e D ω − = Z γ e D ω − + Z γ e D σ ∗ ω − = Z γ e D ω − − Z γ e D ω − = 0as well as2 Z γ R,i ω − = Z γ R,i ω − − Z γ R,i σ ∗ ω − = Z γ R,i ω − − Z σ ♯ γ R,i ω − = Z γ R,i − σ ♯ γ R,i ω − = I C i ω − , i.e. R γ R,i ω − = H C i ω − . Since γ − σ ♯ γ = 2 γ R one hasΩ − γ = (cid:18) Z γ R ω − i (cid:19) g σ + n − k =1 = 12 (cid:18) n − X k =1 I C k ω − k (cid:19) g σ + n − k =1 . Due to the normalization of the holomorphic 1-forms defined in (2) one hasΩ − γ = P n − k =1 Ω − C k . If Ω + γ ⊕ Ω − γ would be contained in one of the translatedcopies of Λ + ⊕ Λ − , then Ω − γ would be contained in the second component ofthe direct sum in one of the translated copies of Λ − introduced in Lemma3.3. This is not possible, since the generators of Λ + ⊕ Λ − are linearly inde-pendent and only integer linear combinations of C -cycles are contained inall translated lattices. Therefore, Ω γ Λ for n >
1. For n ≤
1, there areno C -cycles in H ( X, Z ) and equation (8) would read as D + σ ( D ) ≃
0. Soequation (7) can only hold if n ≤
1. Since P and P are fixed points of σ N A PROPERTY OF FERMI CURVES 11 one has n = 1.Let now P and P be the only fixed points of σ . Then Lemma 2.1 yieldsthat there exists a divisor e K on X with deg( e K ) = 2 g σ − K = e K + σ ( e K ) + P + P . Define D := e K + P + P . The Hurwitz Formulafor n = 1 yields deg( D ) = 2 g σ = g , compare e.g. [Mir, Theorem II.4.16],and one has D + σ ( D ) = e K + σ ( e K ) + 2 P + 2 P ≃ K + P + P . (cid:3) References [Fo] O. Forster: Lectures on Riemann surfaces. Graduate Texts in Mathematics .Springer, New York (1981).[L-B] H. Lange, C. Birkenhake: Complex abelian varieties. Grundlehren der mathema-tische Wissenschaften . Springer, Berlin (1992).[Mu] D. Mumford: Prym Varieties I. Contribution to Analysis, 325-350, AcademicPress, New York (1974)[Mir] R. Miranda: Algebraic Curves and Riemann Surfaces. Graduate Studies in Math-ematics . American Mathematical Society, (1995).[N-V] S. Novikov, A.P. Veselov: Finite-Zone two-dimensional Schr¨odinger Operators.Potential operators. Soviet Math. Dokl. 30, 588-591 (1984) Mathematics Chair III, Universit¨at Mannheim, D-68131 Mannheim, Germany
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