Abelian varieties over finite fields

Frans Oort & Aise Johan de Jong

IX-XII/2008

Seminar at Columbia University, September — December 2008

In this seminar we will prove one theorem:

Theorem [HT] (T. Honda and J. Tate). Fix a finite field K = 𝔽_{q}. The assignment A ↦ π_{A} induces a bijection

W : {simple abelian variety over K}/ ∼_{K}= M(K, s) ⟶^{∼} W (q), A ↦ π_{A}

from the set of K-isogeny classes of K-simple abelian varieties defined over K and the set W (q) of conjugacy classes of Weil q-numbers.

An amazing theorem: on the left hand side we find geometric objects, usually difficult to construct explicitly; on the right hand side we find algebraic objects, easy to construct.

Most material we need can be found in the following basic references. You can find links to all of these documents Johan de Jong’s webpage http://math.columbia.edu/∼dejong at Columbia University. Also see Frans Oort’s webpage http://math.columbia.edu/∼oort.

[73] J. Tate – Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), 134-144.

[74] J. Tate – Classes d’isogénies de variétés abéliennes sur un corps fini (d’àpres T. Honda). Sém. Bourbaki 21 (1968/69), Exp. 352.

[75] 2005-05 VIGRE number theory working group. Organized by Brian Conrad and Chris Skinner.

[60] F. Oort – Abelian varieties over finite fields. Summer School on varieties over finite fields, Göttingen 2007. To appear. Higher-dimensional geometry over finite fields, Advanced Study Institute 2007 Proceedings ( Editors: Y. Tschinkel and D. Kaledin).

Instead of following the seminar, it might be more useful (?) to read the fascinating paper [74]: just 14 pages, sufficient for understanding a proof of this theorem. In the seminar we will basically follow this paper. For more references, for an introduction to this topic and to various methods used you can consult [60].

In the seminar several concepts, definitions, results and proofs will be explained. However we will assume known certain basic concepts; these are surveyed in an appendix. In case you feel you are not enough prepared for following the seminar, in case some of the basic concepts are not familiar to you, please let us know. We can either give more references, or have talks on such a topic, or we can explain things to you in private. Do not hesitate to ask for details.

In every talk in the seminar prerequisites needed in that talk should be stated, explained and discussed. Please indicate clearly of which statement you give a proof, and which statement you use as a black box.

You are welcome to contact us while preparing a talk.

Some notation. In definitions and proofs below we need various fields, in various disguises. We use K, L, M, P, k, 𝔽_{q}, \overline{𝔽_{p}} = 𝔽, ℙ, m.

We write K for an arbitrary field, usually the base field, in some cases of arbitrary characteristic, however most of the times a finite field. We write k for an algebraically closed field. We write g for the dimension of an abelian variety, unless otherwise stated. We write p for a prime number. We write ℓ for a prime number, which usually is different from the characteristic of the base field, respectively invertible in the sheaf of local rings of the base scheme. We write 𝔽 = \overline{𝔽_{p}}. We use the notation M for a field, sometimes a field of definition for an abelian variety in characteristic zero.

We will use L as notation for a field, usually the center of an endomorphism algebra; we will see that in our cases this will be a totally real field or a CM-field. We write P for a CM-field, usually of degree 2g over ℚ. We write ℙ for a prime field: either ℙ = ℚ or ℙ = 𝔽_{p}.

A discrete valuation on a base field usually will be denoted by v, whereas a discrete valuation on a CM-field usually will be denoted by w. If w divides p, the normalization chosen will be given by w(p) = 1.

For a field M we denote by Σ_{M} the set of discrete valuations (finite places) of M. If moreover M is of characteristic zero, we denote by Σ_{M}^{(p)} the set of discrete valuations with residue characteristic equal to p.

We write lim_{←i} for the notion of “projective limit” or “inverse limit”.

We write colim_{i→} for the notion of “inductive limit” or “direct limit”.

Introduction

Here is a sketch of the main lines in the proof (for definitions of the various concepts, see below or consult references).

The basic idea starts with a theorem by A. Weil, a proof for the Weil conjecture for an abelian variety A over simple a finite field K = 𝔽_{q} with q = p^{n}, see (3.3):

the geometric Frobenius π_{A} of A/K is an algebraic integer which for every embedding ψ : ℚ(π_{A}) → ℂ has absolute value | ψ(π_{A}) |= \sqrt{q}.

ONE (Weil) For a simple abelian variety A over a finite field K = 𝔽_{q} the Weil conjecture implies that π_{A} is a Weil q-number, see Theorem (3.3). Hence the map

{simple abelian variety over K} ⟶ W (K), A ↦ π_{A}

is well-defined.

TWO (Tate) For simple abelian varieties A, B defined over a finite field we have:

A ∼ B ⟺ π_{A} ∼ π_{B}.

See (5.2). Note that A ∼ B only makes sense if A and B are defined over the same field. Note that π_{A} ∼ π_{B} implies that A and B are defined over the same finite field. This shows that the map W : M(𝔽_{q}, s) → W (q) is well-defined and injective. See Theorem (5.2).

THREE (Honda) Suppose given π ∈ W (q). There exists a finite extension K = 𝔽_{q} ⊂ K′ := 𝔽_{q^{N}} and an abelian variety B′ over K′ with π^{N} = π_{B′}.

See [29], Theorem 1. This step says that for every Weil q-number there exists N ∈ ℤ_{>0} such that π^{N} is effective. See (10.1).

FOUR (Tate) If π ∈ W (q) and there exists N ∈ ℤ_{>0} such that π^{N} is effective, then π is effective.

This result by Honda plus the last step shows that (A mod ∼) ↦ (π_{A} mod ∼) is surjective. See (11.1) - (11.5).

These four steps together show that the map

W : {simple abelian variety over K}/ ∼_{K} = M(K, s) ⟶^{∼} W (q)

is bijective, thus proving the main theorem of Honda-Tate theory.

(0.1) Quesion / Open Problem. Surjectivity of the map W, see Step 3 and Step 4, is proved by constructing enough complex abelian varieties. Can we give a purely geometricalgebraic proof, not using methods of varieties over the complex numbers?

1 LECTURE I: Weil numbers

See [74], the first three pages; see [60] §2. To make this talk work, please do all of this in great detail.

(1.1) Topic. Give the definition of a Weil q-number. Treat the special cases π ∈ ℚ. Give 2 definitions of a CM-field and prove their equivalence. Give some examples of CM-fields. Find your own! Characterize Weil q-numbers and give examples. Try to convince the audience that it is easy to construct Weil numbers having certain properties. Two versions: finding suitable totally real numbers, and finding suitable monic polynomials with integer coefficients. Find examples (two kinds) of Weil q-numbers π such that ℚ(π) ≠ ℚ(π^{n}) for some n > 1.

2 LECTURE II: Endomorphisms of abelian varieties and Frobenius

(2.1) Recall Frobenius morphisms. Briefly: Discuss absolute Frobenius, denoted F rob for a scheme T over 𝔽_{p}. Discuss relative Frobenius F for a scheme T over a base scheme S over 𝔽_{p}. Discuss geometric Frobenius π_{X} for a scheme X over a finite field K = 𝔽_{q}. In particular, we have π_{A} for an abelian variety over a finite field K = 𝔽_{q}. Why is it an endomorphism?

(2.2) The Tate ℓ-group of an abelian variety. Briefly give the definition. Let A be an abelian variety over a field K. Let ℓ ∈ K^{∗}. Define T_{ℓ}(A) as a pro-finite group scheme over K. Show it is equivalent to give: either T_{ℓ}(A), or T_{ℓ}(A(K^{sep})) endowed with the structure of a continuous Galois module over Gal(K^{sep}/K). Discuss some examples. Discuss the structure of this group (no proofs, black box), density of ℓ-power torsion points.

Optional: Extra on Tate ℓ-groups. Show or mention that a finite flat group scheme N → S of constant rank n, where n is invertible in O_{S} is etale over S. Discuss fundamental groups, Galois modules; e.g. see [10], 10.5. Should this be a separate topic? or material incorporated in other talks?

(2.3) Finite rank of endomorphism rings. For an abelian variety A over a field K and a prime number ℓ ≠ char(K) the natural map

End(A) ⊗_{ℤ} ℤ_{ℓ} ↪ End(T_{ℓ}(A)(\overline{K}))

is injective, as Weil showed. Give a proof. Conclude that the endomorphism ring has finite rank and conclude that in case A is simple, π_{A} is an algebraic integer, etc.

(2.4) Dual abelian varieties, and Rosati. Discuss the dual abelian variety. (Black box.) Define the notion of a polarization. Show how having a polarization gives rise to an involution on the endomorphism algebra, called the Rosati involution.

3 LECTURE III: Positivity of Rosati and the Weil conjecture for an abelian variety over a finite field

(3.1) Positivity of Rosati. Formulate and indicate the proof of this property.

(3.2) Verschiebung. Define the Veschiebung V for an abelian variety A over a field of characteristic p, by dividing [p] by F. Show that the V is the transpose of the Frobenius of the dual abelian variety.

Optional: Extra on Verschiebung. Discuss V_{G} for a finitely presented, flat, commutative group scheme over a base in positive characteristic; see [63], Exp. VII_{A}.4. Show

(F_{B/S} : B → B^{(p)})^{t} = (V_{B^{t}/S} : (B^{(p)})^{t} → B^{t})

for an abelian scheme over a base scheme S in positive characteristic. See [23].

(3.3) Theorem (Weil). Let A be a simple abelian variety over K = 𝔽_{q}; consider the endomorphism π_{A} ∈ End(A), the geometric Frobenius of A/𝔽_{q}. The algebraic number π_{A} is a Weil q-number, i.e. it is an algebraic integer and for every embedding ψ : ℚ(π_{A}) → ℂ we have

| ψ(π) | = \sqrt{q}.

See [79], page 70; [80], page 138; [47], Theorem 4 on page 206. Using properties of Frobenius and Verschiebung give a proof, which is different form the classical approach by Weil, see [23].

Remark. A proof of this Weil conjecture can also be given along the “classical lines”, see [47], Theorem 4 on page 206. Is this an alternative to be presented as the seminar? Or perhaps present both proofs?

4 LECTURE IV: Abelian varieties over finite fields

See [73], [74] and [84] or one of the many other possible references.

(4.1) Theorem (Tate, Faltings, and many others). Suppose K is of finite type over its prime field. (Any characteristic different from ℓ.) The canonical map

End(A) ⊗_{ℤ} ℤ_{ℓ} ⟶^{∼} End(T_{ℓ}(A)) ≅ End_{G_{K}}((ℤ_{ℓ})^{2g})

is an isomorphism. □ This was conjectured by Tate. In 1966 Tate proved this in case K is a finite field, see [73]. The case of function field in characteristic p was proved by Zarhin and by Mori, see [82], [83], [43]; also see [42], pp. 9/10 and VI.5 (pp. 154-161).

(4.2) Moduli spaces of abelian varieties: Existence Formulate as a black box the existence of moduli spaces of polarized abelian varieties. Deduce finiteness properties of the numbers of isomoprhism classes of abelian vareities.

(4.3) Proof of the theorem over finite fields. Suggestion: Show (2.3) e.g. by following arguments in [47]. Then show (4.1) over a finite field, either by following [73], or by using [83].

5 LECTURE V: Full description of endomorphism algebras

(5.1) Central Simple Algebras. Recall briefly the notion of a central simple algebra and the description of them over number fields, including local to global principle.

(5.2) Theorem (Tate). Let A be an abelian variety over the finite field K = 𝔽_{q}. The characteristic polynomial f_{A,π_{A}} = f_{A} ∈ ℤ[T] of π_{A} ∈ End(A) is of degree 2· dim(A), the constant term equals q^{dim(A)} and f_{A}(π_{A}) = 0.

If an abelian variety A is K-simple then f_{A} is a power of the minimum polynomial Irr(π_{A}) ∈ ℤ[T].

Let A and B be abelian variety over K = 𝔽_{q}. Then:

A is K-isogenous to an abelian subvariety of B iff f_{A} divides f_{B}.

In particular

A ∼_{K} B ⟺ f_{A} = f_{B}.

(5.3) Theorem (Tate). Suppose A is a simple abelian variety over the finite field K = 𝔽_{q}.

(1) The center of D := End^{0}(A) equals L := ℚ(π_{A}).

(2) Moreover

2g = [L : ℚ]·\sqrt{[D : L]},

where g is the dimension of A. Hence: every abelian variety over a finite field admits smCM. We have:

f_{A} = (Irr(π_{A}))^{\sqrt{[D:L]}} .

(3)

ℚ ⊂ L := ℚ(π_{A}) ⊂ D = End^{0}(A).

The central simple algebra D/L

• does not split at every real place of L,

• does split at every finite place not above p.

• For a discrete valuation w of L with w | p the invariant of D/L is given by

inv_{w}(D/L) = \frac{w(π_{A})}{w(q)}·[L_{w} : ℚ_{p}] mod ℤ,

where L_{w} is the local field obtained from L by completing at w. Moreover

inv_{w}(D/L) + inv\overline{w}(D/L) = 0 mod ℤ,

where \overline{w} = ρ(w) is the complex conjugate of w.

(5.4) Proofs These theorems should be discussed and proved in the seminar.

(5.5) Note that refstruct1 eables us to compute from a given Weil q-number π the number e = [ℚ(π) : ℚ], the algebra D, and hence d = \sqrt{[D : L]} and g = ed/2.

6 LECTURE VI: Albert classification and endomorphism algebras

(6.1) Explain the Albert classification. This is a classification of central simple algebras finite dimensional over ℚ endowed with a positive involution. See [60, Section 18.2]. Treat this result as a black box.

(6.2) Explain which types occur. Which types of the Albert classification occur as endomorphism algebras with Rosati involution for polarized abelian varieties over finite fields? See [60, Section 15.9].

(6.3) Examples. Going back to our examples of Weil q-numbers describe the associated division algebras with Rosati involution. Mention again (5.5).

(6.4) Rephrase the notions mentioned in (5.5), and compute various examples. E.g. see (6.7).

Examples and extra topics, which can be used in the remaining talks

Here we mention some topics not yet included in despcriptions of various talks. Examples and ideas mentioned here should feature in some of the talks to be given. Please choose from these topics as material for illustation. Or/and, go through some of this material yourself in order to get some background information and feeling for the ideas of the topic of this seminar. – More information on these aspects is to be found in [60].

(6.5) Isotypic. We have defined π_{A}, an algebraic integer, for any simple abelian variety A over a finite field K = 𝔽_{q}. We define B to be isotypic (over K), if there exists a simple A and an isogeny B ∼ A^{m} for some m ∈ ℤ_{>0}. We define π_{B} := π_{A}. Note that if C is isotypic over K then C ⊗ K′ is isotypic for every K ⊂ K′, see (11.4).

(6.6) An application of [HT]: the Manin conjecture. At least one lecture should include the primary motivation and this important application of Honda-Tate theory:

Let ξ be a symmetric Newton polygon and fix p. There exists an abelian variety A over \overline{𝔽_{p}} with Newton polygon equal to ξ.

See [74], the last example in §1; see [60], Section 11. Remark. Another proof (nott using complex uniformization) of the Manin conjecture will be given in the course.

(6.7) Isogeny classes of elliptic curves over finite fields. See [76]; see [60] §14. Note the aspect that under a finite extension 𝔽_{q} = K ⊂ K′ = 𝔽_{q^{m}} in general several Kisogeny clasess get merged into one K′-isogeny class, and, in general there exist K′-isogeny classes which do not come from any abelian variety over K. Discuss examples.

(6.8) Behavior of End(A) under field extension. If A is an abelian variety over Kand K ⊂ K′ is a field extension, then

End(A) ⊂ End(A ⊗ K′).

Moreover

EndA ⊗ K′ / End(A) has no torsion.

Give a proof. Give examples. The fact that End(A) ⫋ EndA ⊗ K′ can have two “reasons”

e.g. A is simple, and A ⊗ K′ is not simple, e.g. A and A ⊗ K′ are both simple, but the endomorphism ring gets larger (or a combination of both).

Give examples.

(6.9) Exercise. Let E be an elliptic curve over a prime field ℙ. Show that:

(1) If ℙ = ℚ show: End(E) = ℤ.

(2) If ℙ = 𝔽_{p} show that End(E) is an imaginary quadratic field.

(3) End(E) is commutative.

(4) Give an example of an abelian variety A simple over ℚ with End(A) ⫌ ℤ.

(5) Give an example of an abelian variety A simple over 𝔽_{p} such that End(A) is not commutative.

(6.10) Behavior of End(A) under reduction modulo p. Let A be an abelian scheme over an integral domain R, and let R → K be a surjective homomorphism onto a field. We obtain a homomorphism of rings End(A) → End(A ⊗_{R} K). Show:

(1) This homomorphism is injective.

(2) Suppose n ∈ ℤ_{>0} is not divisible by the characteristic of K. Then End(A ⊗_{R} K) / End(A) has no n-torsion.

(3) Give examples where End^{0}(A) ⫋ End^{0}(A ⊗_{R} K).

(4) Give examples where End^{0}(A) = End^{0}(A ⊗_{R} K) and End(A) ⫋ End(A ⊗_{R} K)

(6.11) Deligne: classification of ordinary abelian varieties. See [17].

7 LECTURE VII: On p-divisible groups

(7.1) Topic. Give the definition of a p-divisible group. Define homomorphisms and endomorphism rings of p-divisible groups. Define isogenies of p-divisible groups. Give a discussion with examples.

Additional: Give an easy example to show that the “naive Tate-p-group” of an abelian scheme over a base on which p is not invertible is not a good notion.

(7.2) Lemma. Let A be an abelian variety over a field K. Let O ⊂ End^{0}(A) be a subring of the endomorphism algebra of A. Show that if O is a finite ℤ-module, then there exists an isogeny A → B of abelian varieties over K such that O ⊂ End(B).

Additional: Give an example to show an isogeny is needed. Give the analogue of this lemma for p-divisible groups.

(7.3) Lemma. Let A be an abelian variety isotypic over a finite field K = 𝔽_{q}, with q = p^{n}. As above we write π = π_{A}, the geometric Frobenius of A, and L = ℚ(π) with [L : ℚ] = e and D = End^{0}(A) with [D : L] = r^{2} and dim(A) = g = er/2. Show there exists a number field field P, with L ⊂ P ⊂ D, where P is a CM-field of degree 2g over ℚ.

A reference for a proof of this is Tate’s Bourbaki lecture [74], Section 3, Lemma 2. Give a proof of this lemma.

Remark/Notation. We will write L for the center of D = End^{0}(A). We will write P ⊃ L for a splitting field of D/L. Also we write P for a CM-algebra for an abelian variety.

Additional: What would be an analogue of this lemma for p-divisible groups over finite fields? Is it true?

(7.4) Theorem. Let A be an abelian variety isotypic over a finite field K = 𝔽_{q}, with q = p^{n}. As above we write π = π_{A}, the geometric Frobenius of A, and L = ℚ(π) with [L : ℚ] = e and D = End^{0}(A) with [D : L] = r^{2} and dim(A) = g = er/2. Let X = A[p^{∞}]. Assume that O_{L} ⊂ End(A), which we may after replacing A by an isogenous abelian variety because of Lemma (7.2) above. Consider the set Σ_{L}^{(p)} of discrete valuations of L dividing the rational prime number p.

(1) The decomposition

D ⊗ ℚ_{p} = ∏_{w∈Σ_{L}^{(p)}} D_{w}, O_{L} = ∏ O_{L_{w}},

gives a decomposition X = ∏_{w} X_{w}.

(2) The height of X_{w} equals [L_{w} : ℚ_{p}]·r.

(3) The p-divisible group X_{w} is isoclinic of slope γ_{w} equal to w(π_{A})/w(q); note that q = p^{n}.

(4) Let \overline{w} be the discrete valuation of L obtained from w by complex conjugation on the CMfield L; then γ_{w} + γ\overline{w} = 1.

This theorem should have been proved almost completely in the talk giving the precise description of the endomoprhism algebra of an abelian variety over a finite field. Recall the necessary setup if needed. See also: [78], and [60], 9.2 and 21.22.

(7.5) CM-types. Define the notion of a CM-type of a CM field. Suppose that A is an abelian variety over the complex numbers ℂ. Suppose that P is a CM-field contained in End^{0}(A). Assume that [P : ℚ] = 2 dim(A). Show that the set of complex embeddings Φ = {φ_{1}, ⋯, φ_{g}} of P which occur in the P -action on the tangent space of A form a CMtype; i.e. Φ ⊔ Φ·ρ = Hom(P, ℂ), where ρ is the involution of P having the totally real subfield as fixed field.

Additional: Define similarly the notion of a CM-type for a finite separable algebra P over ℚ_{p}. Is there an analogue of the result above for actions of P on p-divisible groups over a field of characteristic p? What about p-divisible groups over complete discrete valuation rings of mixed characteristic (0, p)?

8 LECTURE VIII: The Shimura-Taniyama formula

(8.1) Topic. This formula relates the characteristic p reduction of the p-divisible group of an CM-abelian variety with good reduction to the CM-field + CM-type of the abelian variety in characteristic 0. In this talk we formulate this formula and prove it along the lines of Tate’s argument using CM-theory for p-divisible groups in his Bourbaki lecture.

Please follow Brian Conrad’s notes; here is a link http://math.stanford.edu/∼conrad/vigregroup/vigre04/stformula.pdf.

Other references: [74], Lemma 5, or [60], Section 9, or [70], §13, or [40], Corollary 2.3.

9 LECTURE IX: CM-abelian varieties

In this lecture we treat the following topics.

(9.1) Definition. Give the definition of smCM for an abelian variety.

(9.2) Construction. Recall the definition of a CM-type Φ on a CM-field P. Let Φ = {σ_{1}, …, σ_{g}}. Thus Φ induces a map

σ_{Φ} = (σ_{1}, …, σ_{g}) : P ⟶ ℂ^{g}.

Show that T = ℂ^{g}/σ_{Φ}(O_{P}) is a complex torus with P ⊂ End^{0}(A).

Remark/Notation. We will write L for the center of D = End^{0}(A). We will write P ⊃ L for a splitting field of D/L. Also we write P for a CM-algebra for an abelian variety.

(9.3) Lemma. For any (P, Φ) as above the complex torus T constructed in (9.2) is an abelian variety, i.e., it is the complex manifold associated to an abelian variety A. Also, show that P ⊂ End^{0}(A) and that we can recover Φ from this as in Lemma (7.5).

(9.4) Lemma. Suppose that A is an abelian variety with smCM over an algebraically closed field k. If the characteristic of k is zero, show that there exists a number field M ⊂ k and an abelian variety A_{M} over M such that A and A_{M} ⊗ k are isomorphic over k.

Additonal: What is an analogous statement to this if the characteristic of k is p > 0. Is this analogous statement true? You might want to consult [51]. Or first do:

Exercise. Show there exists an abelian variety A over a field K ⊃ 𝔽_{p} such that A admits smCM, and such that A cannot be defined over a finite field (and here K is not a finite field).

(9.5) Black Box: Néron models. Let R be a complete discrete valuation ring with field of fractions K. Let A_{K} be an abelian variety over K. Define the notion of a Néron model of A_{K} over R. Formulate existence and functoriality of the Néron model. In particular, discuss how the endomorphisms of A_{K} act on the special fibre.

(9.6) Black box: Potential semi-stable reduction Formulate potentially semi-stable reduction.

(9.7) Lemma. Show, using ℓ-torsion points that in the semi-stable reduction case the endomorphism ring of the generic fibre injects into the endomorphism ring of the special fibre.

(9.8) Lemma. Use the above to show that given an abelian variety A with smCM over a number field K, there exists a finite extension K ⊂ K′ such that A_{K′} has everywhere good reduction, i.e., extends to an abelian scheme over the ring of integers O_{K′}.

(9.9) Remark. An abelian variety of CM-type is an abelian variety in characteristic zero with smCM where moreover the inclusion P ⊂ End^{0}(A) is given. However an abelian variety in positive charactersitic will not be called an abelian variety of CM-type. See a discussion in [60], 13.12 why.

10 LECTURE X: The Honda lifting theorem.

A Weil q-number π is said to be effective if there exists an abelian variety A over 𝔽_{q} such that the geometric Frobenius π_{A} corresponds to π. The goal of this lecture is to prove the following theorem, using the material from the preceding lectures.

(10.1) Theorem (Honda). For every Weil q-number π there is an integer N > 0 such that π^{N} is effective.

(10.2) An open problem. The proof we know of the Honda theorem (10.1) is via complex uniformization. Is there a proof of this theorem using only algebraic, and no analytic methods?

Using previous results in the seminar, show the Honda lifting theorem. The rest of the material in this section is not necessary for the general line of thought, but please include some of it in the lecture as time permits! Namely, we think it is instructive to discuss [56], and as well as some examples as in [11]. Here is a definition and theorem that might be discussed.

(10.3) Definition. Suppose A_{0} be an abelian variety over a field K ⊃ 𝔽_{p} such that A_{0} admits smCM.

1. A lifting of A_{0} to characteristic zero is given by an integral domain (R, 𝔪), with a homomorphism R → K, an abelian scheme A over R, such that (a) A_{0} ≅ A ⊗_{R} K, and (b) the characteristic of the fraction field of R is zero.

2. We say A is a CM-lifting of A_{0} to characteristic zero if A/R is a lifting of A_{0}, and if moreover A/R admits smCM. If this is the case we say that A_{0}/K satisfies (CML).

3. Moreover, if P ⊂ End^{0}(A_{0}) is a CM-field of degree 2g over ℚ and End^{0}(A) = P we say that A_{0}/K satisfies (CML) by P.

(10.4) Theorem (Honda). Let K = 𝔽_{q}. Let A_{0} be an abelian variety, defined and simple over K. Let L ⊂ End^{0}(A_{0}) be a CM-field of degree 2g over ℚ. There exists a finite extension K ⊂ K′, an abelian variety B_{0} over K′ and a K′-isogeny A_{0} ⊗_{K} K′ ∼ B_{0} such that B_{0}/K′ satisfies (CML) by L.

See [29], Th. 1 on page 86, see [74], Th. 2 on page 102. For the notion (CML) see (10.3).

The isogeny mentioned in the theorem is necessary in general, as follows from [56]. It is an open problem whether the exension K ⊂ K′ of finite fields is necessary ?!

Additional: Analyzing the road to a proof of this theorem we see that complex uniformization is used. However in [11], Section 5 and Appendix A, a purely algebraic proof for (10.4) is given. Comments?

Note that we do not have an algebraic proof for (10.1), nor do we have an algebraic proof for [HT] on page 1.

11 LECTURE XI: The Weil restriction functor

(11.1) The Weil restriction functor. Suppose given a finite extension K ⊂ K′ of fields We could consider much more general situations, but we will not do that in this seminar. Write S = Spec(K) and S′ = Spec(K′). We have the base change functor

Sch_{S} → Sch_{S′}, T ↦ T_{S′} := T ×_{S} S′.

General nonsense defines a right adjoint functor to the base change functor to be a functor

Π = Π_{S′/S} = Π_{K′/K} : Sch_{S′} → Sch_{S},

characterized by the following universal property:

Mor_{S}(T, Π_{S′/S}(Z)) = Mor_{S′}(T_{S′}, Z).

functorially in T ∈ Sch_{S}, and Z ∈ Sch_{S′}. In other words, the scheme Π(Z) is supposed to represent the functor T ↦ Mor_{S′}(T_{S′}, Z).

Note that, in our case where K′/K is Galois we have Π_{S′/S}(Z) ×_{S} S′ = Z ×_{S′} ⋯ ×_{S′} Z the self-product of [K′ : K]-copies of Z over S′. Please explain why!

In our situation, with K′/K Galois, Weil showed that Π_{S′/S}(Z) exists. In fact, consider Z ×_{S′} ⋯ ×_{S′} Z, the self-product of [K′ : K] copies of Z over S′. It can be shown that, if Z is quasi-projective over K′, then Z ×_{S′} ⋯ ×_{S′} Z can be descended to K in such a way that it solves this problem. Namely, there is a natural action of G = Gal(K′/K) on the self product such that

Π_{S′/S}(Z) = (Z ×_{S′} ⋯ ×_{S′} Z)/G.

Please explain! For a more general situation, see [25], Exp. 195, page 195-13. Also see [75], Nick Ramsey - CM seminar talk, Section 2.

(11.2) Lemma. Let B′ be an abelian variety over a finite field K′. Let K ⊂ K′, with [K′ : K] = N. Write

B := Π_{K′/K} B′; then f_{B}(T) = f_{B′}(T^{N}).

where f_{B}(T), resp. f_{B′}(T) are the characteristic polynomials of Frobenius on B, resp. B′.

See [74], page 100. We make a little detour. From [14], 3.19 we cite:

(11.3) Theorem (Chow). Let K′/K be an extension such that K is separably closed in K′. (For example K′ is finite and purely inseparable over K.) Let A and B be abelian varieties over K. Then

Hom(A, B) ⟶^{∼} Hom(A ⊗ K′, B ⊗ K′)

is an isomorphism. In particular, if A is K-simple, then A ⊗ K′ is K′-simple.

(11.4) Claim.

For an isotypic abelian variety A over a field K, and an extension K ⊂ K′, we have that A ⊗ K′ is isotypic.

Proof. It suffices to this this in case A is K-simple. It suffices to show this in case K′/K is finite. Moreover, by the previous result it suffices to show this in case K′/K is separable.

Let K ⊂ K′ be a separable extension, [K′ : K] = N. Write Π = Π_{Spec(K′)/Spec(K)}. For any abelian variety A over K we have Π(A ⊗_{K} K′) ≅ A^{N}, and for any C over K′ we have Π(C) ⊗_{K} K′ ≅ C^{N}, as can be seen by the construction; e.g. see the original proof by Weil, or see [75], Nick Ramsey - CM seminar talk, Section 2; see (11.1). If there is an isogeny A ⊗_{K} K′ ∼ C_{1} × C_{2}, with non-zero C_{1} and C_{2} we have Π(C_{1} × C_{2}) ∼ A^{N}. Hence we can choose positive integers e and f with Π(C_{1}) ∼ A^{e} and Π(C_{2}) ∼ A^{f}. Hence

Π(C_{1}) ⊗ K′ ≅ (C_{1})^{N} ∼ (A ⊗_{K} K′)^{eN}, (C_{2})^{N} ∼ (A ⊗_{K} K′)^{fN}.

Hence Hom(C_{1}, C_{2}) ≠ 0. Hence: if A is simple, any two isogeny factors of A ⊗_{K} K′ are isogenous. □

By Step 6 and by Lemma (11.2) of [60], Section 10 we conclude:

(11.5) Corollary (Tate). Let π be a Weil q-number and N ∈ ℤ_{>0} such that π^{N} is effective. Then π is effective.

See [74], Lemme 1 on page 100.

12 Checklist topics/talks

Here we can fill in the various topics with speakers.

(1) 12-IX-2008 Speaker:Bhargav Bhatt

(2) 19-IX-2008 Speaker: Matt Deland

(3) 26-IX-2008 Speaker: Sho Tanimoto

(4) 3-X-2008 Speaker: Alon Levy

(5) 10-X-2008 Speaker: Mingmin Shen

(6) 17-X-2008 Speaker: Bin Du

(7) 24 - X - 2008 Speaker:

(8) 31 - X - 2008 Speaker:

(9) 7 - XI - 2008 Speaker:

(10) 14 - XI - 2008 Speaker:

(11) Last talk: 21 - XI - 2008 Speaker:

13 Appendix: prerequisites

In this apendix we indicate some of the definitions, concepts and results we assume you know. please study these, ask for advice or ask for further explanation.

Also in the seminar some “black boxes” will be used: results, technical details, with a a reference, which will be used, but not proved in the seminar.

However, some of the subjects below could be chosen as a “Topic” in the seminar.

Recommended reading: Abelian varieties: [47], [35], [15] Chapter V. Honda-Tate theory: [74], [29], [75]. Abelian varieties over finite fields: [73], [76], [78], [65]. Group schemes: [63], [49]. Endomorphism rings and endomorphism algebras: [69], [24], [73], [76], [54]. CM-liftings: [56], [11].

(13.1) Algebra. We need: standard facts about fields, number fields, valuations, ramification in finite extensions.

(13.2) Central simple algebras: the Brauer group. Basic references: [7], [62], [8] Chapter 7, [66] Chapter 10.

(13.3) Abelian varieties. Basic references: [47], [15], [GM].

(13.4) Endomorphism algebras of abelian varieties. Basic references: [69], [47], [35] Chapt. 5, [54]. Endomorphism algebras of abelian varieties can be classified. In many cases we know which algebras do appear. However it is difficult in general to describe all orders in these algebras which can appear as the endomorphism ring of an abelian variety.

(13.5) Complex tori with smCM and abelian varieties with CM. See [70], [47], [35], [61]; see [60] §19.

(13.6) Abelian varieties with good reduction. References: [48], [12], [68], [64], [6], [53], [13].

(13.7) Dieudonné theory, some properties in positive characteristic. See [39], [19]. For information on group schemes see [49], [63], [77], [10].

14 The Tate-p-conjecture.

Probably we will not use the following results:

(14.1) Exercise. Let A and B be abelian varieties over a field K. We know that Hom(A, B) is of finite rank as ℤ-module. Let p be a prime number. Show that the natural map

Hom(A, B) ⊗_{ℤ} ℤ_{p} ↪ Hom((A)[p^{∞}], B[p^{∞}])

is injective. Also see [78], Theorem 5 on page 56. Also see [84].

(14.2) Remark. On could feel the objects T_{ℓ}(A) and A[p^{∞}] as arithmetic objects in the following sense. If A and B are abelian varieties over a field K which are isomorphic over \overline{K}, then they are isomorphic over a finite extension of K; these are geometric objects. Suppose X and Y are p-divisible groups over a field K which are isomorphic over \overline{K} then they need not be isomorphic over any finite extension of K, these are arithmetic objects. The same statement for pro-ℓ-group schemes.

(14.3) Theorem (Tate and De Jong). Let K be a field finitely generated over 𝔽_{p}. Let A and B be abelian varieties over K. The natural map

Hom(A, B) ⊗ ℤ_{p} ⟶^{∼} Hom(A[p^{∞}], B[p^{∞}])

is an isomorphism. □ This was proved by Tate in case K is a finite field; a proof was written up in [78]. The case of a function field over a finite field was proved by Johan de Jong, see [30], Th. 2.6. This case follows from the result by Tate and from the following result on extending homomorphisms (14.4).

(14.4) Theorem (Tate, De Jong). Let R be an integrally closed, Noetherian integral domain with field of fractions K. (Any characteristic.) Let X, Y be p-divisible group over Spec(R). Let β_{K} : X_{K} → Y_{K} be a homomorphism. There exists (uniquely) β : X → Y over Spec(R) extending β_{K}. □ This was proved by Tate, under the extra assumption that the characteristic of K is zero. For the case char(K) = p, see [30], 1.2 and [31], Th. 2 on page 261.

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Frans Oort

Mathematisch Instituut

P.O. Box. 80.010 NL - 3508 TA Utrecht The Netherlands email: f.oort@uu.nl, oort@math.columbia.edu

Aise Johan de Jong

Columbia University, dejong@math.columbia.edu