Fixed points of endomorphisms on two-dimensional complex tori
aa r X i v : . [ m a t h . AG ] A ug Fixed points of endomorphismson two-dimensional complex tori
Thomas Bauer, Thorsten HerrigMay 15, 2019
Abstract.
In this paper we investigate fixed-point numbers of endomorphisms oncomplex tori. Specifically, motivated by the asymptotic perspective that has turnedout in recent years to be so fruitful in Algebraic Geometry, we study how the numberof fixed points behaves when the endomorphism is iterated. Our first result shows thatthe fixed-points function of an endomorphism on a two-dimensional complex torus canhave only three different kinds of behaviours, and we characterize these behaviours interms of the analytic eigenvalues. Our second result focuses on simple abelian surfacesand provides criteria for the fixed-points behaviour in terms of the possible types ofendomorphism algebras.
Introduction
Given a holomorphic map f : X → X on a complex variety X , one of the naturalquestions about f is how many fixed points it has. This number may, as expected,vary a lot between different endomorphisms, but it is a recurring theme in AlgebraicGeometry that one hopes for much more regularity when adopting an asymptoticperspective. Examples for the fruitfulness of this approach are questions about baseloci [ELMNP], growth of higher cohomology [DKL], syzygies [EL] and Betti numbers[EEL]. Concerning the question of fixed points, a natural asymptotic point of viewconsists in considering large iterates f n of a given map. Specifically, denoting by f ) the number of fixed points of a map f , the question becomes: What is the asymptotic behaviour of the fixed-points function n f n ) where f n = f ◦ . . . ◦ f denotes n -th iterate of f . The growth of the fixed-points function is also of interest in purely analytic contexts(e.g. [SS]). In the present paper we consider it when f is a holomorphic map on acomplex torus. As is customary (cf. [BL]) we set f ) = 0, if the fixed-pointsset is infinite, i.e., if f fixes an analytic subvariety of positive dimension. The second author was supported by Studienstiftung des deutschen Volkes.
Keywords: abelian variety, endomorphism, fixed point.
Mathematics Subject Classification (2010):
Consider for instance the multiplication map m X : X → X , x mx , on acomplex torus X of dimension g , for a given integer m >
2. Its fixed points are the( m − m X ) n ) = ( m n − g grows exponentially with n . It is natural to wonder whether this is typical forendomorphisms on complex tori, and what other behaviour, if any, might occur.For two-dimensional complex tori we provide a complete answer: Theorem 1.
Let X be a two-dimensional complex torus and let f : X → X be anon-zero endomorphism. Then the fixed-points function n f n ) has one ofthe following three behaviours: (B1) It grows exponentially in n , i.e., there are real constants A, B > and aninteger N such that for all n > N , A n f n ) B n . In this case both eigenvalues of f (i.e., of its analytic representation ρ a ( f ) ∈ M ( C ) ) are of absolute value = 1 . (B2) It is a periodic function. In this case the non-zero eigenvalues of f are rootsof unity, and they are contained in the set of k -th roots of unity where k ∈{ , . . . , , , , } . (B3) It is of the form f n ) = ( , if n ≡ r ) h ( n ) , otherwisewhere r > is an integer and h is an exponentially growing function. In thiscase one of the eigenvalues of f is of absolute value > and the other is a rootof unity.All three behaviours occur already in the projective case, i.e., on abelian surfaces. The fact that the eigenvalues govern the behaviour of the fixed-points functionis a consequence of the Holomorphic Lefschetz Fixed-Point Formula (see Prop. 1.1).The main point of the theorem is that the stated cases are the only ones that canoccur.For simple abelian surfaces there are only three non-trivial types of possibleendomorphism algebras, and it is desirable to know about the fixed-points behaviourin terms of these types. Our second result contains this information:
Theorem 2.
Let X be a simple abelian surface. Then the fixed-points function ofany non-zero endomorphism f ∈ End( X ) is either exponential (B1) or periodic (B2) ,but has never behaviour (B3) . Specifically, we have: (a) Suppose that X has real multiplication, i.e., End Q ( X ) = Q ( √ d ) for a square-free integer d > . Then f n ) is periodic if f = ± id X , and it growsexponentially otherwise. (b) Suppose that X has indefinite quaternion multiplication, i.e., End Q ( X ) is ofthe form Q + i Q + j Q + ij Q , where i = α ∈ Q \ { } , j = β ∈ Q \ { } with ij = − ji and α > , α > β . Write f ∈ End( X ) as f = a + bi + cj + dij with a, b, c, d ∈ Q . Then f n ) is periodic if | a + p b α + c β − d αβ | = 1 , andit grows exponentially otherwise. (c) Suppose that X has complex multiplication, and let σ : End( X ) ֒ → C be anembedding. Then f has periodic fixed-point behaviour if | σ ( f ) | = 1 , and it hasexponential fixed-points growth otherwise. We give a more detailed description of the three types in Sect. 2, and we providea list of the finitely many eigenvalues that occur in endomorphisms with periodicfixed-points behaviour (see Prop. 2.1).Fixed points of endomorphisms on complex tori and abelian varieties have beenstudied previously by Birkenhake and Lange [BL] with a focus on the classificationof fixed-point free automorphisms. The question of fixed point numbers for iteratesof endomorphisms on abelian varieties was first addressed in the preprint [Rin];however, Theorem 1.2 in [Rin] is unfortunately erroneous.
1. Fixed points and eigenvalues
Let X be a complex torus of dimension g and let f : X → X be a holomorphic map.The map f is a translate f = h + a of a group endomorphism h ∈ End( X ) by some a ∈ X , and we have (see [BL]) f ) = h ) . So, as far as fixed point numbers are concerned, it is enough to consider endomor-phisms. The following proposition allows one to determine fixed point numbers fromthe eigenvalues of the analytic representation.
Proposition 1.1.
Let f : X → X be an endomorphism of a g -dimensional complextorus, and let λ , . . . , λ g be the eigenvalues of its analytic representation (countedwith algebraic multiplicities). Then we have for every integer n > , f n ) = (cid:12)(cid:12)(cid:12) g Y i =1 (1 − λ ni ) (cid:12)(cid:12)(cid:12) . Proof.
Thanks to the Holomorphic Lefschetz Fixed-Point Formula [LB, 13.1.2], thefixed point number can be computed from the analytic representation ρ a ( f ) ∈ M g ( C ), f n ) = | det(1l g − ρ a ( f ) n ) | . As the eigenvalues of ρ a ( f ) n are λ n , . . . , λ ng , we get the asserted formula. (cid:3) The proposition shows that the fixed-points function n f n ) is governedby the size of the eigenvalues. If, for instance, it were to happen that all eigenvaluesof f are of absolute value bigger than 1, then clearly f n ) grows exponentially.(This is the case for the multiplication maps mentioned in the introduction.) Thefollowing examples show, however, that eigenvalues of absolute value equal to 1 aswell as less than 1 occur, too. Example 1.2. (Eigenvalues of absolute value 1). Take an elliptic curve E andconsider the complex torus X = E × E . The endomorphism f : X → X, ( x, y ) ( x − y, x )has the eigenvalues √− and −√− . Using Prop. 1.1 we find that the fixed-pointsfunction of f is periodic: f n ) = , if n ≡ , if n ≡ n ≡ , if n ≡ n ≡ , if n ≡ . Example 1.3. (Eigenvalues of absolute value < a > P ( t ) = t + at + t + 1occurs as the characteristic polynomial of the rational representation of an endo-morphism on a two-dimensional complex torus. One checks that the zeros of P ( t )appear in conjugate pairs α, α, β, β with | α | < | β | . As the constant term of P equals1, it follows that | α | < Proposition 1.4.
For every ε > there exists a two-dimensional complex toruswith an endomorphism that has a non-zero eigenvalue of absolute value less than ε .Proof. We draw on the complex tori constructed in Example 1.3. An application ofRouch´e’s theorem shows that for all sufficiently large values of a there exists a rootof the polynomial t + at + t + 1in the disk of radius ε around the origin. This proves the proposition. (cid:3) The following example shows that eigenvalues of absolute value < Example 1.5.
Let E be an elliptic curve, and consider the abelian surface X = E × E . Every matrix ( a bc d ) ∈ SL ( Z ) defines an automorphism f of X . Its analyticcharacteristic polynomial is t − ( a + d ) t + 1, and from this one checks that f hasan eigenvalue arbitrarily close to 0 when a + d is sufficiently large.With a little more effort the same behaviour can even be found among simple abelian surfaces: There is a two-dimensional family of principally polarized abeliansurfaces X with endomorphism ring End( X ) = Z [ √
2] (cf. [Bir, Prop. 2.1]). Onsuch a surface consider the endomorphism f = − √
2. Its analytic characteristicpolynomial is t + 2 t −
1, and hence it has − √ f n has an eigenvalue arbitrarily close to 0 when n is sufficiently large.In view of Prop. 1.1, and considering the preceding examples, it becomes appar-ent that the issue, in the general case, is to understand what kind of eigenvaluesmay occur in endomorphisms of complex tori. We focus from now on on the surfacecase, where we show as a first step: Proposition 1.6.
Let X be a two-dimensional complex torus and f : X → X anendomorphism. If f has a non-zero eigenvalue of absolute value < , then it alsohas an eigenvalue of absolute value > .Proof. The characteristic polynomial of the analytic representation ρ a ( f ) is P af ( t ) = det( t − ρ a ( f )) = ( t − λ )( t − λ ) . Since the rational representation ρ r ( f ) is the direct sum of ρ a ( f ) and its conjugate,the characteristic polynomial of ρ r ( f ) is given by P rf ( t ) = P af ( t ) · P af ( t ) = ( t − λ )( t − λ )( t − λ )( t − λ )Assume now by way of contradiction that 0 < λ < λ λ = 0. Then λ λ appears as a coefficient in P rf ( t ),and it must therefore be an integer. So | λ | >
1, a contradiction.Next, suppose λ >
0. We havedet( ρ r ( f )) = λ λ λ λ = | λ | | λ | . and this implies 0 < | det( ρ r ( f )) | <
1, which again is impossible as this number isan integer. (cid:3)
The next statement will be crucial for dealing with the case of eigenvalues ofabsolute value 1.
Proposition 1.7.
Let X be a two-dimensional complex torus and f : X → X anendomorphism. If λ is an eigenvalue of f with | λ | = 1 , then λ is a root of unity.Proof. We may assume that λ = ±
1, as otherwise there is nothing to prove. Let λ and λ be the eigenvalues of f . Suppose to begin with that | λ | = | λ | = 1.The characteristic polynomial P rf of the rational representation of f is then a monicpolynomial over the integers, all of whose roots are of absolute value 1. It followsthat all of its roots are then roots of unity (see [Coh, Prop. 3.3.9]), and we are donein this case.It remains to consider the case that | λ | = 1 and | λ | 6 = 1. By Lemma 1.6 weknow that then necessarily | λ | > λ = 0. Let h be the minimal polynomialof λ over Q . ( λ is a root of P rf , and hence an algebraic integer.) According toLemma 1.8 below, h is a symmetric integer polynomial of degree 2 or 4, whose rootsappear in reciprocal pairs. So there are two cases: Case 1: deg h = 2. In that case, the second root of h is λ . So all roots of h areof absolute value 1. And as above, we conclude that λ is a root of unity. Case 2: deg h = 4. Then h coincides with P rf , so its roots are λ , λ , λ , λ . Butbecause of | λ | > | λ | = 1, there is no way that the roots can occur in reciprocalpairs – so this case does not happen. (cid:3) We very much assume that the following elementary algebraic lemma is well-known. For lack of a reference we include a proof.
Lemma 1.8.
Let a ∈ C be an algebraic integer of absolute value 1 and differentfrom ± . Then its minimal polynomial is a polynomial over Z of even degree withsymmetric coefficients, whose roots occur in reciprocal pairs.Proof. By definition there is a monic polynomial g over Z with g ( a ) = 0. As g is amultiple of the minimal polynomial h of a , it follows from Gauß’ Lemma that h isintegral as well.We now prove the symmetry statement. As a is of absolute value 1, we knowthat 1 /a = a appears as a root of h as well. On the other hand, we have h (1 /a ) = h ( a ) = 0 , hence a is a root of the integral polynomial t n h (1 /t ), which is also of degree n .Therefore t n h (1 /t ) = c · h ( t ) for some c ∈ Q . Setting t = 1, we get h (1) = c · h (1).Since a is irrational, the degree of h is at least 2, and hence h (1) = 0. We concludethat c = 1, i.e., t n h (1 /t ) = h ( t ) , and this shows that the roots of h occur in reciprocal pairs, and hence that its degreeis even. (cid:3) We can now give the
Proof of Theorem 1.
Let f : X → X be an endomorphism of a two-dimensionalcomplex torus, and let λ and λ be the eigenvalues of its analytic representation(counted with algebraic multiplicities). After reordering we assume | λ | | λ | . Wehave then by Prop. 1.1 for every integer n > f n ) = | (1 − λ n )(1 − λ n ) | . Suppose first that | λ | >
1. Then by our setup we have | λ | > f n ) grows exponentially. So we are in Case (B1) ofthe theorem.Suppose next that 0 < | λ | <
1. It follows from Prop. 1.6 that then | λ | > f n ) grows exponentially again.Suppose now that | λ | = 1. Proposition 1.7 tells us that λ is then a root ofunity. If | λ | = 1, then the same is true for λ and hence f n ) is a periodicfunction, so we are in Case (B2) of the theorem. And if | λ | >
1, then the fixed-points function has the form described in Case (B3) of the theorem. In either case,the roots of unity are of algebraic degree
4, since they appear as roots of therational characteristic polynomial P rf ( f ). They are therefore k -th roots of unity,where k ∈ { , . . . , , , , } .Finally, suppose that λ = 0. Then we have behaviour (B1) if | λ | > | λ | = 1.It remains to show that all three behaviours actually occur. Case (B1) happensfor the multiplication map x mx on every complex torus, as soon as | m | > (cid:3) Example 1.9. (One eigenvalue of absolute value >
1, the other a root of unity.)Consider the elliptic curve E with complex multiplication in Z [ i ], and take theabelian surface X = E × E . The endomorphism X → X, ( x, y ) ( ix, iy )has the eigenvalues i and 2 i , and hence the fixed-points function has the behaviourdescribed in Case (B3) of Theorem 1: f n ) = ( , if n ≡ h ( x ) otherwisewhere the function h grows exponentially, h ( n ) ∼ n .
2. Fixed points on simple abelian surfaces
In this section we will explicitly determine the endomorphisms on simple abeliansurfaces whose fixed-points function grows exponentially, thus proving Theorem 2stated in the introduction.We will make use of the fixed-point formula for abelian varieties [CAV, 13.1.4],which for an endomorphism f of a simple abelian surface X takes the following form:Let D = End Q ( X ), K = center( D ), e = [ K : Q ], d = [ D : K ], and let N : D → Q be the reduced norm map. Then for f ∈ End( X ), f ) = (cid:16) N (1 − f ) (cid:17) de . (The formula expresses the fact that the characteristic polynomial of the rationalrepresentation ρ r ( f ) coincides with the map N /de .) Since on a simple abeliansurface every non-zero endomorphism f is an isogeny, both eigenvalues of f arenon-zero.We employ now a strategy as in [BL], i.e., we use Albert’s classification and dealwith the possible types of simple abelian surfaces separately. Type 0: Integer multiplication.
Suppose that End( X ) = Z . In that case everyendomorphism f is a multiplication map x mx for some m ∈ Z . So f hasexponential fixed-points growth if and only if | m | > Type 1: Real multiplication.
Suppose that X has real multiplication, i.e.,that End Q ( X ) = Q ( √ d ) for some square-free integer d >
0. Every endomorphism f ∈ End( X ) is then of the form f = a + bω with a, b ∈ Z , where ω = ( √ d if d ≡ , (1 + √ d ) if d ≡ C , the analytic representation ρ a : End Q ( X ) = Q ( √ d ) → M ( C ) is given by1 and √ d √ d −√ d ! (see [Rup]). So the eigenvalues of ρ a ( f ) = ρ a ( a + bω ) are • a ± b √ d , if d ≡ , • a + b (1 ± √ d ), if d ≡ b = 0, then f is multiplication by a , and hence it has exponential fixed-pointsgrowth if and only if | a | >
1. And if b = 0, then both eigenvalues are of absolutevalue = 1. Using Prop. 1.6 we see that then f has exponential fixed-points growth. Type 2: Indefinite quaternion multiplication.
Suppose that X has indefinitequaternion multiplication, i.e., there are α, β ∈ Q \ { } with α > β and α > Q ( X ) is isomorphic to the quaternion algebra (cid:0) α, β Q (cid:1) , i.e., End Q ( X ) = Q + i Q + j Q + ij Q , where i and j satisfy the relations i = α , j = β and ij = − ji .Using the splitting field Q ( √ α ) of End Q ( X ), one has an isomorphism ψ : End Q ( X ) ⊗ Q Q ( √ α ) → M ( Q ( √ α ))given by i ⊗ √ α −√ α ! and j ⊗ β ! . For an element f ∈ End( X ), written as f = a + bi + cj + dij with a, b, c, d ∈ Q , wehave ψ ( f ) = a + b √ α cβ + dβ √ αc − d √ α a − b √ α ! . The reduced norm of f over Q is given by N ( f ) = det( ψ ( f )) = a − b α − c β + d αβ .After diagonalization of ψ ( f ) the norm of 1 − f n can be written as N (1 − f n )= det − ( a + p b α + c β − d αβ ) n
00 1 − ( a − p b α + c β − d αβ ) n ! = (1 − t n )(1 − t n ) , where t i = a ± p b α + c β − d αβ . So we have f n ) = ((1 − t n )(1 − t n )) .Consider now the reduced characteristic polynomial of f , χ f ( t ) = t − tr( f ) t + N ( f ) = t − tr( ψ ( f )) t + det( ψ ( f )) . Since f is contained in an order, it is an integral element, and hence χ f has integercoefficients. For every integer m we have χ f ( m ) = ( m − t ) ( m − t ) = N ( m − f ) = N (1 − ( f − ( m − = f − ( m − − ρ r ( f − ( m − − ρ r ( f ) + ( m − )= det( m − ρ r ( f )) = P rf ( m )and this implies that χ f ( t ) = P rf ( t ) as polynomials in t . Therefore, if we denote by λ and λ the analytic eigenvalues of f , then t ∈ (cid:8) λ , λ (cid:9) and t ∈ (cid:8) λ , λ (cid:9) . • If | t | > | t | >
1, then f n ) grows asymptotically with | t t | n . • If | t | <
1, then Prop. 1.6 tells us that | λ | >
1. Therefore the number offixed-points grows exponentially in this case. • If | t | = | t | = 1, then the possible real values for t and t are ±
1. If t and t are complex, then they are roots of unity, because they are roots of theintegral polynomial χ f . The function n f n ) is then periodic. • We show now that the case | t | > | t | = 1 does not occur. Assume thecontrary. Since t and t are the roots of a rational polynomial of degree 2, weknow that t has to be ±
1. As t = t , at least one of the coefficients b, c, d isnon-zero (since otherwise a = t = t ). But then t = ± f ∓ Type 3: Complex multiplication.
Suppose that X has complex multiplication,i.e., that End Q ( X ) is isomorphic to an imaginary quadratic extension K of a realquadratic number field Q ( √ d ), where d is a positive square-free integer.Consider an endomorphism f ∈ End( X ) and let λ and λ be the eigenvalues ofits analytic representation ρ a ( f ). By the Cayley-Hamilton theorem, f is annihilatedby its rational characteristic polynomial P rf . As the isomorphism σ : End Q ( X ) → K fixes Q , the complex number e f = σ ( f ) is then a zero of P rf as well. This implies thatit is contained in the set (cid:8) λ , λ , λ , λ (cid:9) . So, after renumbering, we have e f = λ or e f = λ . We distinguish now cases according to e f : • If | e f | <
1, then we know by Prop. 1.6 that | λ | >
1. Therefore in this case thenumber of fixed points grows exponentially, which is behaviour (B1). • If | e f | = 1, then by Prop. 1.7 we know that λ , and hence e f , is a root of unity.The fixed-points function n f n ) = N (1 − f n ) is then periodic (B2). • Let | e f | > e f ∈ R . Then e f is of the form a + b √ d , hence we have behaviour(B1) as in the case of real multiplication. • Let | e f | > e f / ∈ R . So | λ | >
1. Our aim is to show that then | λ | 6 = 1,which implies behaviour (B1). Assuming by way of contradiction that | λ | = 1,note first that the proof of Prop. 1.7 shows that the minimal polynomial of λ must be of degree 2. Therefore λ can be written as a + b √− e with a, b ∈ Q and e a square-free positive integer. For t ∈ Z we compute the norm of t − f , N K/ Q ( t − f ) = N Q ( √− e ) / Q ( N K/ Q ( √− e ) ( t − f ))= N Q ( √− e ) / Q (( t − f ) )= (( t − a ) + b e ) . On the other hand, N K/ Q ( t − f ) coincides as a polynomial in t with P rf ( t ), andtherefore λ equals e f or its conjugate. It cannot be of absolute value 1 then,and so we arrive at a contradiction.We provide a complete list of the possible eigenvalues that occur for endomor-phisms with periodic fixed-points behaviour in quaternion and complex multiplica-tion.0 Proposition 2.1. a) If X is a simple abelian surface with quaternion multiplica-tion and f a non-zero endomorphism on X with periodic fixed-points behaviour,then all eigenvalues of f are roots of unity of algebraic degree , i.e., theyare contained in the following set:in degree 1: ± in degree 2: ± i, ± ± √− Conversely, each of these numbers occurs as an eigenvalue of an endomorphismon some simple abelian surface with quaternion multiplication. b) If X is a simple abelian surface with complex multiplication and f a non-zeroendomorphism on X with periodic fixed-points behaviour, then all eigenvaluesof f are roots of unity of algebraic degree , i.e., they are contained in thefollowing set:in degree 1: ± in degree 2: ± i, ± ± √− in degree 4: ± √ ± √− , ± √ ± √− , ± ( + √ ) ± i q − √ , ± ( − √ ) ± i q + √ Conversely, each of these numbers occurs as an eigenvalue of an endomorphismon some simple abelian surface with complex multiplication.Proof.
One direction is clear by now: If f has periodic fixed-points behaviour, thenwe know that all eigenvalues are roots of unity. In the quaternion case they are rootsof χ f and therefore of degree
2, and in the complex multiplication case they areroots of P rf and therefore of degree ±
1, we nowexhibit quaternion algebras B and B that are skew-fields, and orders O ⊂ B and O ⊂ B containing elements that lead to the required eigenvalues ± i and ± ± √− respectively. By Shimura’s theory (cf. [CAV, § O k is contained in the endomorphism ring of some simple abelian surface.To this end, consider first the quaternion algebra B = (cid:0) , Q (cid:1) . It is a skew fieldand contains the order O = Z + Z i + Z j + Z ij . The element f = i + j + ij hasreduced characteristic polynomial χ f ( t ) = t + 1, whose roots are ±√− B = (cid:0) − , Q (cid:1) and the splitting field L = Q ( i ), where i = − L is S = Z + i Z , and it follows from this that S + jS , with j = 2, is an order in B . The element f = i ∈ S hat reduced characteristicpolynomial χ f ( t ) = t − t + 1, whose roots are ± √−
3. And the element f = − i ∈ S has reduced characteristic polynomial χ f ( t ) = t + t + 1, whose rootsare − ± √− ζ of algebraic degree 4 isan algebraic integer in the CM field Q ( ζ ), and therefore there exists a simple abeliansurface X of CM-type, where the root represents an endomorphism f (cf. [CAV, § L and take the CM field L ( ζ ). (cid:3) References [Bir] Birkenhake, Ch.: Tensor products of ample line bundles on abelian varieties.Manuscripta math. 84 (1994), 21-28[BL] Birkenhake, C., Lange, H.: Fixed-point free automorphisms of abelian varieties. Geom.Dedicata 51, No.3, 201–213 (1994)[CAV] Birkenhake, C., Lange, H.: Complex abelian varieties. Springer, 2004.[Coh] Cohen, H.: Number Theory. Volume I: Tools and Diophantine Equations. Springer,2007.[DKL] De Fernex, T., K¨uronya, A. , Lazarsfeld, R.: Higher cohomology of divisors on aprojective variety. Math. Ann. 337, No. 2, 443-455 (2007)[EEL] Ein, L., Erman, D., Lazarsfeld, R.: Asymptotics of random Betti tables. J. ReineAngew. Math. 702, 55-75 (2015)[EL] Ein, L., Lazarsfeld, R.: Asymptotic syzygies of algebraic varieties. Invent. Math. 190,No. 3, 603-646 (2012)[ELMNP] Ein, L., Lazarsfeld, R., Mustat¸ˇa, M., Nakamaye, M., Popa, M.: Asymptotic invariantsof base loci. Ann. Inst. Fourier 56, No. 6, 1701-1734 (2006)[LB] Lange, H., Birkenhake, Ch.: Complex Abelian Varieties. Grundlehren der math. Wiss.302, Springer-Verlag, 1992.[McM] McMullen, C.: Dynamics on K3 surfaces: Salem numbers and Siegel disks. J. reineangew. Math 545, 201–233 (2001)[Res] Reschke, P.: Salem Numbers and Automorphisms of Complex Surfaces. Math. Res.Lett. 19(2), 475–482 (2012)[Rin] Ringler, A.: Fixed points of smooth varieties with Kodaira dimension zero.arXiv:0708.3587 [math.AG][Rup] Ruppert, W.: Two-dimensional complex tori with multiplication by √ d . Arch. Math.72, 278–281 (1999)[SS] Shub, M., Sullivan, D.: A remark on the Lefschetz fixed point formula for differentiablemaps. Topology 13, 189–191 (1974)Thomas Bauer, Fachbereich Mathematik und Informatik, Philipps-Universit¨at Marburg, Hans-Meerwein-Straße, D-35032 Marburg, Germany. E-mail address: [email protected]
Thorsten Herrig, Fachbereich Mathematik und Informatik, Philipps-Universit¨at Marburg, Hans-Meerwein-Straße, D-35032 Marburg, Germany.