On the Picard numbers of abelian varieties
aa r X i v : . [ m a t h . AG ] D ec ON THE PICARD NUMBERS OF ABELIAN VARIETIES
KLAUS HULEK AND ROBERTO LAFACE
Abstract.
In this paper we study the possible Picard numbers ρ of an abelian variety A of dimension g . It is well known that this satisfies the inequality 1 ≤ ρ ≤ g . We prove thatthe set R g of realizable Picard numbers of abelian varieties of dimension g is not completefor every g ≥
3, namely that R g ( [1 , g ] ∩ N . Moreover, we study the structure of R g as g → + ∞ , and from that we deduce a structure theorem for abelian varieties of largePicard number. In contrast to the non-completeness of any of the sets R g for g ≥
3, wealso show that the Picard numbers of abelian varieties are asymptotically complete, i.e.lim g → + ∞ R g /g = 1. As a byproduct, we deduce a structure theorem for abelian varietiesof large Picard number. Finally we show that all realizable Picard numbers in R g can beobtained by an abelian variety defined over a number field. Introduction
For an algebraic variety X over the field of complex numbers the Lefschetz (1 , X ) = H ( X, Z ) ∩ H , ( X ) . Consequently, the rank ρ ( X ) of the N´eron-Severi group, the so-called Picard number , satisfiesthe inequality 1 ≤ ρ ( X ) ≤ h , ( X ). Computing the Picard number is in general a difficultquestion, as already the case of projective surfaces shows. For example, the Picard numberof a quintic surface S in P satisfies the inequality ρ ( S ) ≤
45. It is known that all numbersbetween 1 and 45 can be obtained if one allows the surface to have
ADE -singularities, butit remains an open problem for smooth surfaces, where the maximum known is 41 [15], [16].In this article we will concentrate on the Picard numbers of abelian varieties. To put thisinto perspective it is worthwhile to recall the situation for surfaces. For abelian surfaces allpossible Picard numbers between 1 (or 0 if one includes the non-algebraic case) and 4 occur.Indeed, a very general abelian surface has ρ = 1, whereas Picard numbers from 2 to 4 can berealized by taking a product E × E of two elliptic curves. If the two elliptic curves are notisogenous, then ρ = 2, if they are isogenous but they do not have complex multiplication,then ρ = 3, while if they also have complex multiplication ρ = 4. For the other surfaces withtrivial canonical bundle the situation is similar: for K3 surfaces all possibilities between 1(respectively 0) and 20 can occur, as can be seen by the Torelli theorem for K3 surfaces.Enriques surfaces and bi-elliptic surfaces have no holomorphic 2-forms, and thus their Picardnumber is 10 and 2 respectively.For higher-dimensional varieties with numerically trivial canonical bundle the situation isas follows. By the Beauville-Bogomolov decomposition theorem [4], every K¨ahler manifoldwith trivial first Chern class admits a finite cover which is a product of tori, Calabi-Yauvarieties and irreducible holomorphic symplectic manifolds (IHSM), also know as hyperk¨ahlermanifolds. For higher dimensional Calabi-Yau varieties Y we always have ρ ( Y ) = b ( Y ) as h , ( Y ) = h , ( Y ) = 0. For irreducible holomorphic symplectic manifolds X one can use uybrechts’ surjectivity of the period map [11] to conclude, as in the case of K3 surfaces,that all values 0 ≤ ρ ( X ) ≤ b ( X ) − A be a complex torus of dimension g . Its cohomology is the exterior algebra overH ( A, Z ) ∼ = Z g . In particular, this implies that the k th Betti numbers are b k ( A ) = (cid:0) gk (cid:1) . AsH p, ( A ) ∼ = H ( A, Ω pA ), we get h p, ( A ) = (cid:0) gp (cid:1) , and thus h , ( A ) = g . We shall from now onexclude the case of non-algebraic tori and concentrate on abelian varieties. By the above weknow that 1 ≤ ρ ≤ g . As we have already seen, any number 1 ≤ ρ ( A ) ≤ A , we can invoke Poincar´e Complete Reducibility The-orem [7, Thm. 5.3.7] to pass to a better representative in its isogeny class, namely A −→ A n × · · · × A n r r , where A i is a simple abelian variety ( i = 1 , . . . , r ), and A i is not isogenous to A j if i = j .Moreover, the abelian varieties A i and the integers n i are uniquely determined up to isogenyand permutations. A result of Murty [12, Lemma 3.3] describes the Picard number of aself-product B k of a simple abelian variety in terms of k and the dimension of B . In light ofthis, we prove in Proposition 2.2 a splitting result concerning the Picard group of varietiesof the form A × B with Hom( A, B ) = 0, which allows us to compute the Picard numberof such products. This, together with Murty’s result in [12], provides us with a theoreticalalgorithm for computing the Picard number of a given abelian variety.It is then a combinatorial question as to determine the set R g of possible Picard numbers ofabelian varieties for a given genus g . Very little seems to be known about this. The purposeof this paper is to take a first step in the analysis of R g . As a first result we show that thereare gap series for the possible Picard numbers of abelian varieties, and therefore that the sets R g are not complete for every g ≥
3. In fact, it is not hard to show that R = { , . . . , , } :indeed, a very general abelian threefold has Picard number ρ = 1; a product S × E , S beinga very general abelian surface and E being an elliptic curve, has Picard number ρ = 2; allother Picard numbers can be obtained by using suitable products of elliptic curves. Thisphenomenon had previously been notice by Shioda in [21, Appendix].As the dimension g grows larger, clear gaps in the set of possible Picard numbers startto appear. Moreover, more and more gaps occur as g → ∞ . The following result showsthe existence of two precise gaps and characterizes the three largest Picard numbers for anabelian variety. Theorem 1.1. (1)
Fix g ≥ . There does not exist any abelian variety of dimension g with Picardnumber ρ in the following range: ( g − + 1 < ρ < g . Fix g ≥ . There does not exist any abelian variety of dimension g with Picardnumber ρ in the following range: ( g − + 4 < ρ < ( g − + 1 . We would like to remark that the conditions on the dimension g given in Part 1 and 2 ofTheorem 1.1 are necessary. In fact, as for Part 1, for g = 2 all Picard numbers occur, andfor g = 3 there exists an abelian threefold of Picard number ρ = 6 (namely, the product ofthree isogenous elliptic curves without complex multiplication). Similar considerations canbe made for Part 2 of Theorem 1.1 and g ≤
6. After some preliminary work in Section 2, weshall prove this theorem in Section 3. As an application of our analysis we derive in Section4 a structure theorem for abelian varieties with large Picard number, namely Theorem 4.2.The above results are a first indication of a much more general phenomenon which westudy more systematically in Section 7, where we consider the behaviour of the set R g asymptotically, namely as g grows. In particular, we define the asymptotic density of Picardnumbers to be the quantity δ := lim g → + ∞ R g g . Contrary to the non-completeness of any of the sets R g , we prove asymptotic completenessin Theorem 7.1, namely Theorem 1.2.
The Picard numbers of abelian varieties are asymptotically complete: δ = lim g → + ∞ R g g = 1 . By using similar techniques, we are also able to describe the distribution of the Picardnumbers within [1 , g ] ∩ N (Theorem 7.4). As a consequence, we obtain a structure theoremfor abelian varieties of large Picard number in Corollary 7.6. We also provide a practicalalgorithm which allows to compute the sets R g inductively.Finally we show that all realizable Picard numbers ρ ∈ R g can be obtained by an abelianvariety defined over a number field. Acknowledgement.
We would like to thank Bert van Geemen for his genuine interest in thisquestion, and Matthias Sch¨utt for discussions around this topic and beyond. RL would like toparticularly thank Fran¸cois Charles for the many fruitful discussions and for his invitationto the IH ´ES, whose excellent working conditions are here gratefully acknowledged. Wewould like to thank Davide Lombardo for suggesting the proof of Lemma 7.2. We gratefullyacknowledge a very fruitful exchange of ideas with Ben Moonen who also supplied the proofof Theorem 8.1. Finally, we are grateful to the anonymous referee for his or her comments,which have helped improving the manuscript. This research was partially funded by theDFG funded GRK 1463 ”Analysis, Geometry and String Theory”.2.
Preliminary work
In this section we will develop the basic tools of our analysis. Some of these results are ofindependent interest in their own right. .1. Additivity of the Picard number for non-isogeneous products.
As the Picardnumber of an abelian variety is invariant under isogenies [6, Ch. 1, Prop. 3.2], we can picka convenient representative in its isogeny class. Such a choice is indicated by the followingresult [7, Thm 5.3.7]:
Theorem 2.1 (Poincar´e’s Complete Reducibility Theorem) . Given an abelian variety A ,there exists an isogeny A −→ A n × · · · × A n r r , where A i is a simple abelian variety ( i = 1 , . . . , r ), and A i is not isogenous to A j if i = j .Moreover, the abelian varieties A i and the integers n i are uniquely determined up to isogenyand permutations. Let us now consider a product of simple abelian varieties as in Theorem 2.1. The fact that A i is not isogenous to A j for i = j yields the following splitting of the Picard group: Proposition 2.2.
Let A , . . . , A r be simple abelian varieties, such that A i is not isogenousto A j for i = j . Then the (exterior) pullback of line bundles yields an isomorphism r Y i =1 Pic( A n i i ) ∼ = Pic r Y i =1 A n i i ! . Clearly, exterior pull-back of line bundles always yields an injective map, but surjectivityis a special feature. In fact, if E is an elliptic curve, the abelian surface E × E has Picardnumber ρ ∈ { , } , depending on the presence of CM. Therefore, the exterior pull-back mapPic( E ) × Pic( E ) −→ Pic( E × E )cannot be surjective, as otherwise we would get a surjective map of the corresponding N´eron-Severi groups, hence yielding a contradiction, since NS( E ) ∼ = Z . Proof.
Exterior pullback of line bundles ψ ( L , . . . , L r ) = L ⊠ · · · ⊠ L r defines the following commutative diagram0 / / Pic ( Q ri =1 A n i i ) / / Pic( Q ri =1 A n i i ) / / NS( Q ri =1 A n i i ) / / / / O O Q ri =1 Pic ( A n i i ) ψ O O / / Q ri =1 Pic( A n i i ) ψ O O / / Q ri =1 NS( A n i i ) ψ NS O O / / . O O We will show that ψ and ψ NS are isomorphisms, thus proving the proposition. Clearly ψ is injective, and since ψ is a homomorphism of abelian varieties of the same dimension itmust be an isomorphism. To prove that ψ NS is an isomorphism we recall from [7, Ch. 2]that a polarization on an abelian variety A is given by a finite isogeny f : A → A ∨ whoseanalytic representation is hermitian. By assumption the abelian varieties A i and A j are notisogeneous for i = j . Hence Hom( A i , A j ) = Hom( A i , A ∨ j ) = 0 and every isogeny f : r Y i =1 A n i i −→ ( r Y i =1 A n i i ) ∨ s of the form f = ( f , . . . , f r ) where f i : A n i i −→ ( A n i i ) ∨ is an isogeny. Since a direct sum ofendomorphisms is hermitian if and only if all its summands are, the claim follows, as everyclass in the N´eron-Severi group can be written as the difference of two ample classes (i.e. twopolarizations). (cid:3) As a consequence, we get that the Picard number is additive (but not strongly additive)for product varieties coming from the Poincar´e’s Complete Reducibility Theorem.
Corollary 2.3.
Let A , . . . , A r be simple abelian varieties, such that A i is not isogenous to A j for i = j . Then, ρ r Y i =1 A n i i ! = r X i =1 ρ ( A n i i ) . Picard numbers of self-products.
Due to additivity, we are left to see how to com-pute the Picard number in the case of a self-product of a simple abelian variety. Theendomorphism ring End( A ) of an abelian variety A is a finitely generated free abelian groupand hence F := End( A ) ⊗ Q ≡ End Q ( A ) is a finite dimensional Q -algebra. Any polarization L on A defines an involution on F , the so-called Rosati involution. If A is a simple abelianvariety, then F is a finite-dimensional skew field admitting a positive anti-involution. Suchpairs were classified by Albert [1], [3], see also [7, Proposition 5.5.7] for a summary.Let K be the centre of F . We will say that F is of the first kind if the Rosati involutionacts trivially on K , and of the second kind otherwise. Let us denote by K the maximal realsubfield of K , and let us consider the following invariants of F :[ F : K ] = d , [ K : Q ] = e, [ K : Q ] = e . Notice that, as K is the center of F , [ F : K ] is always a square.As a useful example, we can consider quaternion algebras over a field K . Let us recallthe reader that any quaternion algebra F/K is either a division ring or it is isomorphic to M ( K ). In light of this, we can define the ramification locus of F asRam( F ) = { p ∈ Spec O K | K p is a division ring } . A quaternion algebra is ramified at a finite even number of places. The ramification locusgoverns the isomorphism classes of quaternion algebras: there is a 1:1 correspondence be-tween isomorphism classes of quaternion algebras and subsets of non-complex places of K of even cardinality.By [7, Proposition 5.5.7] the classification divides into four types, where the first three areof the first kind:(1) Type I : F is a totally real number field, so that d = 1 and e = e .(2) Type II : F is a totally indefinite quaternion algebra over a totally real number field K , i.e. ∅ = Ram( F ) ∩ { archimedean places of K } and Ram( F ) = ∅ . In particular, we have that d = 2 and e = e .(3) Type III : F is a totally definite quaternion algebra over K , i.e. ∅ 6 = Ram( F ) ⊇ { archimedean places of K } holds. Again, we have d = 2 and e = e . Type IV : F is of the second kind, and it center Z ( F ) = K is a CM field with maximalreal subfield K .The following result, due to Murty, gives a complete description of the Picard number ofa self-product of a simple abelian variety. Proposition 2.4 (Lemma 3.3 of [12]) . Let A be a simple abelian variety. Set e := [ K : Q ] , d := [ F : K ] . Then, for k ≥ , one has ρ ( A k ) = ek ( k + 1) Type I ek (2 k + 1) Type II ek (2 k − Type III ed k Type IV.
In fact, Murty’s result is in terms of the maximal commutative subfield E of F , which hasdegree [ E : K ] over K . However, the proof of [12, Lemma 2.2] implies that [ E : K ] = [ F : K ].Proposition 2.4 enables us to compute the following bound for the Picard number of a self-product of a simple abelian variety: Corollary 2.5.
Let A be a simple abelian variety of dimension n , and let k ≥ . Then ρ ( A k ) ≤ nk (2 k + 1) .Proof. Proposition 2.4 applied with k = 1 allows us to compute the Picard number of A : ρ = ρ ( A ) = e Type I e Type II e Type III ed Type IV.
Now, plugging this back in Proposition 2.4 gives the following reformulation in terms of thePicard number of A : ρ ( A k ) = ρk ( k + 1) Type I ρk (2 k + 1) Type II ρk (2 k − Type III ρk Type IV.
The divisibility conditions for ρ given by [7, Prop. 5.5.7] imply that ρ ≤ n Type I n Type II n Type III n Type IV and, therefore, we see that ρ ( A k ) ≤ nk ( k + 1) Type I nk (2 k + 1) Type II nk (2 k − Type III nk Type IV from which the result follows. (cid:3) n the case of a self-product of an elliptic curve this gives the well known Corollary 2.6. If E is an elliptic curve, then ρ ( E k ) = ( k ( k + 1) E has no CM k E has CM. We will use these results, in particularly the case of a self-product of elliptic curves,frequently in the proof of Theorem 1.1. Notice that Corollary 2.5 provides us with a boundon the Picard number of A k which is independent of the type of the endomorphism ring of A . 3. Restrictions on the Picard number
Some bounds on the Picard number.
We would like to show that there are betterbounds on the Picard number, if one is given a partition of the dimension. More precisely,letting A be an abelian variety, we define r ( A ) to be the length of a decomposition accordingto Poincar´e Complete Reducibility Theorem. In other words, given an abelian variety A ,Theorem 2.1 gives an isogeny A −→ A n × · · · × A n r r , and we set r ( A ) := r . Notice that this quantity is well-defined because the factors A i andthe powers n i are determined up to permutations and isogenies. Then, for r ≤ g , we define M r,g as M r,g := max { ρ ( A ) | dim A = g, r ( A ) = r } . In other words, M r,g is the largest Picard number that can be realized by a g -dimensionalabelian variety that splits into a product of r non-isogenous pieces in its isogeny class. Proposition 3.1.
For integers r, g ∈ N such that r ≤ g , one has M r,g = [ g − ( r − +( r − .This value is attained as the Picard number of E g − r +1 × E × · · · × E r − , where E is a CMelliptic curve not isogenous to any of the E i ’s, and E i and E j are not isogenous for i = j .Proof. If A ∼ A × · · · × A r , Hom( A i , A j ) = 0 for i = j , then ρ ( A ) ≤ k + · · · + k r where k i := dim A i ( i = 1 , . . . , r ) and k + · · · + k r = g . Hence we are looking for the maximaof the function h ( x , . . . , x r ) = x + · · · + x r − + x r on the integral points of the simplexΩ r,g = { ( x , . . . , x r ) | x i ≥ , x + . . . x r = g } . These points are precisely the vertices (cid:8) ( g − r + 1 , , . . . , , (1 , g − r + 1 , , . . . , , . . . , (1 , . . . , , g − r + 1) (cid:9) . By the symmetry of h , the maximum is attained at any of these points, with value h ( g − r + 1 , , . . . ,
1) = [ g − ( r − + ( r − . Therefore, we conclude that ρ ( A ) ≤ [ g − ( r − + ( r − E g − r +1 × E × · · · × E r − ith E a CM curve and E, E i , E j for i = j not pairwise mutually isogeneous, has Picardnumber [ g − ( r − + ( r − (cid:3) Corollary 3.2.
Let A be an abelian variety. Then, ρ ( A ) = M r ( A ) ,g ⇐⇒ A ∼ E g − ( r − × E × · · · × E r − , where E is a CM elliptic curve not isogenous to any of the E i ’s, and E i and E j are notisogenous for i = j .Remark . The numbers M r,g give the following (strictly) increasing sequence of positiveintegers: g = M g,g < M g − ,g < · · · < M ,g < M ,g < M ,g = g . We will now proceed with the proof of Theorem 1.1, which we divide into two parts.3.2.
Proof of part (1).
Let A be an abelian variety of dimension g ≥ ρ = ρ ( A ). We will divide our analysis of the Picard number ρ into the following cases:(a) A has length at least two, i.e. r ( A ) ≥ A is a self-product of a lower dimensional abelian variety. Case (a).
Since r ( A ) ≥
2, we have that A ∼ A × A with Hom( A , A ) = 0. Let n := dim A , so that dim A = g − n . Then, ρ ( A ) ≤ n + ( g − n ) . Consider the function f ( x ) := x + ( g − x ) on Ω = [1 , g − x = 1 and x = g −
1, with value f (1) = f ( g −
1) = ( g − + 1. Therefore, ρ ( A ) ≤ ( g − + 1. Case (b).
Let B be an m -dimensional simple abelian variety, and suppose A is isogenousto B k , for k := g/m . If m = 1 (i.e. B is an elliptic curve), then again by Corollary 2.6 ρ ( B g ) = ((cid:0) g +12 (cid:1) B has no CM g B has CM.If B has CM, then A attains the maximal Picard number g ; if B does not have CM, then ρ ( A ) = (cid:18) g + 12 (cid:19) ≤ g − because g ≥
4. The case of a self-product of an elliptic curve being dealt with, we can assume k ≤ g/
2. Then, by Corollary 2.5 we have ρ ( B k ) ≤ g (2 k + 1) ≤ g ( g + 1)and the claim follows, since the equality12 g ( g + 1) ≤ ( g − + 1holds for g ≥ (cid:3) .3. Proof of part (2).
To start with observe that, if r ( A ) ≥
3, then ρ ( A ) ≤ M r ( A ) ,g ≤ M ,g < ( g − + 4 . Therefore, we can assume r ( A ) ≤
2. Suppose that A is an abelian variety with r ( A ) = 1, i.e. A ∼ B k with dim B = b and bk = g . If b = 1, then B is an elliptic curve and we have two casesaccording to whether B has complex multiplication. If B does have complex multiplication,then ρ ( A ) = g (the maximal Picard number), otherwise ρ ( A ) = g ( g + 1) < ( g − + 4(as g ≥ b >
1, then k ≤ g/ ρ ( B k ) ≤ g (2 k + 1) ≤ g ( g + 1) ≤ ( g − + 4again because g ≥
7. The last remaining case is r ( A ) = 2, which we will divide into threesteps. Step 1.
We deal with abelian varieties of the form E n × E g − n , where E and E are ellipticcurves, and 1 ≤ n ≤ g − n . If n = 1, then, by Proposition 2.3 ρ ( E × E g − ) = 1 + ρ ( E g − )which equals M ,g if E has complex multiplication, and 1 + g ( g −
1) otherwise. In the CMcase, we obtain the second largest attainable Picard number, in the non-CM case insteadone sees that it is always the case that 1 + g ( g − ≤ ( g − + 4. Suppose now that n ≥ ρ ( E n × E g − n ) ≤ n + ( g − n ) , and we want to bound the right-hand side. Thefunction f ( x ) := x + ( g − x ) attains its maximum on the interval Ω = [2 , g −
2] at x = 2 and x = g − f (2) = f ( g −
2) = ( g − + 4. This implies that ρ ( E n × E g − n ) ≤ ( g − + 4. Step 2.
We now consider abelian varieties of the form E k × A l , with E an elliptic curve,dim A = a > k ≥ l ≥ g = k + al . Notice that, by Proposition 2.3 and Lemma2.5, one has ρ ( E k × A l ) ≤ k + 12 al (2 l + 1) = k + 12 ( g − k )(2 l + 1) . Consider the function f ( x, y ) = x + 12 ( g − x )(2 y + 1) , in the domain Ω := { ( x, y ) ∈ R | x ≥ , y ≥ , x + 2 y ≤ g } . We will prove that f isbounded from above by ( g − + 4 in Ω.By looking at the partials ∂f∂x ( x, y ) = 2 x − y − , ∂f∂y ( x, y ) = g − x, we see that f is increasing on the lines where x is constant. Thus the maximum of f in Ωwill lie on the line x + 2 y = g . Therefore, we have reduced ourselves to studying the function g ( y ) := f ( g − y, y ) = ( g − y ) + 2 y + y on [1 , ( g − / y max = 1, with value g ( y max ) = ( g − + 3 < ( g − + 4 . tep 3. The last case is that of products of the form A k × B l , with dim A = a > B = b > k ≥ l ≥ g = ak + bl . One has, ρ ( A k × B l ) ≤ ak (2 k + 1) + 12 bl (2 l + 1) ≤ ak (2 k + 1) + 12 bl (2 k + 1) == 12 g (2 k + 1) ≤ g ( g − < ( g − + 4 . (cid:3) Structure of abelian varieties with large Picard number
As an application of Theorem 1.1, we will now derive a structure result for abelian varietiesof large Picard number (up to isogeny). Our starting point is the following result:
Theorem 4.1 (Exercise 5.6.10 of [7]) . Let A be an abelian variety of dimension g . Thefollowing are equivalent (1) ρ ( A ) = g ; (2) A ∼ E g , for some elliptic curve E with complex multiplication; (3) A ∼ = E ×· · ·× E g , for some pairwise isogenous elliptic curves E , . . . , E g with complexmultiplication. This result points out how the Picard number can force the structure of an algebraicvariety to be in some sense rigid. Algebraic varieties with the maximum Picard numberpossible have shown to possess interesting arithmetic and geometric properties: for example,see [20] and [19], or [5] for a recent account.The aim of this section is to prove a similar statement for abelian varieties whose Picardnumber is the second or third largest attainable according to Theorem 1.1, namely ( g − +1or ( g − + 4. However, unlike in the case of maximal Picard number, one cannot expect astatement which is analogous to Theorem 4.1(3). Already for ρ ( A ) = ( g − + 1, one canconstruct abelian varieties which are isogenous to E g − × E , but which are not isomorphicto a product of elliptic curves.The following result describes the structure of these abelian varieties up to isogeny. Itshould be noticed that, in contrast with Theorem 4.1, the result depends on the dimensionof the abelian varieties we consider: on one hand we need Theorem 1.1, and on the otherwe need to guarantee that the abelian varieties having such Picard numbers all belong to aunique isogeny class. Theorem 4.2.
Let A be an abelian variety of dimension g . (1) Suppose g ≥ . Then, ρ ( A ) = ( g − + 1 ⇐⇒ A ∼ E g − × E , where E has complex multiplication and E and E are not isogeneous. (2) Suppose g ≥ . Then, ρ ( A ) = ( g − + 4 ⇐⇒ A ∼ E g − × E , where E and E both have complex multiplication but are not isogeneous. roof. Recall that we have the following (strictly) increasing sequence of positive integers: g = M g,g < M g − ,g < · · · < M ,g < M ,g < M ,g = g . Assume ρ ( A ) = ( g − + 1 = M ,g and g ≥
5. By definition of M r,g it follows that r ( A ) ≤ r ( A ) = 2. Indeed, if r ( A ) = 1, then necessarily A ∼ E g . But then either ρ ( A ) = g if E has CM (by Theorem 4.1) or ρ ( A ) = (cid:0) g +12 (cid:1) otherwise, either of which is acontradiction. Therefore r ( A ) = 2 and (1) follows from Corollary 3.2.Now let A have Picard number ρ ( A ) = ( g − + 4, and let g ≥
7. As ρ ( A ) > M ,g , wededuce r ( A ) ≤
2. If r ( A ) = 1 we again get a contradiction as above (this time, we also use g ≥ r ( A ) = 2. In a similar fashion to the proof of the Theorem 1.1, we distinguishthree cases:(a) Let A ∼ A k × B l , with dim A >
B >
1. Then, as we have seen in Step 3of the proof of Theorem 1.1 ρ ( A ) ≤ g ( g − < ( g − + 4 , which gives a contradiction.(b) Let A ∼ E k × B l , with dim B > E = 1. Then, as we have seen in Step 2 ρ ( A ) ≤ ( g − + 3 < ( g − + 4 , again a contradiction.(c) Let A ∼ E n × E g − n , for two elliptic curves E and E . Then, the cases n = 1and n = g − ≤ n ≤ g −
2. We claim that ρ ( A ) < ( g − +4, unless both E and E have complexmultiplication. Indeed, if one of the factors does not have complex multiplication then ρ ( A ) ≤ g − < g − . Therefore both E and E must have complexmultiplication, and so ρ ( A ) = ρ ( E n × E g − n ) = n + ( g − n ) . The maximum of thisexpression is achieved for n = 2 or n = g −
2, and this corresponds to a product E × E g − .We have thus shown that the only possible case is A ∼ E × E g − , for two non-isogeneouselliptic curves E and E with complex multiplication, hence proving (2), and in this casethe Picard number is as stated. (cid:3) Clearly, one can continue this analysis along arguments used above. However, one cannot,in general, expect to obtain a unique decomposition for a given Picard number. Already for ρ = ( g − + 3 there are two possible isogeny decompositions, namely:(1) E g − × E , E being an elliptic curve with complex multiplication and E not havingcomplex multiplication;(2) E g − × S , E being an elliptic curve with complex multiplication and S being asimple abelian surface of type II (these do exist by results of Shimura [18], see alsothe discussion in Section 5). Remark . The Picard numbers g and ( g − + 4 both lead to cases which have nocomplex moduli, whereas the intermediate Picard number ( g − + 1 leads to 1-dimensionalfamilies. This is in striking contrast to the case of K3 surfaces where increasing the Picardnumber by one corresponds to a decrease in the number of moduli by one. This is clear fromthe Torelli theorem for K3 surfaces. The difference lies in the fact that the Torelli theorem or K3 surfaces works with a weight 2 Hodge structure, wheres abelian varieties are governedby weight 1 Hodge structures.5. Computing Picard numbers
In this section we approach the question how to compute the set R g of possible Picardnumbers of abelian varieties of a given dimension g . To this end, let us fix a positive integer G , such that we are interested in computing R G . Because of the structure of R G , we will infact have to compute the sets R g for all g ≤ G . In order to do this, we need to compute thePicard numbers of simple abelian varieties of dimension g , for every g ≤ G .5.1. Picard numbers of simple abelian varieties.
Let g ≥ X is a simple abelian variety of dimension g , its Picard number ρ = ρ ( X ) must respect somedivisibilty conditions [7, Proposition 5.5.7], namely( I ) Type I : ρ | g ;( II ) Type II : ρ ∈ N and ρ | g ;( III ) Type III : 2 ρ | g ;( IV ) Type IV : ρ | g .For a fixed dimension g , we would like to understand which ρ satisfying condition ( I ), ( II ),( III or ( IV ) above can actually appear as the Picard number of a simple abelian variety ofthe corresponding type. For the notation used in the statement and in the proof of the nextresult, we refer the reader to Section 2 and to [7, Ch. 9] (in particular Section 9.6). Proposition 5.1.
Let g be a fixed positive integer. For all positive integers ρ that satisfyone of the conditions above, there exists a simple abelian variety X of the corresponding typesuch that ρ ( X ) = ρ , unless we are in one of the five following exceptional cases: (1) F is of type III, and m := g/ e = 1 ; (2) F is of type III, m := g/ e = 2 , and there exists a totally positive element α ∈ K such that N ( T ) = α ( N being the reduced norm of M ( F ) to K ); (3) F is of type IV, P e ν =1 r ν s ν = 0 ; (4) F is of type IV, m := g/d e = 2 , d = 1 and r ν = s ν = 1 for all ν = 1 , . . . , e ; (5) F is of type IV, m := g/d e = 1 , d = 2 and r ν = s ν = 1 for all ν = 1 , . . . , e .Proof. It is a theorem of Shimura that given an endomorphism structure ( F, ′ , ι ) one has thata general member ( X, H, ι ) of the moduli space A ( M , T ) has the property End Q ( X ) = ι ( F ),except in the cases above (for example see [18], or [7] for a modern approach).In fact, under the assumption that our abelian variety X be simple, one can show thatthese cases never occur:(1) X is isogenous to a square Y , where Y is an abelian variety of dimension e , con-tradicting the fact that X is simple;(2) same argument as above;(3) X is isogenous to Y d m , where Y is an abelian variety of dimension e , thus d = m = 1,while d = 2 because F is a quaternion algebra over its center;(4) End Q ( X ) contains a totally indefinite quaternion algebra ˜ F over K with F = K ⊂ ˜ F ,so that F = K ⊂ ˜ F ⊂ End Q ( X ) = F , contradiction;(5) as in (1) and (2). or details, consult [7, Ch. 9, Ex. 9.10(1)–(5)], or see the original paper by Shimura [18].By work of Gerritzen [9] and Albert [1–3], this implies that given an involutorial divisionalgebra F of type I–IV outside of the five exceptional cases above, there exists a simpleabelian variety whose endomorphism algebra is F . For a survey on these results, see [14].We are now left to show that for any integer ρ satisfying one of the conditions (I-IV)and outside of (1)-(5), we can actually construct an involutorial division algebra F of thecorresponding type, such that there exists an abelian variety X with End Q ( X ) ∼ = F and ρ ( X ) = ρ . We will divide our analysis according to the type.5.1.1. Type I.
Let ρ be a positive integer such that ρ | g . In this case, it is enough to constructa totally real number field F of degree ρ over Q . However, given a finite abelian group G ,it is always possible to construct a totally real number field F such that Gal( F/ Q ) ∼ = G asa subfield of a suitable cyclotomic field. This implies, in particular, that we can exhibit atotally real number field of degree ρ over Q .5.1.2. Type II.
It is enough to construct a totally indefinite quaternion algebra F over atotally real number field K of degree e = [ K : Q ] over Q , such that F is a division ring,as then ρ = 3 e satisfies the required condition. We are able to exhibit such an algebra forany e | g by simply considering a quaternion algebra F whose ramification is non-empty anddisjoint from the archimedean place of K , i.e. ∅ = Ram( F ) ∩ { archimedean places of K } and Ram( F ) = ∅ . Type III.
In this situation, we aim at constructing a totally definite quaternion algebra F over a totally real number field K of degree e = [ K : Q ] over Q , such that F is in facta division ring. It is enough to consider a quaternion algebra F whose ramification locus isnon-empty (this ensures the condition of being a division ring) and fulfills the condition ∅ 6 = Ram( F ) ⊇ { archimedean places of K } . Type IV.
We are left with the case corresponding to an involutorial division algebra F of the second kind, whose center K is a CM field with maximal real subfield K . In thiscase, it is enough to consider CM fields K such that the degree e = [ K : Q ] of the maximaltotally real subfield K of K ranges among all divisors of g (i.e. we are considering the case d = 1). (cid:3) Additivity of the range of Picard numbers
Additivity.
As before we denote by R g the set of realizable Picard numbers of g -dimensional abelian varieties, i.e. R g := (cid:8) ρ | ∃ X abelian variety, dim X = g , ρ ( X ) = ρ (cid:9) . Conventionally, let us set R = { } . The main result of this section is Proposition 6.1.
For any integers g, h ≥ we have an inclusion R g + R h := { x + y | x ∈ R g , y ∈ R h } ⊂ R g + h . learly, the idea is to use the additivity of Picard numbers for products of non-isogeneousabelian varieties, as proven in Corollary 2.3. In order to be able to us this we need that forany k ∈ R g we find countably many abelian varieties of dimension g and Picard number k in different isogeny classes. The case of elliptic curves illustrates how this can be proved:Suppose E and E ′ are two elliptic curves, and let F := End Q ( E ) and End Q ( E ′ ) be theirendormorphism algebras. If E and E ′ have CM, then they are mutually isogenous if andonly if F ∼ = F ′ . However, if they don’t have CM, then F ∼ = F ′ ∼ = Q but E and E ′ arenot necessarily isogenous. However, by removing CM elliptic curves from M , , we get anuncountable set of isomorphism classes of elliptic curves. Since each isogeny class consistsof countably many (isomorphism classes of) elliptic curves, we must have infinitely manyisogeny classes of non-CM elliptic curves. We first note Proposition 6.2.
Let X and X ′ be two simple abelian varieties of dimension g , and let F and F ′ be the corresponding endomorphisms algebras. Then, if X is isogenous to X ′ , then F ∼ = F ′ .Proof. The isomorphism ι : End Q X ′ −→ End Q X is defined by sending α ψ ◦ α ◦ φ , where φ : X −→ X ′ is an isogeny, and ψ : X ′ −→ X is the unique isogeny such that ψ ◦ φ = e X and φ ◦ ψ = e Y , e being the exponent of φ (or ψ respectively). Surjectivity and injectivityof ι follow from the fact that multiplication maps are invertible in the endomorphism Q -algebra. (cid:3) The key result of this section is the
Proposition 6.3.
Given an integer k ∈ R g , then there exist at least countably many isogenyclasses of abelian varietiss of dimension g and Picard number k .Proof. Assume that k ∈ R g and that this integer is realized by an abelian variety A ofdimension g . We first assume that F := End Q ( X ) = Q . We will now go through the varioustypes of endomorphism algebras and start with type I and [ F : Q ] > F , and the Picardnumber of a simple abelian variety with such an endomorphism algebra is the degree e :=[ F : Q ] of F . One can find infinitely many totally real number fields of a fixed degree e > F over a totally realnumber field K . In this case, the Picard number is ρ = 3 e for type II and ρ = e for typeIII, where e := [ K : Q ]. Since by the previous argument there exist infinitely many totallyreal number fields of degree e , it follows that we can find infinitely many (totally definite ortotally indefinite) quaternion algebras. By considering the corresponding abelian varieties,we have shown the claim for types II-III.For type IV, the endomorphism algebra F has degree [ F : K ] = d over a CM field K , e := [ K : Q ]. If K is the totally real subfield of K , of degree e := [ K : Q ], the Picardnumber is ρ = e d . Similarly to the argument in the proof of Proposition 5.1, we can restrictourselves to consider abelian varieties whose endomorphism algebra satisfies the condition d = 1: under this assumption, ρ = e . Since there are infinitely many totally real numberfields, we find infinitely many CM fields (by just adding the imaginary unit), and the sameargument as in the previous cases shows the statement for type IV abelian varieties.This leaves us with the general situation where F = End Q ( X ) ∼ = Q . The ppav of thistype are given by removing from A g a countable union of proper Shimura varieties. Since g has positive dimension and since the set of ppav isogeneous to a given abelian variety iscountable, the claim follows. (cid:3) Proof of Proposition 6.1.
This now follows immediately from Proposition 6.3 and the addi-tivity proved in Corollary 2.3. (cid:3)
Computing R g . Our final aim is to find all realizable Picard number of abelian varietiesof a given dimension. We can use Proposition 6.1 to easily show that some of the lower oneindeed occur.
Proposition 6.4.
Given g ≥ , consider the set R g of Picard numbers of abelian varietiesof dimension g . Then, { , . . . , g } ⊂ R g .Proof. As R = { , . . . , } is complete and R ⊃ { , . . . , } , the result easily follows byinduction on g . (cid:3) Remark . In fact, it is not hard to prove that all Picard numbers ρ satisfying the inequality g ≤ ρ ≤ g are attained by products of elliptic curves.This now allows us to formulate an algorithm which computes the ranges R g inductively. • set R = { } and R = { , , , } ; • for all g in the range 3 ≤ g ≤ G , we compute R g as follows:(i) by Proposition 6.4, R g ⊃ { , . . . , g − } (in particular, all Picard numbers ofsimple abelian varieties of dimension g are in this range);(ii) compute all possible Picard numbers of self-product abelian varieties A k , wheredim A = g/k ;(iiii) for every pair ( g , g ) of positive integers such that g + g = g , compute R g + R g ;(iv) assemble everything in light of R g = [ k | g (cid:8) ρ ( A k ) | A simple, dim A = g/k } ∪ [ ≤ n ≤ g − (cid:0) R n + R g − n (cid:1) . Asymptotic behaviour of Picard numbers of abelian varieties
Asymptotic completeness of Picard numbers.
In the course of this note, we haveshown that for every g ≥ R g of Picard numbers of g -dimensional abelian varieties isnot complete, or in other words that R g < g . The ratio δ g := R g /g is the density of R g in [1 , g ] ∩ N , and it describes how many admissible Picard numbers (according to LefschetzTheorem of (1 , asymptoticdensity of Picard numbers of abelian varieties: this is the quantity defined as δ := lim g → + ∞ δ g . We now show that the Picard numbers of abelian varieties are asymptotically complete,namely that δ = 1, contrary to the fact that δ g < g ≥ Theorem 7.1 (Asymptotic completeness) . The sets of Picard numbers of abelian varietiesare asymptotically dense, i.e. δ = 1 . The proof relies on Lagrange’s four-square theorem and the following lemma, whose prooffollows readily from the additivity of the Picard number. emma 7.2. Suppose g ≥ and ≤ n ≤ g , where g and n are two integers. Assume thatthere exist positive integers n , . . . , n k such that n − n + · · · + n k and n + · · · + n k ≤ g − . Then, there exists a g -dimensional abelian variety X with ρ ( X ) = n .Proof of Proposition 7.1. Let n be the largest positive integer such that n ≤ n − < ( n + 1) . Then, 0 ≤ n − − n < ( n + 1) − n = 2 n + 1 , from which it follows that0 ≤ n − − n ≤ n ≤ √ n − < √ n ≤ g. Lagrange’s four-square theorem implies that m := n − − n = n + n + n + n , for some n , n , n , n ∈ N . We will now show that n + · · · + n < g for g ≫
0: indeed, bylooking at the power means of n , . . . , n one has that n + · · · + n ≤ r n + n + n + n √ m ≤ p g. Therefore, n + · · · + n ≤ √ n − p g, and the right-hand side is strictly smaller than g if and only if n < g + 8 g + 1 − √ g / =: b g . This implies that all Picard numbers in the range [1 , b g ) indeed occur, by virtue of the lemmaabove. Hence, we have that asymptotically R g ≥ b g − g + 8 g − √ g / , and thus δ = 1. (cid:3) Distribution of large Picard numbers.
We are interested in describing the distri-bution of large Picard numbers within [1 , g ] ∩ N . As we have already observed, for every g ≥
1, the set R g has the following structure R g = [ k | g (cid:8) ρ ( A k ) | A simple, dim A = g/k } ∪ [ ≤ n ≤ g − (cid:0) R n + R g − n (cid:1) . The proof of Theorem 1.1(1) shows that all Picard numbers of abelian varieties of di-mension g that are isogenous to a self-product of a simple abelian variety are bounded by g ( g + 1), unless we are considering the g -fold product of a CM elliptic curve, in which casethe maximal Picard number is attained.In order to begin our analysis, we need to specify what we mean by ”large Picard numbers”.First of all, we will require large Picard numbers to satisfy the inequality ρ > g ( g + 1) / R g : [ ≤ n ≤ g − (cid:0) R n + R g − n (cid:1) . ow we need a little bit of notation. Let us set R g,n := (cid:8) ( g − n ) + x | x ∈ R n (cid:9) . In other words, R g,n is the subset of R g obtained by translating R n to the right by ( g − n ) inside N . Notice that given g, k, n ∈ N , one has by Proposition 6.1 that R g,k + R n ⊂ R g + n,k + n . As we want to consider large Picard numbers only, we will be concerned only with someof the R g,s ’s, namely those for which the inequality12 g ( g + 1) ≤ ( g − s ) + 1 (1)holds (1 ≤ s ≤ g ), which implies that the abelian varieties we are considering are not self-products of simple abelian varieties. This in particular implies that g ≥ s ≥ s ≤ g − p g + g − . Finally, let us look at the mutual interaction of the R g,s ’s. For a fixed g , there might existpositive integers a and b such that R g,a ∩ R g,b = ∅ . However, if a and b are distinct and smallenough with respect to g , then R g,a ∩ R g,b = ∅ . Indeed, for a fixed g , the inequality[ g − ( s + 1)] + ( s + 1) < ( g − s ) + 1 (2)holds for all positive integers s < − √ g + 3. Hence R g,a ∩ R g,b = ∅ for a and b in thisrange.We are now able to define precisely what ”large Picard number” means. We will say thata Picard number ρ is large if ρ ∈ R g,s with s satisfying conditions (1) and (2). This conditioncan be made explicit: Proposition 7.3.
Let g ≥ . Then, ρ ∈ R g,s is a large Picard number if and only if s ≤ min n g − p g + g − , − p g + 3 o . The theorem we are going to discuss next describes the distribution of the large Picardnumbers inside [1 , g ] ∩ N . The argument we use is inductive and its initial step is a proofof an asymptotic version of Theorem 1.1(2) starting from an asymptotic version of Theorem1.1(1). Theorem 7.4 (Distribution of large Picard numbers) . For every positive integer ℓ thereexists a genus g ℓ such that for all g ≥ g ℓ large Picard numbers in R g are distributed asfollows: R g,ℓ · · · R g, R g, R g, • ( g − +1 • g . In other words, for all g ≥ g ℓ , we have that [( g − ℓ ) + 1 , g ] ∩ R g = R g,ℓ ⊔ R g,ℓ − ⊔ · · · ⊔ R g, ⊔ R g, ⊔ R g, . Remark . In particular this shows that, as g → ∞ more and more gaps arise in R g as wego down from the maximum Picard number g . roof. We will give a proof by induction. To start with, note that we can (and will) alwaysassume that g is large enough, so that there is no overlapping between the sets R g,n that wewish to consider.Let us start with a pair ( ℓ, g ℓ ) such that the following holds: there is no abelian varietyof dimension g ≥ g ℓ and Picard number ρ such that ( g − t ) + t < ρ < ( g − t + 1) + 1for 2 ≤ t ≤ ℓ or ( g − + 1 < ρ < g . Then, the claim is that we can find g ℓ +1 ≥ g ℓ such that there is no abelian variety of dimension g ≥ g ℓ +1 and Picard number ρ such that[ g − ( ℓ + 1)] + ( ℓ + 1) < ρ < ( g − ℓ ) + 1.As the start of the induction, we will now show how to recover the second part of Theorem1.1 from the first one, at least asymptotically (we will not be able to get any bound on g ,but of course it is always possible to do so). Suppose that for all g ≥ g there is no abelianvariety X of dimension g and Picard number ( g − + 1 < ρ ( X ) < g (notice that in lightof Theorem 1.1, we can choose g = 4). We will now prove that there exists g , g ≥ g ,such that for every g ≥ g there is no abelian variety Y of dimension g and Picard number( g − + 4 < ρ ( Y ) < ( g − + 1.Let g be such that for all g ≥ g conditions (1) and (2) above are satisfied (i.e. thePicard numbers ρ in the range ( g − + 4 < ρ < ( g − + 1 are large according to ourdefinition). Suppose Y is an abelian variety whose Picard number contradicts the statementwe want to prove, namely ( g − + 4 < ρ ( Y ) < ( g − + 1. Then, as ρ ( Y ) is large, Y isisogenous to a product of two abelian varieties, i.e. Y ∼ A n × A g − n , where n ≤ g − n andHom( A n , A g − n ) = 0, (here the subscripts indicate the dimension). Since ρ > ( g − + 4,we have that n = 1 necessarily. Therefore Y ∼ E × A g − , where E is an elliptic curve andHom( E, A g − ) = 0. As ρ ( Y ) = 1 + ρ ( A g − ), we readily see that( g − + 1 < ρ ( A g − ) < ( g − , a contradiction. This is first step of the induction.Now, let us assume that there exists g ℓ such that for all g ≥ g ℓ there is no abelian variety X of dimension g and Picard number in the following ranges:(1) ( g − + 1 < ρ ( X ) < g ;(2) ( g − + 4 < ρ ( X ) < ( g − + 1;...( ℓ ) ( g − ℓ ) + ℓ < ρ ( X ) < [ g − ( ℓ − + 1.We claim that there exists g ℓ +1 such that for all g ≥ g ℓ +1 there is no abelian variety Y ofdimension g and Picard number satisfying( ℓ + 1) [ g − ( ℓ + 1)] + ( ℓ + 1) < ρ ( Y ) < [ g − ℓ ] + 1.Again, let us let g grow bigger so that the Picard numbers we wish to consider can onlybe realized by abelian varieties that are not a self-product of a simple abelian variety. Bycontradition, let Y be an abelian variety that contradicts the statement we want to prove.Then Y ∼ A n × A g − n , where n ≤ g − n and Hom( A n , A g − n ) = 0. It is straightforward to seethat n ≤ ℓ , as ρ ( Y ) > [ g − ( ℓ + 1)] + ( ℓ + 1) . By additivity of the Picard number ρ ( Y ) = ρ ( A n × A g − n ) = ρ ( A n ) | {z } ρ n + ρ ( A g − n ) | {z } ρ g − n . s ρ ( Y ) > [ g − ( ℓ + 1)] + ( ℓ + 1) , we see that ρ g − n > [ g − ( ℓ + 1)] + ( ℓ + 1) − ρ n > [ g − ( ℓ + 1)] + ( ℓ + 1) − n > [( g − n ) − ( ℓ − n + 1)] + ( ℓ + 1) − n > [( g − n ) − ( ℓ − n + 1)] + ( ℓ − n + 1) . Similarly, ρ g − n < ( g − ℓ ) + 1 − ρ n ≤ ( g − ℓ ) = [( g − n ) − ( ℓ − n )] < [( g − n ) − ( ℓ − n )] + 1 , and summing up we have shown that[( g − n ) − ( ℓ − n + 1)] + ( ℓ − n + 1) < ρ ( A g − n ) < [( g − n ) − ( ℓ − n )] + 1 , which contradicts the ( ℓ − n + 1)-st condition above. (cid:3) As a striking consequence of Theorem 7.4, we get the following structure theorem forabelian varieties of large Picard number up to isogeny, which generalizes the results in Section4. As we had already noticed in Section 4, we cannot expect a structure theorem up toisomorphism, hence this is the strongest result we could hope for.
Corollary 7.6 (Structure theorem for abelian varieties of large Picard number) . For everypositive integer ℓ there exists a genus g ℓ such that for all g ≥ g ℓ the following are equivalent: (1) ρ ( X ) ∈ R g,n for some n ≤ ℓ ; (2) X ∼ E g − n × A n , where E is an elliptic curve with complex multiplication, A n is anabelian variety of dimension n , and Hom(
E, A n ) = 0 .Proof. Let us set g ℓ as in the proof of Theorem 7.4, and let X be an abelian variety of Picardnumber ρ ( X ) ∈ R g,n for some n ≤ ℓ . By means of the Poincar´e reducibility theorem, we canwrite X ∼ E t × A g − t , where E is an elliptic curve with complex multiplication, A g − t is anabelian variety of dimension g − t , Hom( E, A g − t ) = 0, and t is the largest integer appearingas exponent of an elliptic curve with complex multiplication in the isogeny decomposition of X . Let us now set for simplicity t = g − m , so that X ∼ E g − m × A m . In particular, it followsthat ρ ( X ) ∈ R g,m . However, by Theorem 4.2, R g,n cannot intersect R g,m unless n = m , fromwhich the statement follows. (cid:3) Abelian varieties defined over number fields
In this section we show that every realizable Picard number ρ ∈ R g can be obtained byan abelian variety defined over a number field. Theorem 8.1.
Let ( X, λ ) be a polarized complex abelian variety, let D = End ( X ) be theendomorphism algebra of X and let ∗ be the Rosati involution on D . Then there exists apolarized abelian variety over Q , or equivalently over a number field, which has the sameendomorphism algebra with involution ( D, ∗ ) .Proof (Ben Moonen). To prove the assertion, choose a Q -subalgebra R ⊂ C of finite typeand a polarized abelian scheme ( Y, µ ) over S := Spec( R ) such that ( Y, µ ) ⊗ R C is isomorphicto ( X, λ ) and such that all endomorphisms of X are defined over R , in the sense that thenatural map End ( Y /R ) −→ End ( X ) s an isomorphism. The existence of such a model follows from [10, Proposition (8.9.1)]together with the fact that End( X ) is a finitely generated algebra (in fact, it is even finitelygenerated as an abelian group). By construction, if η is the generic point of S , we haveEnd ( Y η ) ∼ = D as algebras with involution. If s is a point of S , we have a specializationhomomorphism i s : End ( Y η ) ֒ → End ( Y s ), and we are done if we can find a closed point s for which i s is an isomorphism.Let ℓ be a prime number. For s a point of S , let T ℓ ( s ) := T ℓ ( Y s ) denote the ℓ -adic Tatemodule of Y s , and let ρ s : Gal (cid:0) κ ( s ) /κ ( s ) (cid:1) → GL( V ℓ ( s ))denote the Galois representation on V ℓ ( s ) = T ℓ ( s ) ⊗ Z Q ℓ . By a result of Faltings [8, Theorem1], End ( Y s ) ⊗ Q ℓ is the endomorphism algebra of V ℓ ( s ) as a Galois representation, namelyEnd ( Y s ) ⊗ Q ℓ ∼ = End Gal( κ ( s ) /κ ( s )) (cid:0) V ℓ ( s ) (cid:1) . For s ∈ S , the image of ρ s may be identified with a subgroup of im( ρ η ); the subgroupwe obtain is independent of choices only up to conjugacy. By a result of Serre [17] (seealso [13, Proposition 1.3]), there exist closed points s ∈ S for which im( ρ s ) = im( ρ η ), andfor all such points the specialization map i s on endomorphism algebras is an isomorphism(see [13, Corollary 1.5]). (cid:3) As the Picard number only depends on ( D, ∗ ) this immediately implies Corollary 8.2.
Every realizabe Picard number ρ ∈ R g can be obtained by an abelian varietydefined over a number field. References [1] A. Adrian Albert. On the construction of Riemann matrices. I.
Ann. of Math. (2) , 35(1):1–28, 1934.[2] A. Adrian Albert. A solution of the principal problem in the theory of Riemann matrices.
Ann. of Math.(2) , 35(3):500–515, 1934.[3] A. Adrian Albert. On the construction of Riemann matrices. II.
Ann. of Math. (2) , 36(2):376–394, 1935.[4] A. Beauville. Vari´et´es K¨ahleriennes dont la premi`ere classe de Chern est nulle.
J. Differential Geom. ,18(4):755–782 (1984), 1983.[5] Arnaud Beauville. Some surfaces with maximal Picard number.
J. ´Ec. polytech. Math. , 1:101–116, 2014.[6] C. Birkenhake and H. Lange.
Complex tori , volume 177 of
Progress in Mathematics . Birkh¨auser Boston,Inc., Boston, MA, 1999.[7] C. Birkenhake and H. Lange.
Complex abelian varieties , volume 302 of
Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, secondedition, 2004.[8] Gerd Faltings. Complements to Mordell. In
Rational points (Bonn, 1983/1984) , Aspects Math., E6,pages 203–227. Vieweg, Braunschweig, 1984.[9] L. Gerritzen. On multiplication algebras of Riemann matrices.
Math Ann , 194:109–122, 1971.[10] A. Grothendieck. ´El´ements de g´eom´etrie alg´ebrique. IV: ´Etude locale des sch´emas et des morphismesde sch´emas. III.
Inst. Hautes ´Etudes Sci. Publ. Math. , (28):255, 1966.[11] D. Huybrechts. Compact hyper-K¨ahler manifolds: basic results.
Invent. Math. , 135(1):63–113, 1999.[12] V. Kumar Murty. Exceptional Hodge classes on certain abelian varieties.
Math. Ann. , 268(2):197–206,1984.[13] Rutger Noot. Abelian varieties—Galois representation and properties of ordinary reduction.
CompositioMath. , 97(1-2):161–171, 1995. Special issue in honour of Frans Oort.[14] Frans Oort. Endomorphism algebras of abelian varieties. In
Algebraic geometry and commutative algebra,Vol. II , pages 469–502. Kinokuniya, Tokyo, 1988.
15] M. Sch¨utt. Quintic surfaces with maximum and other Picard numbers.
J. Math. Soc. Japan , 63(4):1187–1201, 2011.[16] M. Sch¨utt. Picard numbers of quintic surfaces.
Proc. Lond. Math. Soc. (3) , 110(2):428–476, 2015.[17] J.-P. Serre. Lettre ´a Ken Ribet du 1/1/1981. In
Euvres , volume IV.[18] Goro Shimura. On analytic families of polarized abelian varieties and automorphic functions.
Ann. ofMath. (2) , 78:149–192, 1963.[19] T. Shioda and H. Inose. On singular K Complex analysis and algebraic geometry , pages119–136. Iwanami Shoten, Tokyo, 1977.[20] T. Shioda and N. Mitani. Singular abelian surfaces and binary quadratic forms. In
Classification ofalgebraic varieties and compact complex manifolds , pages 259–287. Lecture Notes in Math., Vol. 412.Springer, Berlin, 1974.[21] Tetsuji Shioda. Algebraic cycles on certain K p . pages 357–364, 1975. Klaus HulekInstitut f¨ur Algebraische Geometrie, Leibniz Universit¨at Hannover, Welfengarten 1 30167Hannover (Germany)
E-mail address : [email protected] Roberto LafaceTechnische Universit¨at M¨unchen, Zentrum Mathematik - M11, Boltzmannstraße 3, D-85478Garching bei M¨unchen (Germany)
E-mail address : [email protected]@ma.tum.de