Eigenvalues and dynamical degrees of self-maps on abelian varieties
Fei Hu


Eigenvalues and dynamical degrees on abelian varieties]Eigenvalues and dynamical degrees of self-maps on abelian varieties


Pacific Institute for the Mathematical Sciences, Vancouver, BC V6T1Z4, Canada
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway



mailto:hf@u.nus.eduhf@u.nus.edu



Let X be a smooth projective variety over an algebraically closed field, and f X→ X a surjective self-morphism of X.
The i-th cohomological dynamical degree χ_i(f) is defined as the spectral radius of the pullback f^* on the étale cohomology group H^i_ét(X, 𝐐_ℓ) and the k-th numerical dynamical degree λ_k(f) as the spectral radius of the pullback f^* on the vector space 𝖭^k(X)_𝐑 of real algebraic cycles of codimension k on X modulo numerical equivalence.
Truong conjectured that χ_2k(f) = λ_k(f) for all 0 ≤ k ≤ X as a generalization of Weil's Riemann hypothesis.
We prove this conjecture in the case of abelian varieties.
In the course of the proof, we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.


[2020]
14G17,	
14K05,	
16K20.	






The author was partially supported by a UBC-PIMS Postdoctoral Fellowship and Young Research Talents grant #300814 from the Research Council of Norway.



[
    Fei Hu
    December 30, 2023
=====================










§ INTRODUCTION








 §.§ Parity of eigenvalues



Given a self-morphism f  X → X of a smooth complex projective variety X, the characteristic polynomial P_i(f, t) of the pullback f^* on H^i(X, ) is a monic polynomial of degree b_i(X) with integer coefficients, where b_i(X) denotes the i-th Betti number of X.
As a direct consequence of the Hodge decomposition, for odd i, the b_i(X) complex roots of P_i(f, t) fall into b_i(X)/2 pairs with each one consisting of two conjugate complex numbers; in particular, all real roots of P_i(f, t) are of even multiplicity.
Over a base field of positive characteristic, because of the absence of the Hodge decomposition,
it seems to be unknown whether an analog of the above parity with complex cohomology replaced by ℓ-adic étale cohomology holds in general.

We show that it holds for abelian varieties in arbitrary characteristic.
Henceforth, we fix an algebraically closed field  of arbitrary characteristic.



Let α X → X be an endomorphism of an abelian variety X of dimension g over .
Let P_α(t) be the characteristic polynomial of the pullback α^* on the first étale cohomology group H^1_(X, _ℓ) of X, which is a monic polynomial of degree 2g with integer coefficients.
Then there exists a monic polynomial P_α^(t) ∈[t] of degree g such that P_α(t) = P_α^(t)  P_α^(t).

In particular, the 2g complex roots of P_α(t) fall into g pairs with each one consisting of two conjugate complex numbers; all real roots of P_α(t) are of even multiplicity.


Here, the characteristic polynomial P_α(t) of α is equivalently defined as the characteristic polynomial of the induced endomorphism T_ℓ(α) on the Tate module T_ℓ(X) of X, independent of ℓ≠(); see <ref> for details on abelian varieties.



With notation as in <ref>, we notice that the pullback α^* on the i-th étale cohomology group H^i_(X, _ℓ) is completely determined by its pullback α^* on H^1_(X, _ℓ) (see, e.g., <ref>).
Hence it is straightforward to see that the same conclusion of <ref> holds for all odd i.


The remark below says that the above even multiplicity type phenomenon has occurred in positive characteristic, even though there is no Hodge decomposition/symmetry.



Let X_0 be a smooth projective variety over a finite field _q of characteristic p,
and X = X_0 ×__q_q the base change of X_0 to an algebraic closure _q of _q.
Let  denote the Frobenius endomorphism of X relative to _q.
Deligne proved that the characteristic polynomial Φ_i(t) of the pullback ^* on H^i_(X, _ℓ) has integer coefficients independent of ℓ, and all of its roots are of modulus q^i/2 (see <cit.>).
Later, building on his earlier work, he also proved the hard Lefschetz theorem for étale cohomology (see <cit.>).
Then combining with Poincaré duality, there is a nondegenerate pairing

    H^i_(X, _ℓ) × H^i_(X, _ℓ) →_ℓ(-i),

which is compatible with the Frobenius action.
When i is odd, the pairing is alternating and hence the i-th Betti number b_i(X) is even (see <cit.>).
Putting all together, we see that the b_i(X) complex roots of Φ_i(t) fall into b_i(X)/2 pairs with each one consisting of two conjugate complex numbers and all real roots of Φ_i(t) are of even multiplicity (see also <cit.>).
One may ask if an analog of this even multiplicity type result holds for arbitrary correspondences.
It turns out to follow from the standard conjectures, as kindly pointed out to the author by Deligne.


For an arbitrary smooth projective variety X over , applying <ref> to its Albanese variety (X) which is an abelian variety of dimension equal to the geometric irregularity q(X) b_1(X)/2, we have the following direct corollary.



Let f  X → X be a self-morphism of a smooth projective variety X of dimension n over .
Let P_i(f,t) denote the characteristic polynomial of the pullback f^* on H^i_(X, _ℓ).
Then for i=1 and 2n-1, the b_i(X) complex roots of P_i(f,t) fall into b_i(X)/2 pairs with each one consisting of two conjugate complex numbers and all real roots of P_i(f,t) are of even multiplicity.


Since our proof of <ref> relies on the classification of the endomorphism -algebras of simple abelian varieties by Albert in the 1930s (see <cit.> and references therein), we introduce the following notion.



Let α X→ X be an endomorphism of an abelian variety X of dimension g over .
Let P_α(t) denote the characteristic polynomial of the pullback α^* on H^1_(X, _ℓ).
A (monic) complex polynomial P_α^(t) ∈[t] is called an Albert polynomial of α, if P_α(t) = P_α^(t)  P_α^(t).


We have seen in <ref> that there always exists an Albert polynomial P_α^(t) of α;
however, it may not be unique because we are free to switch those conjugate complex roots.

Note that when X is a complex abelian variety, there is a canonical choice of an Albert polynomial of an endomorphism α, namely, the characteristic polynomial of the pullback α^* on the Dolbeault cohomology group H^1,0(X, ); this is also equal to the characteristic polynomial of the analytic representation ρ_a(α) of α, where ρ_a(α) is the induced linear map of α on the universal cover ^g of X.
So the notion of Albert polynomial could be regarded as a characteristic-free substitute of the characteristic polynomial of the analytic representation.



In the category of abelian varieties, it is inevitable to work on endomorphisms with rational coefficients, namely, the endomorphism -algebra ^0(X) (X) ⊗_.
For any α∈^0(X) with kα∈(X) for some positive integer k, the characteristic polynomial P_α(t) of α is defined as k^-2g P_kα(kt) ∈[t], which is still monic (see <cit.>).
Hence if P_kα^(t) is an Albert polynomial of kα∈(X), we call k^-g P_kα^(kt) ∈[t] an Albert polynomial P_α^(t) of α∈^0(X).
Clearly, this is independent of the choice of k.


We henceforth fix a polarization ϕ = ϕ_ X → X, where _X(H_X) is an ample line bundle on X associated to a fixed ample divisor H_X on X.
Let

    α^†ϕ^-1∘α∘ϕ∈^0(X)

be the Rosati involution of α (see <cit.> and <cit.> for more details).
It would follow from <ref> that for a symmetric element α∈^0(X), i.e., α^† = α, its Albert polynomial P_α^(t), in the sense of <ref>, is unique and lies in [t].
We now give a geometric characterization of the coefficients of the Albert polynomial of α^†∘α.



Let α X → X be an endomorphism of an abelian variety X of dimension g over .
Then the Albert polynomial P_α^†∘α^(t) of α^†∘α is unique and has rational coefficients.
Moreover, if we write

    P_α^†∘α^(t) = ∑_k=0^g (-1)^k c_k   t^g-k,

then for any 0≤ k ≤ g, one has

    c_k = gk α^* H_X^k · H_X^g-k /H_X^g∈,

where both α^* H_X^k · H_X^g-k,   H_X^g ∈ are intersection numbers.


The above result generalizes <cit.>, where k = 1, to more general intersection products.
One might be able to use the exterior product calculation, as in his proof, to deduce the formula.
However, our proof is geometric in nature and essentially relies on the positivity of the Rosati involution.




 §.§ An application to algebraic dynamics




The second main result is an application of Albert polynomials to the study of certain dynamical problems on abelian varieties.
Indeed, our <ref> confirms Truong's dynamical degree comparison <ref> for abelian varieties, which we shall state in detail now.

Let f X  X be a dominant rational self-map of a smooth projective variety X of dimension n over .
In order to measure the dynamical complexity of f under iterations, one can associate to this f two dynamical invariants as follows.
Let ℓ be a prime different from the characteristic of .
Fix a field isomorphism ι_ℓ so that we may speak of the complex absolute value of an element of _ℓ:

    |a|_ι |ι(a)|,  for any  a ∈_ℓ.

We endow a norm ·_ι on the finite-dimensional _ℓ-vector space H^∙_(X, _ℓ).
For any 0≤ i≤ 2n, the i-th cohomological dynamical degree χ_i(f)_ι of f (with respect to ι) is defined by

    χ_i(f)_ιlim sup_m→∞(f^m)^*|_H^i_(X, _ℓ)^1/m_ι,

where the pullback is always defined in the sense of correspondences (see, e.g., <cit.>).

One can define another dynamical degree of f using algebraic cycles.
Indeed, let ^k(X) denote the group of algebraic cycles of codimension k on X modulo numerical equivalence.
Note that ^k(X) is a finitely generated free abelian group (see, e.g., <cit.>).
For any 0≤ k≤ n, we define the k-th numerical dynamical degree λ_k(f) of f as

    λ_k(f) lim_m→∞(f^m)^*|_^k(X)_^1/m,

where we fix an arbitrary norm · on the -vector space ^k(X)_^k(X) ⊗_.
The above limit defining λ_k(f) exists by <cit.>.
Note that when f is a self-morphism, the pullback action is compatible with the iteration, that is, (f^m)^*=(f^*)^m; it thus follows that χ_i(f)_ι = ρ(f^*|_H^i_(X, _ℓ))_ι and λ_k(f) = ρ(f^*|_^k(X)_), where ρ(φ) denotes the spectral radius of certain linear operator φ.

Inspired by results in complex dynamics (see <cit.> for a survey), in Esnault–Srinivas <cit.>, and Weil's Riemann hypothesis (now Deligne's theorem <cit.>), Truong proposed the following dynamical degree comparison conjecture (in fact, his version is formulated for more general dynamical correspondences which are natural generalizations of dominant rational maps).



Let f X X be a dominant rational self-map of a smooth projective variety X of dimension n over .
Then for any 0≤ k≤ n, we have χ_2k(f)_ι = λ_k(f).


As an application of Albert polynomials (see <ref>), we give an affirmative answer to <ref> for surjective self-morphisms of abelian varieties, extending the main result of <cit.> from the case k=1 to the general case.



Let f be a surjective self-morphism of an abelian variety X of dimension g over .
Then for any 0≤ k≤ g, we have χ_2k(f)_ι = λ_k(f), or equivalently,

    ρ(f^*|_H^2k_(X, _ℓ))_ι = ρ(f^*|_^k(X)_).



We conclude this section with some remarks on <ref>.



We first discuss some related work and known results on <ref>.


  * When ⊆, we may associate to (X, f) a projective (and hence compact Kähler) manifold X_ and a dominant meromorphic self-map f_. Then by Artin's comparison theorem and Hodge theory, it is not hard to show that χ_2k(f)_ι = λ_k(f) (see, e.g., <ref>); both of them agree with the usual dynamical degree defined using the Dolbeault cohomology group H^k,k(X_, ) in the context of complex dynamics (see <cit.>).


  * For an arbitrary algebraically closed field (in particular, of positive characteristic), Esnault and Srinivas <cit.> proved that for an automorphism of a smooth projective surface, the second cohomological dynamical degree coincides with the first numerical dynamical degree.

However, as far as we know, <ref> is still open even for polarized endomorphisms of K3 surfaces.
Recently, in joint work with Truong <cit.>, we verify this for polarized endomorphisms of Kummer surfaces using dynamical correspondences.


  * It turns out that <ref> is related to Weil's Riemann hypothesis.
Precisely, as we have seen in <ref>, if X = X_0 ×__q_q is the base change of a smooth projective variety X_0 over a finite field _q to _q and  is the Frobenius endomorphism of X relative to _q, then Deligne's celebrated theorem asserts that all eigenvalues of the pullback ^* on H^i_(X, _ℓ) are algebraic integers of modulus q^i/2.
In particular, we have that χ_i()_ι = q^i/2 for all i and ι.
On the other hand, it is easy to see that the k-th numerical dynamical degree λ_k() of  is equal to q^k (see <ref> for a more general treatment of polarized endomorphisms).
It follows that χ_2k()_ι = λ_k() for all k.

Conversely, using a standard product trick (i.e., consider the product morphism of the product variety), the Künneth formula, and Poincaré duality, one can show that <ref> (for X× X) implies Weil's Riemann hypothesis (for X; see <cit.>).
Recently, joint with Truong <cit.>, we systematically investigate the relations between various comparison conjectures on correspondences, Weil's Riemann hypothesis, and the standard conjectures.


  * When f is a surjective self-morphism of X, Truong <cit.> proved a weaker statement that

    h_(f) max_0≤ i≤ 2nlogχ_i(f)_ι = max_0≤ k≤ nlogλ_k(f)  h_(f),

which asserts that the (étale) entropy h_(f) coincides with the algebraic entropy h_(f) in the sense of <cit.>.
As a consequence, the spectral radius of the pullback f^* on the even-degree cohomology H^2∙_(X, _ℓ) coincides with the spectral radius of f^* on the total cohomology H^∙_(X, _ℓ).
See also <cit.> for a different approach towards this equality using dynamical zeta functions.
In particular, when X = 2, if one has λ_1(f) ≥λ_2(f) (which holds for automorphisms of surfaces as considered in <cit.>), then χ_2(f)_ι≤λ_1(f) and hence χ_2(f)_ι = λ_1(f); see <cit.>.
Note that when ⊆, by the fundamental work of Gromov <cit.> and Yomdin <cit.>, the algebraic entropy also equals the topological entropy h_top(f_) of the holomorphic dynamical system (X_, f_); see <cit.> for details.







  * Note that our <ref> implies Weil's Riemann hypothesis for abelian varieties (see <ref>(3) and <ref>).
Hence borrowing the ideas from dynamical systems, we extend a classical result of Weil <cit.>.
This also answers a http://aimpl.org/cohomabelian/1/question of Esnault raised in the AIM workshop https://aimath.org/pastworkshops/cohomabelian.htmlCohomological methods in abelian varieties, 2012.
For an automorphism f of an abelian variety X, she actually asked whether the spectral radius of f^*|_H^∙_(X, _ℓ) is achieved on the subring in H^∙_(X, _ℓ) generated by all (f^m)^*H_X with m∈.
Our <ref> first asserts that ρ(f^*|_H^∙_(X, _ℓ))_ι = ρ(f^*|_H^2k_(X, _ℓ))_ι for certain 0<k<g and then <ref> applies.


  * As mentioned before, when f is an automorphism of an abelian surface, <ref> is already known by Esnault and Srinivas (see <cit.>).
Even in this two-dimensional case, their proof is quite involved and different from ours (actually, after the standard spreading out and specialization argument, they invoke a celebrated theorem of Tate <cit.>).
Previously, the author <cit.> has dealt with the case of abelian varieties but only when k = 1.
In that case, we can naturally embed the Néron–Severi group (X) of an abelian variety X into the semisimple -algebra ^0(X) of endomorphisms (with -coefficients) and extend the pullback f^* on (X) to ^0(X).
Thanks to the structure theorem of ^0(X), the later action f^* on ^0(X) can be reinterpreted in terms of matrices related to the matrix representing the pullback f^* on H^1_(X, _ℓ).
However, since algebraic cycles of higher codimensions are much more subtle <cit.>, we are not aware of any embedding of ^k(X)_ into certain nice spaces for k≥ 2, on which the pullback f^* can be extended and easily understood.
The proof for general k presented in this paper is different from <cit.>.


  * If an endomorphism α X→ X is not surjective, one can also proceed by replacing X (possibly several times) by the image Yα(X), which is then a lower-dimensional abelian subvariety of X such that α|_Y Y→ Y is surjective and X is isogenous to Y× Z for some abelian subvariety Z of X.










§ PRELIMINARIES ON ABELIAN VARIETIES







We refer to <cit.> and <cit.> for standard notations and terminologies on abelian varieties.


The following notations remain in force throughout the rest of this paper unless otherwise stated.

p2cm p12cm
     an algebraically closed field of arbitrary characteristic 

ℓ     a prime different from the characteristic of  

X     an abelian variety of dimension g over  

X     the dual abelian variety ^0(X) of X 

^0(X)     (X) ⊗_, the endomorphism -algebra of X 

α     an endomorphism of X, or an element of ^0(X) 

^k(X)_     ^k(X) ⊗_, the -vector space of numerical equivalence classes of codimension-k real algebraic cycles on X 

H^i_(X, _ℓ)     H^i_(X, _ℓ) ⊗__ℓ_ℓ, the ℓ-adic étale cohomology group of degree i 

λ_k(α)     the k-th numerical dynamical degree of α 

χ_i(α)_ι     the i-th cohomological dynamical degree of α with respect to ι_ℓ 

P_α(t)     the characteristic polynomial of α, which has degree 2g and integer (resp. rational) coefficients if α∈(X) (resp. if α∈^0(X)) 

χ_α^(t)     the reduced characteristic polynomial of α 

P_α^(t)     an Albert polynomial of α, which has degree g and complex coefficients 

ϕ_     the induced homomorphism of a line bundle  on X: 

         ϕ_ X → X,   x ↦ t_x^*⊗^-1 

ϕ = ϕ_     the fixed polarization of X induced from some fixed ample line bundle = _X(H_X) 

^†     the Rosati involution on ^0(X) defined in the following way: 

         α↦α^†ϕ^-1∘α∘ϕ, for any α∈^0(X) 

     the standard quaternion algebra over  whose basis is { 1, , , }



For the convenience of the reader, we include several important structure theorems on the étale cohomology groups, the endomorphism algebras, and the Néron–Severi groups of abelian varieties.
We refer to <cit.> and <cit.> for details and to <cit.> for more on abelian varieties over finite fields.

First, the étale cohomology groups of abelian varieties are simple to describe.



Let X be an abelian variety of dimension g over , and let ℓ be a prime different from the characteristic of . Let T_ℓ(X) _n X()[ℓ^n] be the Tate module of X, which is a free _ℓ-module of rank 2g.


  (a) There is a canonical isomorphism

    H^1_(X, _ℓ)  T_ℓ(X)^∨__ℓ(T_ℓ(X), _ℓ).


  (b) The cup-product pairing induces isomorphisms

    H^i_(X, _ℓ) ⋀^ i H^1_(X, _ℓ),

for all i.
In particular, H^i_(X, _ℓ) is a free _ℓ-module of rank 2gi.



Furthermore, the functor T_ℓ induces an ℓ-adic representation of the endomorphism algebra. In general, we have:



For any two abelian varieties X and Y, the group (X,Y) of homomorphisms of X into Y is a finitely generated free abelian group, and the natural homomorphism of _ℓ-modules

    (X, Y) ⊗__ℓ→__ℓ(T_ℓ(X), T_ℓ(Y))

induced by T_ℓ(X, Y) →__ℓ(T_ℓ(X), T_ℓ(Y)) is injective.


For a homomorphism f  X → Y of abelian varieties, its degree f is defined to be the order of the kernel f, if it is finite, and 0 otherwise.
In particular, the degree of an isogeny is always a positive integer.
We can extend this notion to any α∈^0(X) by setting α = n^-2g(nα) if nα∈(X).



For any α∈^0(X), there exists a monic polynomial P_α(t) ∈[t] of degree 2g such that P_α(r) = (r_X - α) for all rational numbers r.
Moreover, if α∈(X), then P_α(t) has only integer coefficients; it is also equal to the characteristic polynomial (t  _2g - V_ℓ(α)) of the induced linear map V_ℓ(α) of α on the Tate space V_ℓ(X)  T_ℓ(X) ⊗__ℓ_ℓ.


We call P_α(t) as in <ref> the characteristic polynomial of α.
On the other hand, we can assign to each α the characteristic polynomial χ_α(t) of α as an element of the semisimple -algebra ^0(X).
Namely, we define χ_α(t) to be the characteristic polynomial of the left multiplication α_L β↦α∘β for β∈^0(X) which is a -linear transformation on ^0(X).
Note that the above definition of χ_α(t) makes no use of the fact that ^0(X) is semisimple.
Actually, for semisimple -algebras, it is much more useful to consider the so-called reduced characteristic polynomials.

We recall some basic definitions on semisimple algebras (see <cit.> for details).



Let R be a finite-dimensional semisimple algebra over a field F of characteristic zero, and write

    R = ⊕_i=1^k R_i,

where each R_i is a simple F-algebra.
For any element r ∈ R, as above, we denote by χ_r(t) the characteristic polynomial of r.
Namely, χ_r(t) is the characteristic polynomial of the left multiplication r_L  s ↦ rs for s∈ R.
Let K_i be the center of R_i.
Then there exists a finite field extension E_i/K_i splitting R_i (see <cit.>), i.e., we have

    h_i  R_i ⊗_K_i E_i _d_i(E_i),   where  [R_i:K_i] = d_i^2.

Write r = r_1 + ⋯ + r_k with each r_i ∈ R_i.
We first define the reduced characteristic polynomial χ_r_i^(t) of each r_i as follows (see <cit.>):

    χ_r_i^(t) _K_i/F ((t  _d_i - h_i(r_i ⊗_K_i 1_E_i)) ) ∈ F[t].

Note that (t  _d_i - h_i(r_i ⊗_K_i 1_E_i)) lies in K_i[t], and is independent of the choice of the splitting field E_i of R_i (see, e.g., <cit.>).
The reduced norm of r_i is defined by

    _R_i/F^(r_i) _K_i/F ( (h_i(r_i ⊗_K_i 1_E_i)) ) ∈ F.

Finally, as one expects, the reduced characteristic polynomial χ_r^(t) and the reduced norm _R/F^(r) of r=∑_i r_i are defined by the products:

    χ_r^(t) ∏_i=1^kχ_r_i^(t)   and  _R/F^(r) ∏_i=1^k_R_i/F^(r_i).







  * It follows from <cit.> that

    χ_r(t) = ∏_i=1^kχ_r_i(t) = ∏_i=1^kχ_r_i^(t)^d_i.



  * Note that reduced characteristic polynomials and norms are not affected by change of ground field (see, e.g., <cit.>).



We now apply the above algebraic set-up to R = ^0(X), the semisimple -algebra of endomorphisms (with -coefficients) of an abelian variety X over .
For any α∈^0(X), let χ_α^(t) ∈[t] denote its reduced characteristic polynomial.
For simplicity, let us first consider the case when

    X = A^n

is a power of a simple abelian variety A over .
Hence ^0(X) _n(D), where D^0(A) is a division ring.
Let K denote the center of D which is a field, and K_0 the maximal totally real subfield of K. Set

    d^2 = [D:K],  e = [K:],   and  e_0 = [K_0:].

Then the equality (<ref>) reads as

    χ_α(t) = χ_α^(t)^dn.


Let V_1, …, V_e denote the e nonisomorphic irreducible representations of ^0(X) over , where each one has degree dn.
Note that for any α∈^0(X), the reduced characteristic polynomial χ_α^(t) defined above is exactly the same as the characteristic polynomial of α acting on ⊕_i V_i.
We thus call V^⊕_i V_i the reduced representation of ^0(X).
The proposition below shows that the two characteristic polynomials P_α(t) and χ_α^(t) are closely related.



With notation as above, the representation T_ℓ(X) ⊗__ℓ_ℓ = V_ℓ(X) ⊗__ℓ_ℓ of ^0(X) induced from <ref> is isomorphic to a direct sum of m  2g/(edn) copies of V^⊗__ℓ.
In particular, for any α∈^0(X), we have

    [t] ∋ P_α(t) = χ_α^(t)^m.

Moreover, if α∈(X), then by Gauss's primitivity lemma, χ_α^(t) ∈[t].


It is not hard to extend <ref> to the case when X is an arbitrary abelian variety over  (which is not necessarily of the form A^n).
Indeed, by Poincaré's complete reducibility theorem (see, e.g., <cit.>), X is isogenous to a product X_1 ×⋯× X_s, where the X_j = A_j^n_j are powers of mutually non-isogenous simple abelian varieties A_j, and

    ^0(X) ∏_j=1^s^0(X_j) ∏_j=1^s_n_j(^0(A_j)).

Write the image of α∈^0(X) as the product α_1 ×⋯×α_s with each α_j ∈^0(X_j).
Let V_j,1, …, V_j,e_j denote the e_j nonisomorphic irreducible representations of ^0(X_j) over , where each one has degree d_jn_j.
Denote by V_j^ V_j,1⊕⋯⊕ V_j,e_j the reduced representation of ^0(X_j).



With notation as above, the representation T_ℓ(X) ⊗__ℓ_ℓ = V_ℓ(X) ⊗__ℓ_ℓ of ^0(X) is isomorphic to a direct sum of the m_j copies of V_j^⊗__ℓ as follows:

    V_ℓ(X) ⊗__ℓ_ℓ⊕_j=1^s   (V_j^)^⊕ m_j⊗__ℓ,

where m_j = 2 X_j /(e_jd_jn_j) is a positive integer depending only on X_j (and ^0(X_j); see <ref>).
In particular, for any α∈^0(X), we have

    [t] ∋ P_α(t) = ∏_j=1^s P_α_j(t) = ∏_j=1^s χ_α_j^(t)^m_j.

Moreover, if α∈(X), then χ_α_j^(t) ∈[t] for all j.


We recall the following useful structure theorem on the endomorphism -algebras of abelian varieties which would be used very frequently.



The endomorphism -algebra (X)_(X) ⊗_ is isomorphic to a product of copies of _r(), _r(), or _r().
Moreover, one can fix an isomorphism so that it carries the Rosati involution into the standard involution ↦^* = ^.
In particular, the real Néron–Severi space (X)_(X) ⊗_ is isomorphic to a product of Jordan algebras of the following types:

    _r()    = r × r  symmetric real matrices,
    _r()    = r × r  Hermitian complex matrices,
    _r()    = r × r  Hermitian quaternionic matrices.



For convenience, we include the following <Ref>, which gives the numerical constraints on the endomorphism -algebras of simple abelian varieties A; see, e.g., <cit.>.
Recall that e, d, and e_0 are positive integers defined in <ref>.








§ PROOF OF THEOREM <REF>








By Poincaré's complete reducibility theorem, it suffices to consider the case when X is a power of a simple abelian variety A, say X = A^n, and α∈^0(X).
Indeed, with notation as in <ref>, suppose that for each α_j ∈^0(X_j) we have found a complex polynomial P_α_j^(t) such that P_α_j(t) = P_α_j^(t)  P_α_j^(t).
Note that here the characteristic polynomial P_α_j(t) of each α_j ∈^0(X_j) is defined as k_j^-2 X_jP_k_j α_j(k_j t) if k_j α_j∈(X_j) (see <cit.>).
Then we simply take ∏_j=1^s P_α_j^(t) as our P_α^(t).
Now, after the above reduction, ^0(X) is isomorphic to the simple -algebra _n(D) of all n× n matrices with entries in the division ring D ^0(A).
Let K denote the center of D which is a field, and K_0 the maximal totally real subfield of K.
As usual, we set

    d^2 = [D:K],  e = [K:],   and  e_0 = [K_0:].

Let χ_α^(t) ∈[t] be the reduced characteristic polynomial of α (see <ref>).
Then by <ref>, we have

    P_α(t) = χ_α^(t)^m,

where m = 2g / edn = 2  A / ed is a positive integer.
We also note that

    (X)_^0(X) ⊗__n(D) ⊗__n(D ⊗_)

is isomorphic to a product of copies of _r(), _r(), or _r() (see <ref>).
According to Albert's classification of the endomorphism -algebras of simple abelian varieties (see, e.g., <cit.>), we have the following four cases.



D is of Type I(e): d=1, e=e_0 and D=K=K_0 is a totally real algebraic number field and the involution on D is the identity.
In this case, we have the following restriction:

    e   |  A

in any characteristic (see <Ref>).
It follows that the m in the equation (<ref>) is an even number.
Since χ_α^(t) lies in [t] by the definition of reduced characteristic polynomials, we simply take

    P_α^(t) = χ_α^(t)^m/2.





D is of Type II(e): d=2, e=e_0, K=K_0 is a totally real algebraic number field and D is an indefinite quaternion division algebra over K.
In this case, we have 2e   |  A in any characteristic (see <Ref>) so that m is also even.
We can take the same P_α^(t) as in <ref> because χ_α^(t) has only rational coefficients.




D is of Type III(e): d=2, e=e_0, K=K_0 is a totally real algebraic number field and D is a definite quaternion division algebra over K.
In this case, we have 2e   |  A in characteristic zero and e   |  A in positive characteristic (see <Ref>).
The characteristic zero case is exactly the same as before.
However, when the characteristic of the ground field  is positive, the restriction e   |  A does not automatically guarantee the parity of m as in the previous cases (e.g., it may happen that e =  A and hence m=1; see <ref>).
Nevertheless, we shall construct P_α^(t) using certain special property of quaternion algebras.

First, we have the following isomorphism

    (X)_⊕_i=1^e_n(),

where (-1,   -1/) is the standard quaternion algebra over .
Clearly,  can be embedded, in a standard way (see, e.g., <cit.>), into _2() ⊗_.
This would induce a natural embedding of _n() into _2n() _n() ⊗_ as follows (see, e.g., <cit.>):

    ι_n() _2n()   via   = _1 + _2  ⟼ι () [  _1  _2; -_2  _1 ].

In particular, a quaternionic matrix  is Hermitian if and only if its image ι() is a Hermitian complex matrix.

Denote the image α⊗_ 1_ of α in (X)_ by block diagonal matrix _α = _α, 1⊕⋯⊕_α, e with each _α, i∈_n().
Also, we note that

    (X)_(X)_⊗_⊕_i=1^e_2n(),

is a semisimple -algebra with each summand _2n() being a central simple -algebra.
Then by <ref>, the reduced characteristic polynomial χ_α^(t) of α is equal to the product of the characteristic polynomials (t  _2n - ι(_α, i)) of ι(_α, i).
Thanks to <cit.>, the 2n complex eigenvalues of each ι(_α, i) fall into n pairs with each pair consisting of two conjugate complex numbers;
regardless the multiplicity, denote them by π_i,1, …, π_i,n, π_i,1, …, π_i,n.
In fact, one can easily verify that if π_i,j∈ is an eigenvalue of ι(_α,i) so that

    ι(_α,i) [ _i,j; _i,j ]
    = π_i,j[ _i,j; _i,j ],
     then ι(_α,i) [ -_i,j;  _i,j ]
    = π_i,j[ -_i,j;  _i,j ],

i.e., π_i,j is also an eigenvalue of ι(_α,i) corresponding to the eigenvector (-_i,j^, _i,j^)^.
This yields that

    χ_α^(t) = ∏_i=1^e∏_j=1^n (t - π_i,j)(t - π_i,j).


Clearly, by the equation (<ref>),

    P_α^(t) ∏_i=1^e∏_j=1^n (t - π_i,j)^m ∈[t]

is what we want, though the choice of π_i,j or π_i,j may not be canonical (see <ref>).




D is of Type IV(e_0, d): e=2e_0 and D is a division algebra over the CM-field K ⊋ K_0
(i.e., K is a totally imaginary quadratic extension of a totally real algebraic number field K_0).
In this case, neither the restriction e_0d^2   |  A in characteristic zero nor the restriction e_0d   |  A in characteristic p ensures the parity of the integer m as in <ref>.
However, this last remaining case is also special enough so that the reduced characteristic polynomial of α is canonically equal to the product of a complex polynomial and its complex conjugate (bearing some similarity with <ref>).

In fact, in this case, the endomorphism -algebra

    (X)_⊕_i=1^e_0_dn(),

so that the image α⊗_ 1_ of α in (X)_ could be represented by the block diagonal matrix _α = _α, 1⊕⋯⊕_α, e_0 with each _α, i∈_dn().
We now need to note that (X)_ is a semisimple -algebra, while the center of each component _dn() is .
Then by <ref>, the reduced characteristic polynomial χ_α^(t) of α is equal to

    ∏_i=1^e_0_/ ((t  _dn - _α, i) ) = ∏_i=1^e_0(t  _dn - _α, i) ·(t  _dn - _α, i).

By the equation (<ref>) again, we now just take

    P_α^(t) = ∏_i=1^e_0 ((t  _dn - _α, i) )^m.



We thus complete the proof of <ref>.




It follows from the proof of <ref> that if all factors X_j of X are of Type I(e) or II(e) then the Albert polynomial P_α^(t) of a genuine endomorphism α∈(X) has integer coefficients.
In fact, to reduce potential inaccuracies, let us start with <ref> from the beginning.
So for each factor X_j of Type I(e) or II(e), we have shown that

    P_α_j^(t) = χ_α_j^(t)^m_j/2∈[t]

is monic, where m_j is an appropriate positive even integer depending only on X_j.
Now, the Albert polynomial P_α^(t) of α∈(X) constructed in <ref> is just the product

    ∏_j=1^s P_α_j^(t),

so is a monic polynomial with rational coefficients.
Note, however, that P_α^(t)^2 = P_α(t) ∈[t].
This yields that all P_α_j^(t) and hence P_α^(t) itself have only integer coefficients either by Gauss's primitivity lemma or by noting that all roots are algebraic integers.






  * Our construction of an Albert polynomial P_α^(t) is canonical in characteristic zero, but unfortunately, not canonical in positive characteristic once the endomorphism -algebra of Type III(e) occurs,
in which case our <ref> of supersingular elliptic curves reveals that the canonical construction of an Albert polynomial seems not likely to exist.
This seems to be a big difference between characteristic zero and prime characteristic.
We also hope that there exists an intrinsic and classification-free construction.
Nevertheless, it turns out that for a symmetric endomorphism α, its Albert polynomial is unique (see <ref>).


  * It could happen that P_α(t) has no multiple roots at all.
Indeed, let X be an abelian variety of dimension g over a finite field _q, and  the Frobenius endomorphism of X relative to _q.
Then according to <cit.>, the characteristic polynomial P_ (t) of  has no multiple root if and only if ^0(X) = [] is commutative.
For instance, we can choose X as a simple CM abelian variety of Type IV(g, 1), i.e., ^0(X) is a CM-field of degree 2g over .
See also <cit.> for a discussion on the realization.



The example below indicates that in positive characteristic the canonical construction of an Albert polynomial does not seem likely to exist in <ref>.
Note that by <cit.>, if a simple abelian variety A of dimension g is of Type III(g), it must be a supersingular elliptic curve, i.e., g = 1.
See also <cit.> for a discussion about elliptic curves in positive characteristic.



Let E be a supersingular elliptic curve over an algebraically closed field of positive characteristic, i.e., ^0(E) is of Type III(1).
Equivalently, (E) ⊗_ is the standard quaternion algebra  over , whose basis is { 1, , , }.
Let α be an endomorphism of E such that α⊗_ 1_ = a + b   + c   + d  ∈.
Clearly, the characteristic polynomial P_α(t) of α and the reduced characteristic polynomial χ_α^(t) of α coincide.
Both of them are equal to

    t^2 - 2at + (a^2+b^2+c^2+d^2) = (t - π_1)(t - π_1) ∈[t].

Hence either t - π_1 or its complex conjugate is an Albert polynomial of α by <ref>.
In other words, there seems no way to obtain a canonical choice for P_α^(t) if b^2+c^2+d^2  0.







§ PROOF OF COROLLARIES <REF> AND <REF>







Let (X) denote the Albanese variety of X, which is an abelian variety over  of dimension equal to the geometric irregularity q(X)  b_1(X)/2.
Let _X denote the Albanese morphism from X to (X).
Then there exists an induced morphism a_f (X) →(X) such that _X is f-equivariant, i.e., _X ∘ f = a_f ∘_X.
Note that the Albanese morphism _X also induces an isomorphism

    _X^*  H^1_((X), _ℓ)  H^1_(X, _ℓ)

between the first étale cohomology groups via the pullback (see, e.g., <cit.>).
We thus have the following commutative diagram of isomorphic _ℓ-vector spaces:

    H^1_((X), _ℓ) [rr]^a_f^*[d]__X^*^       H^1_((X), _ℓ) [d]^_X^*_
    
    H^1_(X, _ℓ) [rr]^f^*       H^1_(X, _ℓ).

This yields that

    f^*|_H^1_(X, _ℓ)∼ a_f^*|_H^1_((X), _ℓ),

where ∼ means that these two linear transformations are similar.
Hence both have the same characteristic polynomial satisfying the required property by <ref>.
The case i=2n-1 follows readily from Poincaré duality.


Before proving the next corollary, we prepare a lemma on Albert polynomials of symmetric endomorphisms of abelian varieties with respect to the fixed polarization ϕ = ϕ_ X → X.



Let α∈^0(X).
Suppose that α is symmetric under the Rosati involution, i.e., α^† = α.
Then its Albert polynomial P_α^(t) ∈[t] is unique.


The lemma basically follows from <ref> which asserts the existence of P_α^(t) and the fact that a symmetric/Hermitian matrix has only real eigenvalues.
More precisely, according to <ref>, we could denote the corresponding block diagonal matrix of α⊗_ 1_ by _α = ⊕_i _α,i, where each _α,i is either a symmetric real matrix, a Hermitian complex matrix, or a Hermitian quaternionic matrix.
In the first two cases, we know that all eigenvalues of _α,i are real numbers.
For the last case, if _α,i∈_r(), then the natural embedding ι(_α,i) of _α,i in _2r() is a Hermitian complex matrix and hence has only real eigenvalues as well (see, e.g., <cit.>).
Therefore, by <ref>, all complex roots of the characteristic polynomial P_α(t) are real.
It follows that our Albert polynomial P_α^(t) constructed in <ref> lies in [t].
The uniqueness of P_α^(t) thus follows readily.



Thanks to <ref>, P_α^†∘α^(t) ∈[t] is unique because α^†∘α = (α^†∘α)^† is symmetric.
Clearly, by the definition of the Albert polynomial (see <ref>), we have P_α^†∘α(t) = P_α^†∘α^(t)^2.
Note that P_α^†∘α(t) itself lies in [t].
Then one can easily deduce that P_α^†∘α^(t) ∈[t] by the induction and comparing the coefficients.
Indeed, the constant term P_α^†∘α(0) =  (α^†∘α) = (α)^2 is a square (see <ref>), which yields that the constant term of P_α^†∘α^(t) is ±α∈.
The first part of the corollary follows (though one can also see rationality of coefficients from below).

For the second part, note that the characteristic polynomial P_α^†∘α(t) of α^†∘α is now just the square of its Albert polynomial P_α^†∘α^(t).
As usual, we denote the Euler characteristic of a coherent sheaf  of _X-modules by χ() ∑_i=0^g (-1)^i _ H^i(X, ).
Then by the Riemann–Roch theorem (see, e.g., <cit.>), we have

    χ(^n ⊗α^* ^-1)^2    = ϕ_^n ⊗α^* ^-1 =  (n  ϕ_ - ϕ_α^* ) 
       =  (n  ϕ_ - α∘ϕ_∘α) 
       = (n  ϕ_ - ϕ_∘α^†∘α) 
       = ϕ_·(n_X - α^†∘α) 
       = χ()^2   P_α^†∘α(n) 
       = (χ()   P_α^†∘α^(n))^2.

For the equality ϕ_α^*  = α∘ϕ_∘α, see <cit.>;
for the second-last equality, see also <ref>.
Note that _X(H_X) is an ample line bundle on X.
Hence, for sufficiently large n, the line bundle ^n ⊗α^* ^-1 is still ample.
It follows that χ(^n ⊗α^* ^-1) coincides with χ()   P_α^†∘α^(n) as polynomials in n since P_α^†∘α^(t) (∈[t]) is monic.
Applying the Riemann–Roch theorem again, we have

    P_α^†∘α^(n)    =  χ(^n ⊗α^* ^-1)  /χ() =   (nH_X - α^* H_X)^g/g!  /H_X^g/g!
       = ∑_k=0^g (-1)^k gk α^* H_X^k · H_X^g-k /H_X^g  n^g-k.

The description about c_k follows by comparing the coefficients.







§ PROOF OF THEOREM <REF>






The following lemma should be well known.
We include its proof here for the convenience of the reader.
It says that <ref> has an affirmative answer for complex varieties.



Let f X X be a dominant rational self-map of a smooth complex projective variety X of dimension n.
Then for any 0 ≤ k ≤ n, we have χ_2k(f)_ι = λ_k(f).


By Artin's comparison theorem, in the definition of the i-th cohomological dynamical degree χ_i we can replace the étale cohomology H^i_(X, _ℓ) by the singular/Betti cohomology and hence the de Rham cohomology H^i_(X, ).
It is well-known that the last one admits the following Hodge decomposition

    H^i_(X, ) = ⊕_p+q=i H^p,q(X, ),

where H^p,q(X, )  H^q(X, Ω_X^p).
Let d_k(f) denote the usual k-th dynamical degree of f in the context of complex dynamics, i.e.,

    d_k(f)    lim sup_m→∞(f^m)^*|_H^k,k(X, )^1/m
       = lim_m→∞( (f^m)^* ω_X^k ·ω_X^n-k)^1/m,

where ω_X is a Kähler class on X (see <cit.> or <cit.>).
Note that the above d_k(f) does not depend on the choice of ω_X since the Kähler cone is open.
On the other hand, it is known that the k-th numerical dynamical degree λ_k(f) of f is also equal to

    lim_m→∞( (f^m)^* H_X^k · H_X^n-k)^1/m,

where H_X is an ample divisor on X (see <cit.>).
By the openness of the ample cone, our λ_k(f) is independent of the choice of H_X too.
This yields that λ_k(f) = d_k(f) for any k.

It remains to show the equality d_k(f) = χ_2k(f).
First, by the Hodge decomposition, we have

    χ_2k(f) = max_p+q=2k r_p,q(f) ≥ r_k,k(f) = d_k(f),

where

    r_p,q(f) lim sup_m→∞(f^m)^*|_H^p,q(X, )^1/m.

On the other hand, Dinh's inequality <cit.> (see also <cit.>) asserts that r_p,q(f) ≤√(d_p(f) d_q(f)).
We thus have

    χ_2k(f) ≤max_p+q=2k√(d_p(f) d_q(f))≤ d_k(f).

The last inequality follows from the log-concavity property of dynamical degrees: the function k ↦log d_k(f) is concave in k (see, e.g., <cit.>).
Equivalently, we have

    d_k-1(f)d_k+1(f)≤ d_k(f)^2,  for any  1≤ k≤ n-1.

We thus prove <ref>.




Let X be a smooth projective variety of dimension n over .
Let f be a polarized endomorphism of X, i.e., f^*H_X ∼_ q H_X for some ample divisor H_X on X and an integer q > 0.
Then for any 0≤ k ≤ n, the pullback f^* on ^k(X)_ is diagonalizable with all eigenvalues of modulus q^k;
in particular, the k-th numerical dynamical degree λ_k(f) = q^k.


Let ^k(X) denote the nef cone in ^k(X)_ consisting of nef classes of codimension k on X, which is a salient closed convex cone of full dimension preserved by the pullback f^* (see <cit.> or <cit.>).
Note that the k-th self-intersection H^k of the ample divisor H is an interior point in ^k(X) satisfying that f^*H^k = q^k H^k in ^k(X)_.
The lemma thus follows by applying <cit.> to the nef cone ^k(X) ⊂^k(X)_.




In the above lemma, it would be a challenge to show that all eigenvalues of the pullback f^* on H^i_(X, _ℓ) have absolute value q^i/2.
When ⊆, this is true by a result due to Serre <cit.> using Hodge theory.
In positive characteristic, it is a consequence of the standard conjectures (in particular, of Lefschetz type and of Hodge type; see, e.g., <cit.>, respectively).
See <cit.> for more details.
Without assuming the standard conjectures, to the best of our knowledge, it is only known in the following cases:


  * X is a curve or an abelian variety due to Weil's pioneering work <cit.>;

  * X = X_0 ×__q_q for some X_0 over _q and f= is the Frobenius endomorphism of X relative to _q by Deligne <cit.>.

Note that it is still open even for arbitrary polarized endomorphisms of (K3) surfaces in positive characteristic.
Recently, in joint work with Truong <cit.>, we verify this for Kummer surfaces using dynamical correspondences.


Towards the proof of <ref> we still need one more lemma below, which relates the asymptotic behavior of the singular values of the power matrix ^m to the spectrum of .
See also <cit.> for a different proof using compound matrices.
Recall that the singular values of a square matrix ∈_n() are the square roots of the eigenvalues of ^* (or equivalently, ^*), where ^* is the Hermitian transpose of .
As a convention, we always denote by σ_i() the i-th largest singular value of a general complex matrix .



Let ∈_n(), whose eigenvalues are π_1, …, π_n ∈ so that |π_1| ≥⋯≥ |π_n|.
For each m∈, let σ_1(^m) ≥⋯≥σ_n(^m) denote the singular values of ^m.
Then for any 1≤ i ≤ n, we have

    lim_m→∞σ_i(^m)^1/m = |π_i|.



It suffices to consider the case when  is an upper triangular matrix.
Indeed, let = ^* be the Schur decomposition of  (see, e.g., <cit.>), where  is unitary and  is upper triangular with diagonal entries π_1, …, π_n.
Then ^m = ^m ^* so that ^m (^m)^* = ^m (^m)^* ^* for any m ∈.
Hence, σ_i(^m) = σ_i(^m) for any i.
We now assume that = is upper triangular.
Without loss of generality, we may assume further that π_n  0, i.e.,  is non-singular, since otherwise we would easily have σ_n(^m) = 0.

We first prove that lim_m→∞σ_i(^m)^1/m = |π_i| for i = 1 and n.
Note that σ_1(^m) is equal to the 2-norm (also known as the spectral norm) ^m_2 of ^m (see, e.g., <cit.>).
Thus the well-known spectral radius formula asserts that

    lim_m→∞σ_1(^m)^1/m = lim_m→∞^m_2^1/m = ρ() = |π_1|.

On the other hand, we can repeat the above argument to the inverse ^-m of ^m.
More precisely, we note that the largest eigenvalue of (^-m)^* ^-m, as the inverse of ^m (^m)^*, is just σ_n(^m)^-2.
In other words, the largest singular value of ^-m, which coincides with the 2-norm ^-m_2 of ^-m, is equal to σ_n(^m)^-1.
It follows that

    lim_m→∞σ_n(^m)^1/m = lim_m→∞^-m_2^-1/m = ρ(^-1)^-1 = |π_n|.


In particular, the lemma has been proved when n = 2.
We shall prove the general case by induction on the matrix size n.
Let n ≥ 3.
Suppose that the lemma holds for any matrix in _n-1().
By the preceding discussion, it remains to consider the intermediate case i=2, …, n-1.
First of all, let us fix the following notation: for a general matrix , we use _k to denote the principal submatrix of  obtained by removing both k-th row and k-th column from .
Therefore, we can rewrite  as follows:

    = 
    [ π_1 _1^;   0  _1 ]  or  [  _n  _n;   0 π_n ],

where _1 (resp. _n) is an upper triangular matrix with π_2, …, π_n (resp. π_1, …, π_n-1) on the diagonal.
It is easy to verify that _1^m = (^m)_1 and _n^m = (^m)_n for any m, since  is upper triangular.
As usual, let σ_1(_1^m) ≥⋯≥σ_n-1(_1^m) and σ_1(_n^m) ≥⋯≥σ_n-1(_n^m) denote the singular values of _1^m and _n^m, respectively.
Now, for each m ∈, applying Cauchy's interlacing theorem to the principal submatrix _1^m of ^m (see, e.g., <cit.>), we get

    σ_1(^m) ≥σ_1(_1^m) ≥σ_2(^m) ≥σ_2(_1^m) ≥⋯≥σ_n-1(^m) ≥σ_n-1(_1^m) ≥σ_n(^m).

The induction hypothesis asserts that lim_m→∞σ_i(_1^m)^1/m = |π_i+1| for any 1 ≤ i ≤ n-1.
Taking the limsup of the m-th roots of the above sequences, we thus obtain that

    |π_2| ≥lim sup_m→∞σ_2(^m)^1/m≥ |π_3| ≥⋯≥lim sup_m→∞σ_n-1(^m)^1/m≥ |π_n|.

Similarly, the same argument works for _n^m, which yields the following inequalities

    σ_1(^m) ≥σ_1(_n^m) ≥σ_2(^m) ≥σ_2(_n^m) ≥⋯≥σ_n-1(^m) ≥σ_n-1(_n^m) ≥σ_n(^m).

It thus follows from the induction hypothesis that

    |π_1| ≥lim inf_m→∞σ_2(^m)^1/m≥ |π_2| ≥⋯≥lim inf_m→∞σ_n-1(^m)^1/m≥ |π_n-1|.

Combining inequalities (<ref>) and (<ref>), we have shown that lim_m→∞σ_i(^m)^1/m = |π_i| for all intermediate i = 2, …, n-1.
This concludes the proof of <ref>.


We are now ready to prove our main theorem on the comparison of the cohomological dynamical degrees with the numerical ones on abelian varieties, extending the main result of <cit.>.


It suffices to consider the case when f is a surjective endomorphism of X.
Indeed, any morphism between abelian varieties is a composite of a homomorphism with a translation (see, e.g., <cit.>).
Hence we can write f as t_a ∘α for a surjective endomorphism α∈(X) and a translation t_a for some a∈ X().
Note however that t_a acts as identity on H^1_(X, _ℓ) and hence on H^i_(X, _ℓ) for all i.
It follows from the functoriality of the pullback map on ℓ-adic étale cohomology that χ_i(f) = χ_i(α).
Similarly, we also get that λ_k(f) = λ_k(α) for all k.
So from now on, our f = α∈(X) is an isogeny.

Let P_α(t) ∈[t] be the characteristic polynomial of α.
By <ref>, we can denote its 2g complex roots by π_1, …, π_g, π_1, …, π_g.
Without loss of generality, we may assume that

    |π_1| ≥⋯≥ |π_g| > 0.

It thus follows from <ref> and the spectral radius formula that the 2k-th cohomological dynamical degree of α is

    χ_2k(α)_ι = ∏_i=1^k |π_i|^2.


We shall use <ref> to compute λ_k(α) on the other side.
For each m ∈, let us first consider the characteristic polynomial P_(α^m)^†∘α^m(t) of the symmetric element (α^m)^†∘α^m.
In virtue of <ref>, we have

    P_α(t) = ∏_j=1^s χ_α_j^(t)^m_j  and   P_(α^m)^†∘α^m(t) = ∏_j=1^s χ_(α_j^m)^†∘α_j^m^(t)^m_j.

Note that by the definition, for each j, the reduced characteristic polynomial χ_α_j^(t) is nothing but the characteristic polynomial of the corresponding matrix _α_j of α_j ⊗_ 1_∈(X_j)_;
it is also the characteristic polynomial of α_j acting on the reduced representation V_j^.
Similarly, χ_(α_j^m)^†∘α_j^m^(t) is the characteristic polynomial of the Hermitian matrix _(α_j^m)^†∘α_j^m = (_α_j^m)^*_α_j^m.
In particular, apart from multiplicities, the π_i coincide with the eigenvalues of _α = ⊕_j _α_j,
and the roots of P_(α^m)^†∘α^m(t) the eigenvalues of (_α^m)^*_α^m (i.e., the squares of the singular values of _α^m).
Thus, without loss of generality, for each m ∈, we can denote the 2g real roots of P_(α^m)^†∘α^m(t) by
σ_1(α^m)^2, …, σ_g(α^m)^2, σ_1(α^m)^2, …, σ_g(α^m)^2, where

    σ_1(α^m) ≥⋯≥σ_g(α^m) > 0

coincide with the singular values of _α^m (apart from multiplicities).

However, we note that for each j, the multiplicity m_j depends only on X_j (or rather, X_j and ^0(X_j), but not the index m).
It thus follows from <ref> that for each 1 ≤ i ≤ g,

    lim_m→∞σ_i(α^m)^1/m = |π_i|.

According to <ref>, the Albert polynomial of (α^m)^†∘α^m can be written as

    P_(α^m)^†∘α^m^(t) = ∏_i=1^g (t - σ_i(α^m)^2 ).

Hence applying <ref> to α^m yields that

    gk  (α^m)^* H_X^k · H_X^g-k /H_X^g = e_k ( σ_1(α^m)^2, …, σ_g(α^m)^2 ),

where e_k is the k-th elementary symmetric polynomial.
Note that by <cit.>, the k-th numerical dynamical degree λ_k(α) of α can be reinterpreted by the formula

    λ_k(α) = lim_m→∞( (α^m)^* H_X^k · H_X^g-k)^1/m;

see also <cit.> and <cit.> for more general cases of dominant rational maps and dominant correspondences, respectively.

Putting together all <ref>, we have that

    λ_k(α)    = lim_m→∞( e_k( σ_1(α^m)^2, …, σ_g(α^m)^2 ) )^1/m
       = max_1≤ i_1 < … < i_k ≤ g lim_m→∞σ_i_1(α^m)^2/m⋯lim_m→∞σ_i_k(α^m)^2/m
       = max_1≤ i_1 < … < i_k ≤ g  |π_i_1|^2 · |π_i_2|^2 ⋯ |π_i_k |^2 
       = |π_1|^2 · |π_2|^2 ⋯ |π_k|^2 
       = χ_2k(α)_ι.

We thus complete the proof of <ref>.




With notation as above, one can easily verify that if i = 2k is even, then 

    χ_i(α)_ι^2 = ∏_j=1^k |π_j|^4 = λ_k(α)^2 = max_p+q=2kλ_p(α) λ_q(α);

and if i = 2k - 1 is odd, then

    χ_i(α)_ι^2 = |π_k|^2 ·∏_j=1^k-1 |π_j|^4= λ_k-1(α) λ_k(α) = max_p+q=2k-1λ_p(α) λ_q(α).

This yields an analog of Dinh's inequality and hence answers <cit.> in the case of abelian varieties.


In the end, as a by-product, we deduce the following norm comparison for self-morphisms of abelian varieties, which motivates the so-called norm comparison conjecture considered in <cit.>.
Note that by the triangle inequality, the same also holds for any correspondence f∑_i=1^n a_i Γ_f_i, where a_i∈_>0 and Γ_f_i is the graph of a self-morphism f_i of X.



Let X be an abelian variety of dimension g over .
Then there exists a positive constant C>0 such that for any 0≤ k≤ g and for any self-morphism f of X, we have

    f^*|_H^2k_(X, _ℓ)_ι≤ C  f^*|_^k(X)_.



It suffices to consider the case when f = α is an endomorphism of X.
Also, by the equivalence of norms on the finite-dimensional vector spaces, we are free to choose any norms.
It is well-known that the spectral norm α^*|_^k(X)__2 of α^*|_^k(X)_ is equal to σ_1(α^*|_^k(X)_), the largest singular value of α^*|_^k(X)_.
The latter turns out to be the square root of the spectral radius ρ((α^†∘α)^*|_^k(X)_),
since (α^†)^*|_^k(X)_ is represented by the transpose of α^*|_^k(X)_ which is defined over .
It thus follows from <ref> that

    α^*|_^k(X)__2 = ρ( (α^†∘α)^*|_^k(X)_ )^1/2 = ρ( (α^†∘α)^*|_H^2k_(X, _ℓ) )^1/2.


We shall prove that the right-hand side of <ref> also gives a norm of α^*|_H^2k_(X, _ℓ).
Notice that by <ref>, one has

    α^*|_H^2k_(X, _ℓ) = ⋀^ 2kα^*|_H^1_(X, _ℓ)⋀^ 2k V_ℓ(α)^∨ |_V_ℓ(X)^∨.

On the other hand, by <ref>, we have the following decomposition of V_ℓ(X) ⊗__ℓ_ℓ as a representation of ^0(X) (for brevity, here we suppress the multiplicity m_j):

    V_ℓ(X) ⊗__ℓ_ℓ⊕_j V_j^⊗__ℓ,

which yields that

    V_ℓ(α) |_V_ℓ(X) ⊗__ℓ_ℓα |_⊕_j V_j^⊗__ℓ⊕_jα_j |_V_j^⊗__ℓ.

For each j, we know that α_j|_V_j^ is represented by a matrix _α_j∈_e_jd_jn_j(); further, viewed as a complex matrix, _α_j is nothing but the corresponding matrix of α_j ⊗_ 1_∈(X_j)_.
Then we see from <ref> that α_j^†|_V_j^ is represented by the Hermitian transpose _α_j^* of _α_j.
It follows that the linear maps α^*|_H^2k_(X, _ℓ) and (α^†∘α)^*|_H^2k_(X, _ℓ) could be represented by the complex matrices

    ⋀^ 2k⊕_j_α_j^    ⋀^ 2k⊕_j_α_j_α_j^ = (⋀^ 2k⊕_j_α_j^)^* ·(⋀^ 2k⊕_j_α_j^),

respectively.
This thus yields that the spectral radius of (α^†∘α)^*|_H^2k_(X, _ℓ) is just the square of the largest singular value of α^*|_H^2k_(X, _ℓ).
In other words, we show that

    ρ( (α^†∘α)^*|_H^2k_(X, _ℓ) )^1/2 = σ_1(α^*|_H^2k_(X, _ℓ)) = α^*|_H^2k_(X, _ℓ)_2.

This concludes the proof of <ref>.



It is worth mentioning that in <cit.>, Zarhin proves that for an endomorphism α of an abelian variety X of dimension g over , the matrix representing V_ℓ(α)|_V_ℓ(X) (or equivalently, α^*|_H^1_(X, _ℓ)), lies in _2g(), under an appropriate basis of V_ℓ(X).
Later, Poonen and Rybakov <cit.> refine his result to integer matrices.
Hence the above 2-norm argument applies to the pullback action on étale cohomology as well, which immediately shows that the right-hand side of <ref> gives a norm of α^*|_H^2k_(X, _ℓ).






tocsectionAcknowledgments
Acknowledgments. 
I am greatly obliged to Dragos Ghioca and Zinovy Reichstein for their support during my stay at the University of British Columbia.
I would like to thank Hélène Esnault, Nathan Grieve, Bruno Kahn, Nicholas Katz, Zinovy Reichstein, Tuyen Truong, Yuri Zarhin, Yishu Zeng, and De-Qi Zhang for stimulating discussions and valuable suggestions.
I would also like to thank Pierre Deligne for kindly explaining to me that our <ref> is actually a consequence of the standard conjectures.
My special thanks goes to Tuyen Truong for helpful comments and for pointing out several inaccuracies in an earlier version.
Finally, I am grateful to the referee for his/her many helpful and invaluable suggestions.








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