WHITEHEAD’S LEMMAS AND GALOIS COHOMOLOGY
OF ABELIAN VARIETIES
MICHAEL LARSEN AND RENÉ SCHOOF
Abstract. Whitehead’s lemmas for Lie algebra cohomology translate
into vanishing theorems for H 1 and H 2 in Galois cohomology. Via
inflation-restriction, the H 1 vanishing theorem leads to a simple formula for H 1 (K, T` ), where T` is the `-adic Tate module of an abelian
variety over a number field K. We apply this formula to the “support
problem” for abelian varieties. Under a suitable semisimplicity hypothesis for the Tate modules, we can give a refined version of the theorem
of [6].
1. Introduction
Let A be an abelian variety over a number field K and P and Q K-rational
points of A. By inverting a suitable element in the ring of integers of K,
one can always find a Dedekind domain O with fraction field K such that
A extends to an abelian scheme A over O and P and Q extend to O-points
of A. Therefore, one can speak of reducing P and Q (mod p) for almost
all (i.e., all but finitely many) primes p. In [2], C. Corrales-Rodrigáñez and
R. Schoof proved that when dim A = 1, the condition
(1.1)
nP ≡ 0
(mod p) ⇒ nQ ≡ 0
(mod p)
for all integers n and almost all prime ideals p implies
(1.2)
Q ∈ EndK (A)P.
In [6], M. Larsen proved that (1.1) does not imply (1.2) for general abelian
varieties but that it does imply
(1.3)
kQ ∈ EndK (A)P
for some positive integer k. In this paper, under the same hypotheses, we
consider a refinement of (1.3):
(1.4)
Q ∈ EndK (A)P + A(K)tor .
We say the abelian variety A/K satisfies the refined support conjecture if
(1.1) implies (1.4). An early draft of [7] (version 2) claimed that every
abelian variety satisfies the refined support conjecture. The proof given
was incorrect, and the statement was removed from subsequent versions.
(Version 3 is essentially the same as the published version [6], while version
The first named author was partially supported by NSF grant DMS-0100537.
1
2
MICHAEL LARSEN AND RENÉ SCHOOF
4 corrects a series of misprints, in which P was written for Q and vice versa
throughout several paragraphs of the proof of the main theorem.)
In this paper, we give a condition on the Tate modules of A which guarantees that A satisfies the refined support conjecture. It is satisfied by at
least one abelian variety in every isogeny class. On the other hand, in §6 we
give an example which shows that some abelian varieties do not satisfy the
refined support conjecture.
This paper improves on the method of [6] by working systematically with
`-adic representations where the earlier paper uses properties of (mod `) representations for ` 0. What is needed to make this work is a result about
Galois cohomology of abelian varieties, which is proved via a comparison
theorem between (continuous) `-adic group cohomology and cohomology of
Lie algebras over Q` , analogous to de Rham’s comparison theorem between
the cohomology of real Lie groups and their Lie algebras. By the semisimplicity of the Q` -cohomology of abelian varieties, the Lie algebras in question
are reductive, and the vanishing theorem we need is just one of Whitehead’s
lemmas.
2. Lie algebra cohomology
Let k be a field of characteristic zero, L a Lie algebra over k, and U the
universal enveloping algebra of L. A representation space V of L is the same
thing as a left U -module, and we define the Lie algebra cohomology groups
H i (L, V ) := ExtiU (V, k).
This can be computed from the Koszul resolution of k. In particular,
H 0 (L, V ) = V L = {v ∈ V | X.v = 0 ∀X ∈ L},
while H 1 (L, V ) is the cokernel of the map
(2.1)
V → {f ∈ Hom(L, V ) | X.f (Y ) − Y.f (X) = f ([X Y ])}
given by v 7→ (X 7→ X.v). If L acts trivially on V , (2.1) is the zero-map, so
H 1 (L, V ) = Hom(L/[L L], V ).
The Hochschild-Serre spectral sequence for an ideal I ⊂ L says that
E2pq = H p (L/I, H q (I, V )) ⇒ H p+q (L, V ).
Proposition 2.1. Let L be a reductive Lie algebra and V a semisimple
L-representation. If V L = 0, then
H 1 (L, V ) = H 2 (L, V ) = 0.
Proof. First we note that for every ideal I of a reductive Lie algebra L, we
have an isomorphism of Lie algebras L ∼
= I ⊕ L/I, where both summands
are again reductive. This is a standard fact when L is semisimple. Writing
L = Z ⊕ S, where Z is trivial and S is semisimple, IZ := I ∩ Z is trivial and
IS = I/IZ , being an ideal of S, is semisimple. The action of IS on IZ , arising
from the action of S on Z, is trivial, so by Levi’s theorem, I = IS ⊕ IZ . As
WHITEHEAD’S LEMMAS AND GALOIS COHOMOLOGY OF ABELIAN VARIETIES 3
Z∼
= IZ ⊕ Z/IZ and S ∼
= IS ⊕ S/IS , so I is a direct summand of L = Z ⊕ S.
As Lie algebra cohomology commutes with direct sums, it suffices to prove
the proposition when V is irreducible. Let I be the kernel of the Lie algebra
homomorphism L → End(V ) and M = L/I ⊂ End(V ) the image. If M
consists only of scalars, then dim V = dim M = 1, so H 2 (M, V ) = 0, and
H 1 (M, V ) = 0 since the map (2.1) is an isomorphism of 1-dimensional spaces
in this case. Otherwise the semisimple summand MS of M is non-trivial, so
the hypothesis V L = 0 and the Whitehead lemmas imply H q (MS , V ) = 0 for
q ≤ 2. As L is reductive, MS is an ideal, and the Hochschild-Serre spectral
sequence gives
H p (L/MS , H q (MS , V )) = 0
for p + q = 2. The proposition follows.
Next we prove an analogue of Proposition 2.1 for continuous group cohomology of `-adic analytic groups. Let G be such a group and L the Lie
algebra of G ([8] V 2.4.2). Let V be a finite-dimensional vector space over Q`
on which G operates continuously. There is a corresponding “infinitesimal”
action of L on V ([8] V 2.4.6). By a theorem of D. Lazard ([8] V 2.4.10 (ii)),
if K is a small enough open subgroup of G,
(2.2)
H p (K, V )→H
˜ p (L, V ).
Proposition 2.2. Let G be a compact topological group and V a finite dimensional vector space over Q` on which G admits a faithful and semisimple
action. Suppose that no subrepresentation of G factors through a finite quotient of G. Then
H 1 (G, V ) = H 2 (G, V ) = 0.
Proof. At several points in the proof it will be convenient to replace G by an
open normal subgroup H. The restriction of a semisimple representation to
such a subgroup is again semisimple, and by hypothesis, V H = 0. Suppose
for some open normal subgroup H, H 0 (H, V ) = H 1 (H, V ) = H 2 (H, V ) = 0.
The Hochschild-Serre spectral sequence for modules finite dimensional over
Q` (which follows from the usual Hochschild-Serre spectral sequence with
finite coefficients together with [9] Corollary 2.2 and [9] Proposition 2.3)
asserts
H p (G/H, H q (H, V )) ⇒ H p+q (G, V ).
Thus, H 0 (G, V ) = H 1 (G, V ) = H 2 (G, V ) = 0. Without loss of generality, therefore, we may assume G = H is such that (2.2) holds for all
p ≤ 2. Every G-subrepresentation is also an L-subrepresentation, and every L-subrepresentation comes from a H-subrepresentation for some open
subgroup of G ([8] V 2.4.6). Without loss of generality, therefore, we may
assume that the G-subrepresentations and L-subrepresentations of V are in
one-to-one correspondence and therefore V is a semisimple representation
of L. Applying (2.2) for p = 0, we see that V H = 0 implies V L = 0. The
proposition now follows from Proposition 2.1.
4
MICHAEL LARSEN AND RENÉ SCHOOF
3. Galois cohomology
Theorem 3.1. Let K be a global field, K s a separable closure of K, GK =
Gal(K s /K), ` a rational prime, V a finite dimensional Q` -vector space, and
ρ` : GK → Aut(V )
a continuous semisimple representation which is pure of weight 6= 0. Let
G` = ker ρ` , G` = im ρ` .
Then
H 1 (K, V ) = HomG` (G` , V ).
Proof. As ρ` is pure of non-zero weight, ρ` (Frobp ) cannot have 1 as eigenvalue for any prime p where ρ` is unramified. Therefore, V G` = 0. Applying
Proposition 2.2 with G = G` , we obtain
H 1 (G` , V ) = H 2 (G` , V ) = 0.
Applying the Hochschild-Serre spectral sequence, to G` / Gal(K s /K), we
obtain
H 1 (K, V ) = H 0 (G` , H 1 (G` , V )) = H 1 (G` , V )G`
= Hom(G` , V )G` = HomG` (G` , V ).
Corollary 3.2. Let T be a free Z` module on which Gal(K s /K) acts in such
a way that V = T ⊗ Q` satisfies the hypotheses of Theorem 3.1. Then the
kernel of the restriction map H 1 (GK , T ) → H 1 (G` , T )G` is a torsion group.
`
Proof. The kernel of the restriction map is H 1 (G` , T G ) = H 1 (G` , T ). As
H 1 (G` , T ) ⊗Z Z[1/`] = H 1 (G` , T ⊗ Z[1/`]) = H 1 (G` , V ) = 0,
it follows that every element of H 1 (G` , T ) is of `-power order.
If X is a non-singular projective variety over a global field K, i is a positive
integer, and ` is a rational prime not equal to the characteristic of K, we
expect V = H i (X, Q` ) to satisfy the hypotheses of Theorem 3.1. This is
known to be true in a few important cases, notably when the characteristic
of K is positive ([4] 3.4.1 (iii)) and when X is an abelian variety ([5] Satz 3).
It is the latter case and the cohomology group H 1 , or more precisely, its
dual, which interests us for the rest of the paper.
For A an abelian variety over a number field K and n an integer, we write
A[n] for the kernel of multiplication by n on A. It is a finite group scheme
defined over K. The short exact sequence
n
0 → A[n] → A−→A → 0
gives rise to the Kummer sequence in Galois cohomology:
n
0 → A[n](K) → A(K)−→A(K) → H 1 (K, A[n])
WHITEHEAD’S LEMMAS AND GALOIS COHOMOLOGY OF ABELIAN VARIETIES 5
and therefore to the inclusion A(K) ⊗ Z/nZ ,→ H 1 (K, A[n]). If m and n
are integers, the chain map of short exact sequences
mn
0 −−−−→ A[mn] −−−−→ A −−−−→ A −−−−→ 0






my
my
1y
n
0 −−−−→ A[n] −−−−→ A −−−−→ A −−−−→ 0
gives rise to a commutative square
A(K) ⊗ Z/mnZ −−−−→ H 1 (K, A[mn])




y
y
A(K) ⊗ Z/nZ −−−−→ H 1 (K, A[n])
and therefore to a homomorphism A(K)⊗Z` ,→ H 1 (K, T` ), where T` denotes
the Tate module lim A[`n ]. Let f denote the composition of maps
←
A(K) → A(K) ⊗ Z` ,→ H 1 (K, T` A) → H 1 (G` , T` )G` .
Proposition 3.3. The kernel of f consists exactly of the torsion elements
of A(K).
Proof. As
H 1 (G` , T` )G` = HomG` (G` , T` ) ⊂ Hom(G` , T` )
and T` is a free Z` -module, the same is true of H 1 (G` , T` )G` . If P ∈ A(K)
and nP = 0, then nf (P ) = 0, so f (P ) = 0. Conversely, by Corollary 3.2,
the kernel of H 1 (K, T` ) → H 1 (G` , T` )G` is torsion, and of course the kernel
of A(K) → A(K) ⊗ Z` is torsion, so it follows that ker f is torsion.
4. Semisimple Galois modules over Z`
Definition 4.1. Let T be a free Z` -module with a continuous GK -action
and let V = T ⊗ Q` be the corresponding Galois representation. We say T is
integrally semisimple if for all GK -subrepresentations W ⊂ V , the morphism
T → T /(T ∩ W ) admits a Z` -linear GK -equivariant splitting.
It is clear that T integrally semisimple implies V semisimple, since by
tensoring a complement to T ∩ W with Q` , we obtain a complement of W
in V . We also have a partial converse:
Lemma 4.2. Every semisimple finite dimensional Q` -representation of GK
contains an integrally semisimple lattice T .
Proof. Since every GK -invariant subspace W admits an decomposition into
isotypic components corresponding to the isotypic decomposition of V , without loss of generality we may assume that V is isotypic, i.e., V = V1 ⊗ Qk`
for some irreducible representation V1 . Let T1 be any GK -stable lattice in
V1 ; the existence of such a lattice follows from the compactness of GK . Let
6
MICHAEL LARSEN AND RENÉ SCHOOF
T = T1 ⊗ Zk` ⊂ V . Any GK invariant W is of the form V1 ⊗ W0 , W0 a
subspace of Qk` . Thus,
W ∩ T = (V1 ⊗ W0 ) ∩ (T1 ⊗ Zk` ) = (T1 ⊗ W0 ) ∩ (T1 ⊗ Zk` ) = T1 ⊗ (Zk` ∩ W0 ).
As W0 is `-divisible, the quotient Q = Zk` /Zk` ∩ W0 is torsion-free and therefore free. It follows that the short exact sequence of trivial GK -modules
0 → Zk` ∩ W0 → Zk` → Q → 0
is split. Thus T1 ⊗ Q is a complement for W ∩ T .
Proposition 4.3. If A is an abelian variety over K, for all ` 0, the
`-adic Tate module T` is integrally semisimple.
Proof. Fix an embedding K ⊂ C, and let Λ = H1sing (A, Z). Then E acts
on M . Taking the centralizer of E in End(M ) commutes with flat base
change because the centralizer can be interpreted as the kernel of the commutator homomorphism M → Hom(E, M ). Let Λ = E ∗ . In particular,
the centralizer of E ⊗ Z` in M ⊗ Z` is Λ ⊗ Z` . By the comparison theorem
between singular cohomology and étale cohomology, End(T` ) = M ⊗ Z` . By
a theorem of G. Faltings [5] Remark (2) (a consequence of his proof of the
Shafarevich conjecture), the centralizer of E` in End(T` ) is the Z` -span of
ρ` (GK ) for all ` 0. On the other hand, ` 0 implies that Λ⊗Z` is a maximal order in Λ⊗Q` , and this implies that any finitely generated torsion-free
module over Λ⊗ Z` is projective [3] Theorem 26.12. If (W ∩ T` ) ⊗ Q` is GK stable, then T` /(W ∩ T` ) is a finitely generated torsion-free Λ ⊗ Z` -module,
so
ExtΛ⊗Z` (T` /(W ∩ T` ), W ∩ T` ) = 0
which means that there is a Λ` -splitting and therefore a GK -splitting of
0 → W ∩ T` → T` → T` /(W ∩ T` ) → 0.
Corollary 4.4. If A is an abelian variety over a number field K, there exists
an abelian variety B/K and a K-isogeny A → B such that T` B is integrally
semisimple for all rational primes `.
Proof. Any `-isogeny leaves T`0 unchanged for `0 6= `, so its suffices to show
that for each prime ` there exists B with T` B integrally semisimple. By
Lemma 4.2, T` A ⊗ Q` contains a lattice Λ which is semisimple; multiplying
by a high enough power of `, we may assume Λ ⊂ T` A. This T` /Λ defines a
finite GK -stable `-torsion subgroup D of A, and we define B = A/D.
Lemma 4.5. Let M and N be free Z` -modules of finite rank with continuous
GK -action, N integrally semisimple. Let α, β : M → N be GK -equivariant
Z` -linear homomorphisms such that for positive integers k, α(m) ∈ `k N
implies β(m) ∈ `k N . Then there exists a GK -equivariant Z` -linear endomorphism γ of N such that β = γ ◦α.
WHITEHEAD’S LEMMAS AND GALOIS COHOMOLOGY OF ABELIAN VARIETIES 7
Proof. Let V = N ⊗Q` , Wα = α(M )⊗Q` ⊂ N ⊗Q` , and Wβ = β(M )⊗Q` ⊂
N ⊗Q` . By hypothesis, α(m) = 0 implies β(m) = 0, so Wβ = V / ker(β)⊗Q`
is a quotient of Wα = V / ker(α) ⊗ Q` . Let ξ denote the quotient map.
By integral semisimplicity, there exists a GK -stable Z` -submodule P ⊂ N
which is a complement of W ∩ N in N . Let π : N → Wα ∩ N denote the
corresponding projection. Let γ : N → N ⊗ Q` denote composition of π, the
natural inclusion Wα ∩ N ,→ Wα , and ξ. By construction, γ(α(n)) = β(n)
for all n ∈ N . All that remains is to prove that γ(N ) ⊂ N . As π (and
therefore γ) vanishes on P , it suffices to prove γ(Wα ∩ N ) ⊂ N . However, if
n ∈ Wα ∩ N , then `k n ∈ α(M ) for some non-negative integer k, and we can
write `k n = α(m). By hypothesis, β(m) ∈ `k N , so
γ(n) = `−k γ(`k n) = `−k β(m) ∈ N.
5. The support problem for abelian varieties
This section essentially recapitulates the argument of [6], using `-adic
Galois representations instead of (mod `) representations.
Lemma 5.1. Let M be a finitely generated abelian group, L ⊂ M a subgroup, and x ∈ M . If for every prime `,
x ⊗ 1 ∈ L ⊗ Z` + (M ⊗ Z` )tor ,
then x ∈ L + Mtor .
Proof. Consider the short exact sequence
0 → L + Mtor → M → N → 0.
Tensoring by the flat module Z` , we get
0 → (L + Mtor ) ⊗ Z` → M ⊗ Z` → N ⊗ Z` → 0.
Now,
(L + Mtor ) ⊗ Z` = L ⊗ Z` + Mtor ⊗ Z` = L ⊗ Z` + (M ⊗ Z` )tor ,
so if x denotes the image of x in N , then x ⊗ 1 ∈ L ⊗ Z` + (M ⊗ Z` )tor if and
only if x ⊗ 1 = 0. Of course, x ⊗ 1 = 0 in N ⊗ Z` if and only if x is torsion
of prime-to-` order, and if this holds for all `, then x = 0.
Theorem 5.2. Let A be an abelian variety over a number field K, and let
P and Q be K-points of A. Assume every Tate module of A is integrally
semisimple. Then A satisfies the refined support conjecture, i.e., if
nP ≡ 0
(mod p) ⇒ nQ ≡ 0
(mod p)
for all but finitely many primes p of K, then
Q ∈ EndK (A)P + A(K)tor .
8
MICHAEL LARSEN AND RENÉ SCHOOF
Proof. Let ` be any rational prime. Let K` = K(A[`∞ ]) be the Galois
extension of K generated by all coordinates of all `-power torsion points
on A. Let K` (P, Q) = K(`−∞ P, `−∞ Q) = K` (`−∞ P, `−∞ Q) denote the
field extension of K given by all coordinates of all points on A which when
multiplied by a suitable power of ` give either P or Q. Thus Gal(K` /K) =
G` , and we have the diagram
1
/ Gal(K` (P, Q)/K` )
_
1
δ
/ T` × T`
/ Gal(K` (P, Q)/K)
/ G`
/1
/ (T` × T` ) o Aut(T` )
/ Aut(T` )
/ 1.
Let W ` = (G` /[G` , G` ])⊗Z` . Note that the abelianization of G` is abelian
profinite, and therefore its tensor product with Z` can be embedded as a
closed subgroup in the abelianization itself. For P ∈ A(K), we regard
F (P ) ∈ HomG` (G` , T` ) as a G` -equivariant function from W ` to T` . The
pair (f (P ), f (Q)) defines a GK -map W ` → T` × T` , and we let I` denote the
image of this map. Thus I` = δ(Gal(K` (P, Q)/K` )) in T` × T` .
Suppose that for all `, all positive integers n, and all σ ∈ W ` , f (P )(σ) ∈
n
` T` implies f (Q)(σ) ∈ `n T` . Applying Lemma 4.5 to M = I` and N = T` ,
we conclude that there exists γ ∈ EndGK (T` ) such that γ ◦f (P ) = f (Q). By
the Tate conjecture for abelian varieties [5], there exists φ ∈ EndK (A) ⊗ Z`
such that f (Q) = f (φ(P )). Applying Lemma 5.1 where M = A(K), L =
EndK (A)Q, and x = P , we conclude that Q ∈ EndK (A)Q + A(K)tor .
We may therefore assume that there exists a rational prime `, a positive
integer n, and σ ∈ W ` such that f (P )(σ) ∈ `n T` and f (Q)(σ) 6∈ `n T` .
Lifting σ to σ̃ ∈ G` ⊂ GK , there exists a non-zero element t ∈ T` such
that σ̃ acts trivially on all `-power torsion of A and on all `-power divisors
of P but acts by t on `−∞ Q. Thus σ̃ acts trivially on every point of the
form `−n P but non-trivially on every points of the form `−n Q. Consider
the open set in Gal(K` (P, Q)/K` ) consisting of elements which do the same.
We claim that there exists an integer N ≥ n and an element τ in this open
set such that τ acts as a scalar congruent to 1 (mod `N ) but not (mod `N +1
on the Tate module of A. Indeed, by a theorem of F. Bogomolov [1], G`
contains a non-trivial homothety r which is congruent to 1 (mod `). Lifting
k
to ρ ∈ Gal(K` (P, Q)/K` ), τ = ρ` σ̃ has the desired property when k is
sufficiently large.
By the Cebotarev approximation theorem, there exists a prime p of K
whose Frobenius conjugacy class includes an element Frobp close enough
to τ that it fixes A[`N ] and every element of `−n P but does not fix any
element of `−n Q. Reducing (mod p), we obtain a finite field Fp such that
the `-primary part of A(Fp ) is isomorphic to (Z/`N Z)2 dim A , the image of P
is `n -divisible, and the image of Q is not `n -divisible. It follows that the `
part of the order of P is at most `N −n , while the ` part of the order of Q is
at least `1+N −n , contrary to (1.1). The theorem follows.
WHITEHEAD’S LEMMAS AND GALOIS COHOMOLOGY OF ABELIAN VARIETIES 9
Corollary 5.3. Given a number field K, every non-empty isogeny class of
abelian varieties over K contains at least one element satisfying the refined
support conjecture.
Proof. Immediate from Corollary 4.4.
6. A Counter-example
Proposition 6.1. There exists an abelian variety A over a number field K
which does not satisfy the refined support conjecture.
Proof. Let p be a prime. Let K be a number field and let E be an elliptic
curve over K without complex multiplication that possesses a point R ∈
E(K) of infinite order. Suppose in addition that the p-torsion points of E
are rational over K and let R1 , R2 ∈ E(K) be two independent points of
order p. Consider the abelian surface A obtained by dividing E × E by the
subgroup generated by the point (R1 , R2 ). Then A is defined over K. The
subring End0K (E × E) of endomorphisms of E × E that map the subgroup
generated by (R1 , R2 ) to itself maps naturally to EndK (A) and this map
is a ring isomorphism. If we choose an isomorphism of EndK (E × E) with
the ring of 2 by 2 matrices over Z, then EndK (E × E) corresponds to the
subring of matrices that are congruent to a scalar matrix modulo p.
Let P and Q denote the images of the points (R, 0) and (R, R) in A(K)
respectively. Suppose that nQ ≡ 0 (mod p) for some prime p of good reduction and characteristic different from p. This means that (nR, nR) is
contained in the subgroup generated by (R1 , R2 ) in the group of points on
E × E modulo p. Since the characteristic of p is not p, the torsion points
R1 and R2 are distinct modulo p. This implies that nR ≡ 0 (mod p). It
follows that nP ≡ 0 (mod p). Therefore the conditions of the Theorem are
satisfied. And of course, so is the conclusion
with k = p and f ∈ EndK (A)
0 p
the endomorphism corresponding to
∈ End0K (E × E).
0 0
On the other hand, there is no endomorphism g ∈ EndK (A) for which
P = gQ plus a torsion point. Indeed, this would imply that in (E × E)(K)
one has that
R
λ 0
a b
R
T1
=
+p
+
0
0 λ
c d
R
T2
for some λ ∈ Z and some torsion points T1 , T2 ∈ E(K). Since R has infinite
order, inspection of the second coordinate shows that λ + pc + pd = 0 so
that λ ≡ 0 (mod p). On the other hand, looking at the first coordinate we
see that 1 = pa + pb + λ, a contradiction.
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MICHAEL LARSEN AND RENÉ SCHOOF
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E-mail address: [email protected]
Department of Mathematics, Indiana University, Bloomington, IN 47405,
U.S.A.
E-mail address: [email protected]
Università di Roma “Tor Vergata”, Dipartimento di Matematica, Via della
Ricerca Scientifica, I-00133 Roma, ITALY