ON THE pRANK OF AN ABELIAN VARIETY AND ITS ENDOMORPHISM ALGEBRA

Let A be an abelian variety defined over a finite field. In this paper, we discuss the relationship between the p-rank of A, r(A), and its endomorphism algebra, End(A). As is well known, End(A) determines r(A) when A is an elliptic curve. We show that, under some conditions, the value of r(A) and the structure of End(A) are related. For example, if the center of End(A) is an abelian extension of Q, then A is ordinary if and only if End(A) is a commutative field. Nevertheless, we give an example in dimension 3 which shows that the algebra End(A) does not determine the value r(A).


Introduction
Let k be an algebraically closed field of characteristic p > 0. Given an abelian variety A/k of dimension g, the p-rank of A is defined by where F * denotes the absolute Frobenius.The value r(A) is invariant under isogenies and satisfies r(A × B) = r(A) + r(B), for A, B abelian varieties over k.Thus, if A is isogenous to a product of abelian varieties m i=1 A ni i , then r(A) = m i=1 n i r(A i ).Let C/k be a non-singular projective curve of genus g > 0. Serre [Se 58] characterized the Hasse-Witt invariant, r(C), by means of the action of F * on the first cohomology group: If J denotes the jacobian of C, it is clear that r(J) = r(C).Ordinary abelian varieties are those for which r(A) = g.Supersingular elliptic curves are those for which r(A) = 0.
Let us denote by End 0 (A) := Q ⊗ Z End(A) the endomorphism algebra of A. If E/k is an elliptic curve, E is ordinary if and only if End 0 (E) is a commutative field (it is equal to Q or to an imaginary quadratic extension of Q); E is supersingular if and only if End 0 (E) is a quaternion algebra over Q. If, in addition, the field k is the algebraic closure of a finite field, the case End 0 (E) = Q is excluded.
The asymptotic behaviour of the Hasse-Witt invariants for the fibres of modular curves, resp. of Fermat curves, has been studied in [Ba-Go 97], resp.[Go 97].Both cases are quite diferent.It turns out that, for some projective curves over Q, the distribution of the extreme values of the Hasse-Witt invariant of the fibres seems to depend on the type of the endomorphism algebra over Q of their jacobian variety.
In this paper, we summarize some results which show the relationship between r(A) and End 0 (A) when the abelian variety A is defined over a finite field.Nevertheless, we provide with an example of two abelian varieties which have Q-isomorphic endomorphism algebras but show different p-ranks.One of them is the jacobian of the modular curve X 0 (41)/F 3 .Some of these results are contained in my PhD thesis.I would like to finish this introduction by expressing my gratitude to my dissertation advisor Prof. Pilar Bàyer for her help and encouragement throughout the realization of the work.

Some general facts
We fix a positive integer n and consider a power q = p n of the characteristic of k.Throughout, A denotes an abelian variety of dimension g > 0 defined over the finite field F q , and k = F q .We denote by End Fq (A), resp.End(A), the ring of endomorphisms of A which are defined over F q , resp.k.We write End 0 , where the abelian varieties A i are F q -simple and not F q -isogenous to each other, then End 0 , where M ni denotes the ring of (n i × n i )-matrices.
Let ϕ ∈ End Fq (A) be the relative Frobenius endomorphism, whose action on the variety raises to the q-th power the coordinates of the points of A. For a given prime number = p, we denote by T (A) the Tate module of A, and by V (A) := Q ⊗ Z T (A).Two abelian varieties A, B defined over F q are F q -isogenous if and only if the corresponding Frobenius have the same characteristic polynomial in the -adic representation.
The Q-algebra End 0 Fq (A) has Q(ϕ) as its center.We have that End 0 Fq (A) = Q(ϕ) if and only if the characteristic polynomial of ϕ acting on the Tate module has no double roots.We have that Q(ϕ) = Q if and only if A is F q -isogenous to the g-th power of a supersingular elliptic curve with all its endomorphisms defined over F q .All these assertions can be found in [Ta 66].
Given an F q -polarization λ : A → A, we consider the Rosati involution, defined on End If A is F q -simple, then Q(ϕ) is a number field and the Rosati involution agrees on Q(ϕ) with the complex conjugation c, for all embeddings of Q(ϕ) into Q.The class in the Brauer group of Q(ϕ) of the simple algebra End 0 Fq (A) is characterized by the local invariants i ℘ = f ℘ ord ℘ (ϕ)/n at each prime ℘ over p in Q(ϕ) (here, f ℘ stands for the residual degree at ℘); on each real prime, the local invariant is equal to 1/2; on the remaining primes, the algebra splits.The lowest common denominator e of all the invariants i ℘ is the period of the endomorphism algebra End 0 Fq (A).The characteristic polynomial of ϕ acting on the Tate module equals the e-th power of the Q-irreducible polynomial of ϕ (cf.[Ta 66], [Wa 69]).
We fix an algebraic closure for all archimedian absolute values | • | on Q.Each Weil q-number α determines, up to isogenies, an F q -simple abelian variety A/F q such that the Q-irreducible polynomial of ϕ equals the Q-irreducible polynomial of α.This assignment establishes a one to one correspondence between the conjugation classes of Weil q-numbers and the F q -isogenies classes of F q -simple abelian varieties which are F q -defined (cf.[Ta 68]).
Let α 1 , α 2 be two Weil q-numbers such that Q(α 1 ) = Q(α 2 ).If the ideals (α 1 ), (α 2 ) in the ring of integers of K := Q(α 1 ) coincide, then their associated abelian varieties are F q -isogenous.If (α 1 ) = (α 2 ), then there exists a unit ε ∈ K such that α 2 = εα 1 ; since |ε| = 1, we have that ε must be a root of unity.If ε s = 1, then α s 1 = α s 2 .The characteristic polynomials of the relative Frobenius of both abelian varieties over F q s are equal.Thus, the varieties are F q s -isogenous.We note that the abelian variety associated to a Weil q-number α is F q -isogenous to a power of a supersingular elliptic curve if and only if the ideals (α 2 ) and (q) do coincide.
In the sequel the term isogenous will indicate F q -isogenous.

Some relations between End 0 (A) and r(A)
In this section we give some propositions which relate the p-rank r(A) to the structure of the Q-algebra End 0 Fq (A).Let P (X) := det(ϕ − X Id | V (A)), which is a polynomial with integer coefficients independent of .If α i , 1 ≤ i ≤ g, denote its complex roots, then we have that |α i | = q 1/2 and 2g i=1 α i = q g .The real roots of P (X) have even multiplicity and we can order all the roots so that α i+g = α i = q/α i , for 1 ≤ i ≤ g.For such an order we write , where ℘ is a prime ideal over p in the ring of integers of Q({α i }).
By using the results displayed in the proposition and in the previous section, we obtain 3.2.Proposition.Let A/F q be an F q -simple abelian variety.Then: i) A is ordinary if and only if the ideals (ϕ), (ϕ ) (equivalently, the ideals (ϕ + ϕ ), (p)) are relatively prime in Q(ϕ).
ii) r(A) = 0 if and only if every prime iii) A is isogenous to a power of a supersingular elliptic curve if and only if (ϕ) = (ϕ ).
iv) The period e of End 0 Fq (A) in the Brauer group divides r(A).
On the p-rank of the abelian varieties 123 From now on, we assume that A/F q is F q -simple.As usual, we will say that A is absolutely simple if it is k-simple.

Corollary.
If there exists a prime ideal ℘ | (p) such that ℘ c = ℘, then A is non ordinary.If A is not isogenous to a power of a supersingular elliptic curve, then there exists a prime ideal ) and A will be non ordinary.If every prime ideal over p is invariant under complex conjugation, then (ϕ) = (ϕ ).

Proposition.
Assume that q = p.Then, we have iii) End 0 Fp (A) is a totally imaginary number field if and only if A is not isogenous to the square of a supersingular elliptic curve.
Proof: If A is non ordinary there exists a prime ℘ | (p) such that ℘ | (ϕ) and ℘ | (ϕ ).Since ϕϕ = p, it follows that ℘ ramifies.If r(A) = 0, the condition if fulfilled by all the primes which divide (p).Let us prove iii).We recall that A is F p -simple.Since n = 1, the period e in the Brauer group is 1 or 2, depending on whether Q(ϕ) is totally imaginary or not.Thus, e = 1 if and only if End 0 Fp (A) is a totally imaginary number field.The value e is equal to 2 if and only if ϕ 2 = p.In this case, the characteristic polynomial of ϕ is (X 2 − p) 2 , which has associated an F p -simple abelian variety F p 2 -isogenous to the square of a supersingular elliptic curve defined over F p 2 .

Proposition. If r(A) is prime to g, then End 0
Fq (A) is a commutative field.
Proof: If dim A = 1, the statement is known and, therefore, we may assume that g > 1.The field Q(ϕ) is either totally imaginary, equal to Q( √ p), or equal to Q. Since A is F q -simple, the last possibility is excluded in our case.Thus [Q(ϕ) : Q] is even and e divides g, because e[Q(ϕ) : Q] = 2g.Since e also divides r(A), it follows that e = 1.
The following theorem allows us to characterize, by means of the commutativity of End 0 (A), the ordinary character of those absolutely simple J. González abelian varieties whose Q-algebra End 0 (A) has as center an abelian extension of Q.Note that, since the center of the endomorphism algebra of an elliptic curve is always abelian over Q, the ordinary character of an elliptic curve, E, is equivalent to the commutativity of End 0 (E).

Theorem.
i) If A is ordinary, then End 0 Fq (A) is commutative and, therefore, End 0 Fq (A) = Q(ϕ).In particular, if A is ordinary and absolutely simple, End 0 (A) is commutative.
ii) If p splits completely in Q(ϕ) and End 0 Fq (A) is commutative, then A is ordinary.
iii) If A is absolutely simple and the center K of End 0 (A) is an abelian extension of Q, then p splits completely in K.
Proof: i) Let us assume that A is ordinary.The ideals (ϕ), (ϕ ) are relatively prime in Q(ϕ) by 3.2 i) and ϕϕ = p n .Thus, for all primes ℘ | (p) in Q(ϕ), we have that ord ℘ ϕ is zero or a positive multiple of n and, so, i ℘ ∈ Z.The field Q(ϕ) has no real primes, because if ϕ were real, then ϕ = ±q 1/2 and A would be non ordinary.Since all the local invariants of End 0 Fq (A) are trivial, its Brauer period must be e = 1; i.e., End 0 Fq (A) = Q(ϕ).This line of reasoning parallels that used in [Yu 78] for the case of the jacobian of a curve.
Let us now prove ii).If End 0 Fq (A) is commutative and p splits completely, then for all primes ℘ | (p) in Q(ϕ) we have i ℘ = ord ℘ ϕ/n ∈ Z and ord ℘ ϕ is zero or n.Since n = ord ℘ ϕ + ord ℘ ϕ , we get that (ϕ) and (ϕ ) are relatively prime and A is ordinary.
Let us see iii).We may assume without loss of generality that End 0 (A) is equal to End 0 Fq (A).Then K = Q(ϕ s ), for all s > 0. The Q-irreducible polynomial of ϕ is σ∈G (X−σ(ϕ)), where G = Gal(K/Q).Let ℘ | (p) be a prime in K and let D denote its decomposition group in K/Q.Assume that p does not split completely in K. Then we can take σ ∈ D\{Id} and the ideals (σ(ϕ)) and (ϕ) coincide, since K/Q is abelian.There exists a root of unity ε ∈ K such that ϕ = εσ(ϕ).If s > 1 is the order of ε, then which is a contradiction.
If A/F q and B/F q are absolutely simple abelian varieties of dimension g > 1, then A and B can have Q-isomorphic endomorphism algebras by means of a homomorphism Φ : End 0 (A) If this is the case, A and B are not F q -isogenous.Nevertheless, as the following theorem shows, they have the same p-rank when g = 2.

Theorem.
Let A/F q be an absolutely simple abelian variety of dimension g < 3. We have i) If g = 1, then End 0 (A) determines A up to isogenies.
ii) If g = 2, then End 0 (A) is a commutative field which determines r(A).If, moreover, p does not split completely in End 0 (A), then End 0 (A) determines A up to isogenies.
Proof: The assertion i) is well known.We assume, without loss of generality, that End 0 (A) = End 0 Fq (A).Thus, Q(ϕ) is the center of End 0 (A).For all abelian varieties A/F q of dimension 2 the condition r(A) = 0 is equivalent to the fact that A is isogenous to the square of a supersingular elliptic curve.Thus, if A is absolutely simple, either A is ordinary or r(A) = 1.
Assume that dim A = 2 and that A is absolutely simple.Since, in particular, A is not isogenous to a power of a supersingular elliptic curve, the field K := Q(ϕ) is totally imaginary and the ideals (ϕ), (ϕ ) are different, by 3.2 iii).
In order to show that End 0 (A) is a commutative field, we prove that e = 1.Since e[K : Q] = 4 and K = Q, e = 2 or e = 1.Assume that e = 2. Then [K : Q] = 2 and the prime p splits completely in K by 3.6 iii).We have that (p) = ℘℘ c .The ideal (ϕ) is ℘ i (℘ c ) n−i , for some i such that 0 ≤ i ≤ n, and the corresponding local invariants are i/n, (n − i)/n.Since e = 2, we have that i = n/2 and (ϕ) = (ϕ ), which leads to a contradiction.Thus, e = 1 and [K 1 .This yields the following possibilities for the splitting type of (p) in K: In case a), the ideal (ϕ) is ℘ i (℘ c ) 2n−i , 0 ≤ i ≤ 2n.The local invariants i/n, (2n − i)/n are integers if and only if i ∈ {0, n, 2n}.The case i = n is not possible, since (ϕ), (ϕ ) would coincide.Thus, (ϕ) is equal to ℘ 2n or (℘ c ) 2n .The two possible ideals are conjugated and, therefore, they correspond to isogenous abelian varieties.Thus, the p-rank of A is determined.In this particular case, A is ordinary, since (ϕ) and (ϕ ) are relatively prime in K and we apply 3.2 i).

J. González
In case b), we have that (ϕ) is ℘ n 1 or (℘ c 1 ) n .The two ideals are conjugated and they correspond to isogenous abelian varieties, which are ordinary.
In case c), we have that (ϕ) is . The two solutions are conjugated and they correspond to isogenous abelian varieties, which are not ordinary because ℘ 2 = ℘ c 2 .Thus r(A) = 1.In case d), the ideal (ϕ) is equal to 2 ) n .These four solutions correspond to two possible ordinary abelian varieties which are not isogenous.
We see that in all cases r(A) is determined by End 0 (A).If p does not split completely in End 0 (A) then only cases a), b) or c) are possible.In all of them, A is determined up to isogenies by End 0 (A).We remark that the first claim of ii) can be deduced from [Oo 87, 6.5].
If dim A = 2 and A/F q is not F q -simple, a counting of dimensions in each possible splitting type of A shows that the Q-algebra End 0 (A) also determines r(A).

An example
In this section we will give an example of two absolutely simple abelian varieties of dimension 3 which have isomorphic endomorphism algebras but different p-ranks.
Let α be a Weil q-number.For each positive integer m, we denote by A m an abelian variety associated to the Weil q m -number α m .Let e m be the Brauer period of End 0 F q m (A m ).We have the following equivalent conditions: i) A 1 /F q is absolutely simple.
ii) A 1 /F q m is F q m -simple for all positive integers m.
for all positive integers m.
Since Q(α)/Q is a finite extension, there exists a positive integer t such that Q(α t ) = Q(α tm ) for all positive integers m.For this t, we have that e t = e tm for all m and, thus, A t is absolutely simple.The abelian variety A 1 is absolutely simple if and only if dim and, in this case, we have that End 0 (A 1 ) = End 0 F q t (A t ).In particular, if Q(α m ) = Q(α) for all m, then A 1 is absolutely simple and End 0 (A 1 ) = End 0 Fq (A 1 ).This is the condition which we will use in our example.

J. González
i) H = {Id}.In this case, there exists σ ∈ G such that imaginary because m is odd, and thus, [Q(µ m ) : Q] can only be equal to 6 or 2. Therefore,
From the tables of Wada, we see that the characteristic polynomial of the Hecke operator T 3 acting in S 2 (X 0 (41)) is Q(X) = X 3 − 4X + 2, which is Q-irreducible.We consider the natural action of T 3 as endomorphism of J 0 (41), the jacobian of X 0 (41), and its (mod 3)-reduction, T 3 , as endomorphism of A. The Q-irreducible polynomial of T 3 acting in Ω 1 (A) is Q(X).
Since P (X) is not of type X 6 + aX 3 + 3 3 and Q(ϕ) = Q(µ 7 ), we have by 4.1 that Q(ϕ) = Q(ϕ m ), for all positive integers m.Thus, A and B are absolutely simple and End 0 (A), End 0 (B) are isomorphic to Q(ϕ).