Submitted to Journal of the Mathematical Society of Japan An algebraic construction of an abelian variety with a given Weil number By Ching-Li Chai 1 & Frans Oort 2 Abstract. A classical theorem of Honda and Tate asserts that for every Weil q-number π, there exists an abelian variety over the finite field F q, unique up to F q-isogeny, whose q-frobenius is π. The standard proof (of the existence part in the Honda-Tate theory) uses the the fact that for a given CM field L and a given CM type Φ for L, there exists a CM abelian variety with CM type (L, Φ) over a field of characteristic 0. The usual proof of the last statement uses complex uniformization of (the set of C-points of) abelian varieties over C. In this short note we provide an algebraic proof of the existence of a CM abelian variety over an integral domain of characteristic 0 with a given CM type, resulting in an algebraic proof of the existence part of the Honda-Tate theorem which does not use complex uniformization. Dedicated to the memory of Taira Honda. Introduction. In this note we provide a new proof for a theorem already known for 45 years. We fix a prime number p and we consider q = p n with n Z >0. We say that π is a Weil q-number if π is an algebraic integer such that for every complex embedding ψ : Q(π) C we have ψ(π) = q. A result by A. Weil (which was the starting point of new developments in arithmetic algebraic geometry) states that for any abelian variety A over the finite field F q its associated q-frobenius morphism π A = Fr A,q : A A (q) = A 2010 Mathematics Subject Classification. Primary 11G10; Secondary 11G15, 14G15. Key Words and Phrases. abelian variety, complex multiplication, CM field, Weil number, CM lifting. 1 Partially supported by NSF grant DMS-1200271 2 The Institute of Mathematics of Academia Sinica is gratefully acknowledged for excellent working conditions and for hospitality during a visit of the second-named author in November / December 2012.
2 Ching-Li Chai & Frans Oort is a Weil q-number, in the sense that π A is a root of a monic irreducible polynomial in Z[T ] all of whose roots are Weil q-numbers. See [20, p. 70], [19, p. 138] and [10, Th. 4, p. 206]. This enabled T. Honda and J. Tate to state their theorem: the map A π A defines a bijection {simple abelian variety over F q }/ {Weil q number}/ from the set of isogeny classes of simple abelian varieties over F q to the set of Weil q-numbers up to conjugacy. Here we say that π and π are conjugate if there exists an isomorphism Q(π ) = Q(π ) of fields which carries π into π. This map is well-defined because of the above theorem of Weil, and because isogenous abelian varieties have conjugate Frobenius endomorphisms. The injectivity was proved by Tate, and the surjectivity was proved by Honda and Tate. The surjectivity means: Theorem I. For any Weil q-number π there exists a simple abelian variety A over F q (unique up to F q -isogeny) such that π is conjugate to π A. See [6], Main Theorem; [18, Th. 1]. In the terminology of [18] we say that a Weil q-number is effective if it is conjugate to the q-frobenius of an abelian variety over F q. Theorem I asserts that every Weil number is effective. In the course of the proof of Theorem I we will show, in Theorem II in Step 5, that every CM type for a CM field L is realized by an abelian variety in characteristic zero of dimension [L : Q]/2 with complex multiplication by L. Proofs of these theorems were given by constructing a CM abelian variety over C (using complex uniformization and GAGA) with properties which ensure that the reduction modulo p of this CM abelian variety gives a Weil number which is a power of π A. We give a proof of both theorems without using complex uniformization. The remark in Step 8 gives this proof in the special case when g = 1; that proof is a guideline for the proof below for arbitrary g. In a sense, this answers a question posed in [14, 22,4]. (1) Step 1. A Weil q-number π has exactly one of the following three properties: (Q) It can happen that ψ(π) Q. In this case q = p n = p 2m and π = ± q = ±p m.
Abelian variety with a given Weil number 3 (R) It can happen that ψ(π) Q and ψ(π) R. In this case q = p n = p 2m+1 and π = ± q = ±p m p. In this case every embedding of Q(π) into C lands into R. ( R) If there is one embedding ψ : Q(π) C such that ψ (π) R then for every embedding ψ : Q(π) C we have ψ(π) R and in this case Q(π) is a CM field. The fact that L is a CM field means there is a subfield L 0 L such that L 0 is totally real (every embedding of L 0 into C lands into R) and [L : L 0 ] = 2 and L is totally complex (no embedding of L into C lands into R). We know from [18], page 97 Example (a) that every real Weil q-number comes from an abelian variety over F q, so the first two cases have been taken care of. In order to prove Theorem I we assume from now on that we are in the third case, i.e. ψ(π) R for all ψ : Q(π) C. We show in that case the properties of Q(π) mentioned above indeed hold. As ψ (π) = ψ (π) ψ (π) = q for some embedding ψ : Q(π) C we see that ψ (π) = q ( ψ (π) = q ) ψ. π As q/π π (because ψ (π) R) we see that for every embedding ψ the complex conjugate of ψ(π) equals ψ(q/π) which is different from ψ(π); hence π is totally complex. Moreover we see that β := π + q π is totally real and 2 q < ψ(β) < 2 q. for all ψ : Q(π) C. We note that π is a zero of the quadratic polynomial T 2 β T + q, and β 2 4q is totally negative. We have shown that Q(π) is a CM field. In [18], Th. 1 on page 96, we see a description of a finite dimensional central division algebra M over Q(π) determined by (p-adic) properties of π (denoted by E in [18]); these properties will be used below. We write 2g = [Q(π) : Q] [M : Q(π)]. If ever we would find an abelian variety A over F q with π A π then we would have
4 Ching-Li Chai & Frans Oort End 0 (A) = M and dim(a) = g. By [18], 3, Lemme 2 on page 100 we see that there exists a CM field L with Q(π) L M and [L : Q] = 2g. We fix notation q, and π a totally complex Weil q-number, and a finite dimensional central division algebra M over Q(π) determined by π and Q(π) L M with [L : Q] = 2g as above; Q(π) L M. We write L 0 for the maximal totally real subfield of L. (2) Step 2. Choosing a CM type for L. We follow [18], pp. 103-105. However our notation will be slightly different. A prime above p in Q(π) will be denoted by u. A prime in L 0 above p will be denoted by w and a prime in L above p will be denoted by v. We write ρ for the involution of the quadratic extension L/L 0 (which is the complex conjugation). Following Tate we write H v = Hom(L v, C p ), Hom(L, C p ) = v p H v, where C p is the p-adic completion of an algebraic closure of Q p. Let n v := v(π) v(q)) #(H v) N for each place v of L above p. Using properties of π we choose a suitable p-adic CM type for L, by choosing a subset v w Φ v v w H v for each place w of L 0 above p, as follows. [v = ρ(v)] For any v with v = ρ(v) the map ρ gives a fixed point free involution on H v ; in this case (once π and L are fixed and v is chosen) we choose a subset Φ v H v with #(Φ v ) = (1/2) #(H v ). Note that v(π) = (1/2)v(q) in this case and we have n v = (1/2) #(H v ) = (v(π)/v(q)) #(H v ). [v ρ(v)] For any pair v 1, v 2 above a place w of L 0 dividing p with v 1
Abelian variety with a given Weil number 5 ρ(v 1 ) = v 2, the complex conjugation ρ defines a bijective map? ρ : H v1 H v2. We choose a subset Φ v1 H v1 with #(Φ v1 ) = n v1 and we define Φ v2 := H v2 Φ v1 ρ. Observe that indeed n vi + n ρ(vi) = [L v : Q p ] = #(H vi ) for i = 1, 2. We could as well have chosen first Φ v2 of the right size and then define Φ v1 as Φ v1 := H v1 Φ v2 ρ. Define a p-adic CM type Φ p Hom(L, C p ) = v p H v by Φ p := v p Φ v. By construction we have Φ p (Φ p ρ) =, Φ p (Φ p ρ) = Hom(L, C p ); i.e. Φ p is a p-adic CM type for the CM field L. Let j p : Q C p be the algebraic closure of Q in C p. The injection j p induces a bijection j p?: Hom(L, Q) Hom(L, C p ). The subset Φ := (j p?) 1 (Φ p ) Hom(L, Q) is a CM type in the usual sense, i.e. Φ (Φ ρ) = and Φ (Φ ρ) = Hom(L, Q). We fix the notation Φ p Hom(L, C p ) for the p-adic CM type constructed above, and the corresponding CM type Φ Hom(L, Q). (3) Step 3. Choosing a prime number r. Proposition A. For a given CM field L there exists a rational prime number r unramified in L such that r splits completely in L 0 and every place of L 0 above r is inert in L/L 0. Proof. Let N be the smallest Galois extension of Q containing L, and let G = Gal(N/Q). Note that the element ρ G induced by complex conjugation is a central element of order 2. By Chebotarev s theorem the set of rational primes unramified in N whose Frobenius conjugacy class in G is ρ has Dirichlet density 1/[G : 1] > 0; see [8, VIII.4, Th. 10]. Any prime number r in this subset satisfies the required properties. (4) Step 4. An abelian variety with an action by L.
6 Ching-Li Chai & Frans Oort We know that for every prime number (r in our case) there exists a supersingular elliptic curve E in characteristic r; e.g. for r > 2 we can consider the Legendre family Y 2 = X(X 1)(X λ) and we know from [4, 4.4.2] that there do exist values of λ in F r for which the corresponding elliptic curves over F r are supersingular; in characteristic 2 we can take the elliptic curve given by Y 2 + Y = X 3. Let E be a supersingular elliptic curve over the base field κ := F r ; we know that End(E) is non-commutative. Its endomorphism algebra End 0 (E) is the quaternion division algebra Q r, over Q in the notation of [2], which is ramified exactly at r and. Let B 1 := E g and let D := End 0 (B 1 ) = M g (Q r, ). Proposition B. Let L be a totally imaginary quadratic extension of a totally real number field L 0 such [L v : Q r ] is even for every place v of L above r. Let g = [L 0 : Q]. There exists a positive involution τ on the central simple algebra End Q (L 0) Q Q r, = Mg (Q r, ) over Q and a ring homomorphism ι: E End Q (L 0) Q Q r, such that ι(l ) is stable under the involution τ and τ induces the complex conjugation on L. Proof. Let End Q (L 0) = M g (Q) be the algebra of all endomorphisms of the Q- vector space underlying L 0. The trace form (x, y) Tr L 0 /Q(x y) for x, y L 0 is a positive definite quadratic form on (the Q-vector space underlying) L 0, so its associated involution τ 1 on End Q (L 0) is positive. Multiplication defines a natural embedding L 0 End Q (L 0), and every element of L 0 is fixed by τ 1. Let τ 2 be the canonical involution on Q r,. The involution τ 1 τ 2 on End Q (L 0) Q Q r, is clearly positive because τ 2 is. It is also clear that the subalgebra B := L 0 Q Q r, of End Q (L 0) Q Q r, is stable under τ. Moreover B is a positive definite quaternion division algebra over L 0, so the restriction to B of the positive involution τ is the canonical involution on B. The assumptions on L imply that there exists an L 0-linear embedding L B. From the elementary fact that every R-linear embedding of C in the Hamiltonian quaternions H is stable under the canonical involution on H, we deduce that the subalgebra L Q R B Q R is stable under the canonical involution of B Q R, which implies that L is stable under τ. Corollary C. (i) There exists a polarization µ 1 : B 1 B t 1 and an embedding L End 0 (B 1 ) = D such that the image of L in D = End 0 (B 1 ) is stable under the Rosati involution attached to µ 1. (ii) There exists an isogeny α : B 1 B 0 over F r such that the embedding L End 0 (B 1 ) = End 0 (B 0 ) factors through an action ι 0 : O L End(B 0 )
Abelian variety with a given Weil number 7 of O L on B 0, where O L is the ring of all algebraic integers in L. (iii) There exists a positive integer m such that the isogeny µ 0 := m (α t ) 1 µ 1 α 1 : B 0 B t 0 is a polarization on B 0 and the Rosati involution τ µ0 complex conjugation on the image of L in End 0 (B 0 ). attached to µ 0 induces the Proof. The statement (i) follows from Proposition B in view of the general structure of the Néron-Severi group NS(B) of B and the ample cone in NS(B) explained in [10, 21 pp. 208 210]. The statements (ii) and (iii) follow from (i). From now on we fix (L, Φ) as in Step 1, with r as in Proposition A, and (B 0, ι 0 : O L End(B 0 ), µ 0 : B 0 B t 0) as in Corollary C. We fix an algebraic closure Q r of Q r, an embedding j r : Q Q r, and an embedding i r,ur : W (F r )[1/p] Q r. We have bijections j p? Hom(L, C p ) Hom(L, Q) j r? i r? Hom(L, Q r ) Hom(L, W (F r )[1/r]) The last arrow Hom(L, Q r ) i r? Hom(L, W (F r )[1/r]) is a bijection because r is unramified in L. We regard the p-adic CM type Φ p as an r-adic CM type Φ r Hom(L, W (F r )[1/r]) via the bijection (j r?) (j p?) 1, i.e. Φ r := (j r?) (j p?) 1 (Φ p ) = (j r?)(φ). For each place w of L 0 above r, the w-adic completion L w := L L0 L 0,w of L is an unramified quadratic extension field of the w-adic completion L 0,w = Qr of L 0, and the intersection Φ w := Φ r Hom(L w, W (F r )[1/r]) is a singleton. (5) Step 5. Lifting to a CM abelian variety in characteristic zero. Theorem II. Let (B 0, ι 0 : O L End(B), µ 0 : B 0 B0) t be an ([L : Q]/2)- dimensional polarized supersingular abelian variety with an action by O L such that the subring O L End 0 (B 0 ) is stable under the Rosati involution τ µ0 as in Corollary C. There exists a lifting (B, ι, µ) of the triple (B, ι 0, µ 0 ) to the ring W (F r ) of
8 Ching-Li Chai & Frans Oort r-adic Witt vectors with entries in F r such that the generic fiber B η is an abelian variety whose r-adic CM type is equal to Φ r. Proof. The prime number r was chosen so that for every place w of the totally real subfield L 0 L, the ring of local integers O L0,w of the w-adic completion of L 0 is Z p, and O L,w := O L OL0 O L0,w = W (F r 2). We have a product decomposition O L Z Z p = O L OL0 O L0,w = O L,w, w where w runs over the g places of L 0 above r. The g idempotents associated to the above decomposition of O L Z Z p define a decomposition w B 0 [r ] = w B 0 [w ] of the r-divisible group B 0 [r ] into a product of g factors, where each factor B 0 [w r ] is a height 2 r-divisible group with an action by O w. Similarly we have a decomposition B t 0[r ] = w B t 0[w ] of the r-divisible group attached to the dual B t 0 of B 0. The action of O L on B 0 induces an action of O L on B t 0 by y (ι 0 (ρ(y))) t for every y O L, so that the polarization µ 0 : B 0 B t 0 is O L -linear. The quasi-polarization µ 0 [r ] : B 0 [r ] B t 0[r ] decomposes into a product of quasi-polarizations µ 0 [w ] : B 0 [w ] B t 0[r ] on the O L,w -linear r-divisible groups B 0 [w ] of height 2. It suffices to show that for each place w of L 0 above r, the O L,w -linearly polarized r-divisible group (B 0 [w ], ι 0 [w ], µ 0 [w ]) over F r can be lifted to W (F r ) with r-adic CM type Φ w. For then the Serre-Tate theorem for deformation of abelian schemes tells us that (B 0, ι 0, µ 0 ) can be lifted over W (F r ) to a formal abelian scheme B with an action by O L and an O L -linear polarization whose r-adic CM type is Φ r ; see [7], [9, Ch. V, Th. 2.3] on page 166. Grothendieck s algebraization theorem implies that B comes from a unique abelian scheme B over W (F r ), see [3, III 1.5.4]. For any r-adic place w among the g places of L 0 above r, the existence of a CM lifting to W (F r ) of the O L,w -linear polarized r-divisible group (B 0 [w ], ι 0 [w ], µ 0 [w ]) of height 2 goes back to Deuring who proved that a supersingular elliptic curve with a given endomorphism can be lifted to characteristic zero, see [2, p. 259], and the proof on pp. 259-263; the case we need here is
Abelian variety with a given Weil number 9 [12, 14.7]. Below is a proof using Lubin-Tate formal groups. By [11, Th. 1], there exists a one-dimensional formal group X of height 2, over W (F r ) plus an action β : O L,w End(X) of O L,w on X whose r-adic CM type is Φ w. Let (X 0, β 0 : O L,w End(X 0 )) := (X, β) Spec(W (Fr)) Spec(F r) be the closed fiber of (X, β). It is well-known that (X 0, β 0 ) is isomorphic to (B 0 [w ], ι 0 [w ]); we choose and fix such an isomorphism and identify (B 0 [w ], ι 0 [w ]) with (X 0, β 0 ). The Serre dual X t of X, with the O L,w -action defined by γ : b (β(ρ(b))) t b O L,w, also has CM type Φ w. Let (X0, t γ 0 ) be the closed fiber of (X t, γ). The natural map ξ : Hom ( (X, β), (X t, γ) ) Hom ( (X 0, β 0 ), (X t 0, γ 0 ) ) defined by reduction modulo r is a bijection: [11, Th. 1] implies that (X t, γ) is isomorphic to (X, β), and after identifying them via a chosen isomorphism both the source and the target of ξ are isomorphic to O L,w so that ξ is an O L,w -linear isomorphism. Under our identification of (X 0, β 0 ) with (B 0 [w ], ι 0 [w ]) above, the quasipolarization µ 0 [w ] on B 0 [w ] is identified with a quasi-polarization ν 0 on X 0. The quasi-polarization ν 0 : X 0 X0 t extends over W (κ L,w ) to a quasipolarization ν : X X t because ξ is a bijection. We have shown that the triple (B 0 [w ], ι 0 [w ], µ 0 [w ]) can be lifted over W (F r ). Remark. One can also prove the existence of a lifting of (B 0 [w ], ι 0 [w ], µ 0 [w ]) to W (F r ) using the Grothendieck-Messing deformation theory for abelian schemes. The point is that the deformation functor for (B 0 [w ], ι 0 [w ]) is represented by Spf(W (F r )) because O L,w is unramified over Z p. We fix the generic fiber (B η, µ, ι) of a lifting as in Theorem II over the fraction field W (F r )[1/r] of W (F r ) with an O L -linear action ι : O L End(B η ), whose r-adic CM type is Φ r (6) Step 6. We change to a number field and we reduce modulo p. We have arrived at a situation where we have an abelian variety B η over a field of characteristic zero with an action O L End(B η ) by O L, whose r-adic CM type with respect to an embedding of the base field in Q r is equal to the r-adic CM type Φ r constructed at the end of Step 4. We know that any CM abelian variety in characteristic 0 can be defined over a number field K, see e.g. [16, Prop. 26, p. 109] or [1, Prop. 1.5.4.1]. By [15, Th. 6]
10 Ching-Li Chai & Frans Oort we may assume, after passing to a suitable finite extension of K, that this CM abelian variety has good reduction at every place of K above p. Again we may pass to a finite extension of K, if necessary, to ensure that K has a place with residue field δ of characteristic p with F q δ. We have arrive at the following situation. We have a CM abelian variety (C, L End 0 (C)) of dimension g = [L : Q]/2 over a number field K, of p-adic CM type Φ p with respect to an embedding K C p such that C has good reduction C 0 at a p-adic place of K induced by the embedding K C p and the residue class field of that place contains F q (7) Step 7. Some power of π is effective. Let i Z >0 such that δ = F q i. We have C 0 over δ and π i, π C0 L. We know that π i and π C0 are units at all places of L not dividing p. We know that these two algebraic numbers have the same absolute value under every embedding into C. By the construction of Φ in Step 2 and by [18], Lemme 5 on page 103, we know that π i and π C0 have the same valuation at every place above p. As remarked in [18, pp. 103/104], the essence of this step is the factorization of a Frobenius endomorphism into a product of prime ideals in [16]. This shows that π i /π C0 is a unit locally everywhere and has absolute value equal to one at all infinite places. This implies, by standard finiteness properties for algebraic number fields, that π i /π C0 is a root of unity in O L. See for instance [5, 34 Hilfsatz a)]; the main point is that the set {y O L ψ(y) 1 complex embedding ψ : L C} is finite because the image of O L in ν L ν is a lattice, where the index ν runs through all archimedean places of L, hence π i /π C0 = y O L is a root of unity. We conclude that there exists a positive integer j Z >0 such that π ij = (π C0 ) j. (8) Step 8. End of the proof. The previous step shows that π ij is effective, because it is (conjugate to) the q ij - Frobenius of the base change of C 0 to F q ij. By [18, Lemma 1, p. 100] this implies that π is effective, and this ends the proof of the theorem in the introduction.
Abelian variety with a given Weil number 11 Remark. When g = 1 the proof of Theorem I is easier. This simple proof, sketched below, was the starting point of this note. Suppose that π is a Weil q-number and L = Q(π) is an imaginary quadratic field such that the positive integer g defined by p-adic properties of π is equal to 1. This means (the first case) either that there is an i Z >0 with π i Q, or (the second case) that for every i we have L = Q(π i ), with p split in L/Q and at one place v above p in L we have v(π)/v(q) = 1 while at the other place v above p we have v (π)/v (q) = 0. If π i Q we know that π is the q-frobenius of a supersingular elliptic curve over F q, see Step 1, and π is effective. If the second case occurs, we choose a prime number r which is inert in L/Q, then choose a supersingular elliptic curve in characteristic r, lift it to characteristic zero together with an action of (an order in) L; the reduction modulo p (over some extension of F p ) gives an elliptic curves whose Frobenius is a power of π; by [18, Lemme 1] on page 100 we conclude π is effective. This scheme of the proof of the general case is the same as the simple proof described in the previous paragraph when g = 1, except that (as we do in steps 2, 4 and 5) we specify the CM type in order to keep control of the p-adic properties of the Weil number eventually constructed. Note that the CM lifting problem considered in Step 5 reduces to the g = 1 case by the Serre-Tate theorem and by the Grothendieck algebraization result (as we explained in the proof of Theorem II). References [1] C.-L. Chai, B. Conrad & F. Oort CM liftings. [To appear] Available from www.math.upenn.edu/~chai/papers.html [2] M. Deuring Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hamburg, 14 (1941), 197 272. [3] A. Grothendieck & J. Dieudonné Élements de géométrie algébriue, III1. Étude cohomologique des faisceaux cohérents. Publ. Math. IHES 11, 1961. [4] R. Hartshorne Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer- Verlag, New York-Heidelberg, 1977. [5] E. Hecke Vorlesungen über die Theorie der algebraischen Zahlen. Akademische Verlagsgesellschaft, Leipzig, 1923. [6] T. Honda Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan, 20 (1968), 83 95. [7] N. Katz Appendix to Exposé V: Cristaux ordinaires et coordonnées canoniques. Algebraic surfaces (Orsay, 1976 78), Lecture Notes in Mathematics, vol. 868, Springer, Berlin, 1981, Appendix to an article of P. Deligne, pp. 127 137. [8] S. Lang Algebraic number theory. Addison-Wesley Publishing Co., Inc., Reading, Mass.- London-Don Mills, Ont. 1970. [9] W. Messing The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Mathematics, vol. 264, Springer-Verlag, Berlin, 1972. [10] D. Mumford Abelian varieties. Tata Institute of Fundamental Research Studies in Math. 5, Oxford University Press, London, 1970.
12 Ching-Li Chai & Frans Oort [11] J. Lubin & J. Tate Formal complex multiplication Ann. of Math. 81 (1965), 380 387. [12] F. Oort Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Part 2, pp. 165 195, Amer. Math. Soc., Providence, RI, 1987. [13] F. Oort Moduli of abelian varieties in mixed and in positive characteristic. The Handbook of Moduli (G. Farkas, I, Morrison, editors), Vol. III, pp. 75 134. [To appear in 2012.] [14] F. Oort Abelian varieties over finite fields. Summer School on Varieties over finite fields, Göttingen, 25-VI 6-VII-2007. Higher-dimensional geometry over finite fields. Proceedings of the NATO Advanced Study Institute 2007 (Editors: Dmitry Kaledin, Yuri Tschinkel). IOS Press, 2008, 123 188. [15] J-P. Serre & J. Tate Good reduction of abelian varieties. Ann. of Math. 88 (1968). 492 517. [16] G. Shimura & Y. Taniyama Complex multiplication of abelian varieties and its applications to number theory. Publications of the Mathematical Society of Japan, Vol. 6. The Mathematical Society of Japan, Tokyo 1961. [17] J. Tate Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), 134 144. [18] J. Tate Class d isogenie des variétés abéliennes sur un corps fini (d après T. Honda). Séminaire Bourbaki, 1968/69, no. 352. Lecture Notes Math. 179, Springer-Verlag, 1971, 95 110. [19] W. Waterhouse & J. Milne Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), 53 64. [20] A. Weil Sur les courbes algébriques et les variétés qui s en déduisent. Act. Sci. Ind., no. 1041, Publ. Inst. Math. Univ. Strasbourg 7 (1945). Hermann et Cie., Paris, 1948.
Abelian variety with a given Weil number 13 Ching-Li Chai Institute of Mathematics, Academia Sinica 6F Astronomy-Mathematics Building No. 1, Sec. 4, Roosevelt Rd. Taipei 10617, Taiwan (R.O.C.) chai@ math.sinica.edu.tw and Department of Mathematics University of Pennsylvania 209 S. 33rd St Philadelphia, PA 19104, U.S.A. chai@ math.upenn.edu Frans Oort Mathematisch Instituut Utrecht University P.O. Box. 80.010 NL - 3508 TA Utrecht The Netherlands f.oort@ uu.nl