Tensor Products of Galois
Representations

Ernst Kani
Queen’s University
Kingston, Ontario

A Scientific Celebration in Honour of Professor M. Ram Murty
CRM, Montreal, 15 – 17 October, 2013.

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Tensor Products of Galois Representations
1. Introduction
Motivation 1: The Hasse-Weil ζ-Function of Products
Recall: The zeta function ζX/K (s) of a smooth, projective variety
X over a number field K is given (up to finitely many Euler
factors) by the zeta functionYζX (s) :=(1 − N (x)−s)−1,x∈|X |of any projective model X /OK of X/K. (This product converges for <(s) > dim X + 1.) Write: ζX/K (s) ∼ ζX (s).
Example: If X = Spec(K), then ζX/K (s) ∼ ζSpec(OK )(s) =ζK (s), the Dedekind ζ-function of K.
Main Principle: ζX/K (s) “encodes the arithmetic” of X: thisis expressed in terms of specific conjectures (see below).
Fact (Grothendieck/Artin/Serre): We have the factorization
ζ X/K (s) ∼2d
YmLm(X, s)(−1) =m=0

L0(X, s) · · · L2d(X, s)
L1(X, s) · · · L2d−1(X, s)

where d = dim X and each Lm(X, s) = L(ρm, s) is the Lfunction associated to a suitable (rational) compatible system
ρm = {ρm,`}` of `-adic Galois representations
ρm,` = ρX,m,` : GK = Gal(K/K) → AutQ` (VX,m,`).
(Explicitly: VX,m,` = Hetm(X, Q`).)

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Example 1: If X/K is a curve, then
ζX/K (s) ∼ ζK (s)ζK (s − 1)L1(X, s)−1.
Here L1(X, s) = L(ρJX /K , s) is the L-function associated tothe system ρJX /K = {ρJX /K,`}` of Galois representations
ρJX /K,` : GK → AutQ` (V`(JX ))afforded by the Tate space of the Jacobian JX /K of X:V`(JX ) = T`(JX ) ⊗Z` Q`, where T`(JX ) = lim JX [`n].
←Example 2: If X = X1 × X2 is a product surface, then (by theKünneth formula) we have
ζX/K (s) ∼ζX1 (s)ζX2 (s)ζX1 (s − 1)ζX2 (s − 1)
L(ρ1 ⊗ ρ2, s).
2ζK (s)ζK (s − 1) ζK (s − 2)

Here ρ1 ⊗ ρ2 = {ρJX /K,` ⊗ ρJX /K,`}` is the system of Galois
21representations afforded by the `-adic tensor product modules
V`(JX1 , JX2 ) := V`(JX1 ) ⊗Q` V`(JX1 ).
Thus, the “new contribution” of the ζ-function ζX (s) is thepart coming from the L-function L(ρ1 ⊗ ρ2, s) associated tothe tensor product representation ρ1 ⊗ ρ2.
Remark: In the case that X1 = X2 = E is an elliptic curve, thenV`(E, E) ' S 2(V`(E)) ⊕ det(V`(E)), so ρ1 ⊗ ρ1 is essentiallythe symmetric square representation.

3Motivation 2: The Tate Conjectures
Conjecture 1 (Hasse/Weil): ζX/K (s) has a meromorphic continuation to the entire complex plane C.
Refinement: Each Lm(X, s) has a meromorphic continuation
to C. (Note: Lm(X, s) converges for <(s) > 1 + m2 .)Conjecture 2 (Tate): The order of the pole of L2m(X, s) ats = m+1 equals the rank of the group Am(X/K) = Z m(X)/ ≡of codimension m-cycles on X/K modulo numerical equivalence:−ords=m+1L2m(X, s) = rk(Am(X/K)).
Furthermore, for all primes `,(T m(X))

rk(Am(X/K)) = dimQ` (Het2m(X, Q`)(m))GK .

Remarks: 1) Tate (1963) also has a conjectural interpetation ofords=1L1(X, s) which generalizes the Birch/Swinnerton-Dyer
Conjecture.
2) In the case that X = A × Â, where A/K is an abelianvariety and Â is its dual, Conjecture T 1(A× Â) is equivalent to
the following statement which was proved by Faltings (1983).
Theorem 0 (Faltings): If A/K is an abelian variety, then(1) rk(EndK (A)) = dimQ` EndQ`[GK ](V`(A)),∀ primes `.

More precisely, we have a natural Q`-linear ring isomorphism(2)∼τA/K,` : EndK (A) ⊗ Q` → EndQ`[GK ](V`(A)).

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2. Some Questions
Notation: If ρi : GK → AutQ` (Vi) are two `-adic Galois representations, then put(ρ1, ρ2)GK := dimQ` HomQ`[GK ](V1, V2).
This is often called the intertwining number of ρ1 and ρ2.
Example: If A/K is an abelian variety, then by Faltings(ρA/K,`, ρA/K,`)GK = rk(EndK (A)).
Notation: If A/K and B/K are two abelian varieties, letV`(A, B) = V`(A) ⊗Q` V`(B)
be the tensor product of the Tate spaces and let
ρA,B,K,` := ρA/K,` ⊗ ρB/K,` : GK → AutQ` (V`(A, B))
be the associated tensor product representation.
Question 1: Is there a “Faltings Theorem” for ρA,B,K,`, i.e., isthere an arithmetic interpretation of (ρA,B,K,`, ρA,B,K,`)GK ?
Observation: The Tate Conjecture (T 2(A×B×Â×B̂)) implies:∃ a subgroup AA,B,K ≤ A2((A × B × Â × B̂)/K) such that(ρA,B,K,`, ρA,B,K,`)GK = rk(AA,B,K ),∀`.

In particular, the left hand side is independent of `.
Subproblem: Can AA,B,K be interpreted in terms of endomorphisms of abelian varieties?

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Remark: It is not difficult to see that we have a natural embedding
EndK (A) ⊗ EndK (B) ,→ AA,B,K .
This naturally raises the following question.
Question 2: When is rk(EndK (A) ⊗ EndK (B)) = rk(AA,B,K )?
Combining Questions 1 and 2 leads to:
Question 3: Let τA,B,K,` = τA/K,` ⊗Q` τB/K,`, so τA,B,K,` can beviewed as a ring homomorphism
τA,B,K,` : EndK (A) ⊗ EndK (B) ⊗ Q` → EndQ`[GK ](V`(A, B))
via the identification
EndQ` (V`(A, B)) ' EndQ` (V`(A)) ⊗Q` EndQ` (V`(B)).
When is τA,B,K,` an isomorphism? In other words, when is(3)(ρA,B,K,`, ρA,B,K,`)GK = rk(EndK (A)) rk(EndK (B))?

A first (naive) guess is that the following holds.
Hypothesis HA,B,K : The following are equivalent:(i) Formula (3) holds for all primes `;(i0) Formula (3) holds for one prime `;(ii) HomK (A, B) = 0.
Observation: While HA,B,K holds for some abelian varieties A/K
and B/K, it is not true in general. There are (at least) twoclasses of counterexamples.

6Counterexamples to Hypothesis HA,B,K :1) A/Q and B/Q are modular abelian varieties which have acommon internal twist (in the sense of Ribet);
2) A/K and B/K are CM elliptic curves which are definedover Q and K is a suitable real quadratic field.
Remark: Note that EndK (A) 6= EndQ(A) in both cases. Thus,a better guess is the following:
Hypothesis H A,B : The hypothesis HA,B,K holds whenever K
is large enough, i.e., whenever
EndK (A) = EndK (A) andEndK (B) = EndK (B).

Observation: If H A,B holds for A/K and B/K, and if (ii) holds(i.e., if HomK (A, B) = 0), then for every finite extension L/Kand prime ` we have an induced isomorphism∼τ̃A,B,L : (EndK (A) ⊗ EndK (B))GL ⊗ Q` → EndQ`[GL](V`(A, B)).
From this it follows that
AA,B,K ⊗ Q = (EndK (A) ⊗ EndK (B))GL ⊗ Q,and so we obtain a solution of our subproblem in this case. In
particular, HA,B,L holds if and only if(EndK (A) ⊗ EndK (B))GL ⊗ Q = EndL(A) ⊗ EndL(B) ⊗ Q.

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3. Main Results.
Theorem 1. If A and B are isogenous (over Q) to products ofelliptic curves, then H A,B holds.
Definition: A modular abelian variety A/K is an abelian variety which is isogenous to a quotient of the Jacobian variety
J1(N )K of the modular curve X1(N )K , for a suitable N.
Theorem 2. If A and B are modular abelian varieties, thenH A,B holds.
Remark: Both Theorem 1 and Theorem 2 are special cases of amore general theorem. For this, I introduce the class of abelianvarieties of generalized GL2-type (see below). These include:• products of elliptic curves• K. Murty’s abelian varieties of type (T) (1983)• K. Ribet’s abelian varieties A/Q of GL2-type (1992); theseinclude the Shimura quotients Af , where f ∈ S2(Γ1(N ))new .
Theorem 3. If A and B are abelian varieties of generalized GL2type, then H A,B holds.
Corollary 1: If A/K and B/K are abelian varieties of generalized GL2-type, then HA,B,K holds ⇔(EndK (A) ⊗ EndK (B))GK = EndK (A) ⊗ EndK (B).
Corollary 2: If A/K and B/K are abelian varieties of generalized GL2-type whose Q-endomorphisms are defined over K,then HA,B,K holds.

8Example: If A/K and B/K are isogenous to products of elliptic
curves without CM, then HA,B,K holds by Corollary 2. In otherwords, the following conditions are equivalent:(i) Formula (3) holds for all primes `;(i0) Formula (3) holds for one prime `;(ii) HomK (A, B) = 0;
Remark: Recall that formula (3) was the following identity:(ρA,B,K,`, ρA,B,K,`)GK = rk(EndK (A)) rk(EndK (B)).

94. Analysis of Condition (3).
Notation: If V is a Q`[GK ]-module, let V = V ⊗Q` Q` denotethe associated Q`[GK ]-module. Here Q` denotes an algebraic
closure of Q`.
Lemma 1: τA,B,K,` is an isomorphism (i.e., condition (3) holds
for A, B, K, `) if and only if the following two conditions hold:(1) (Irreducibility) If V ⊂ V `(A) and W ⊂ V `(B) are irreducible Q`[GK ]-submodules, then V ⊗ W is also irreducible.
(2) (Multiplicity 1) If Vi ⊂ V `(A) and Wi ⊂ V `(B) are irreducible Q`[GK ]-submodules (for i = 1, 2), thenV1 ⊗ W1 ' V2 ⊗ W2

⇔V1 ' V2 and W1 ' W2.

Counterexamples to HA,B,K : 1) Let Ei/Q be two elliptic curves
with CM by Fi, where F1 6= F2. If K = (F1F2)+, thenHE1,E2,K does not hold. Here HomQ(E1, E2) = 0, but (3)does not hold (for any `) because dimQ EndK (Ei) = 1 and(ρE1,E2,K,`, ρE1,E2,K,`)GK = 2 6= 1. (Here Property (1) fails.)2) Let Ei/Q be two modular (non-CM) elliptic curves withassociated newforms fi ∈ S2(Γ0(Ni)), and assume that E1
and E2 are not Q-isogenous. Moreover, let χ be a Dirichlet
character of order m > 2, and let gi be the newform associatedto the twist (fi)χ of fi by χ. If Ai = Agi /Q is the Shimura
quotient associated to gi, then HA1,A2,Q does not hold.
φ(m)Indeed, Ai ⊗ Q ∼ Ei
⊗ Q, so HomQ(A1, A2) = 0, but (3)does not hold. (Here Property (1) holds, but (2) fails because
A1 and A2 have “simultaneous inner twists”).

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5. Abelian Varieties of Generalized GL2-type
Definition: A Q`[GK ]-module V has restricted GL2-type if V =⊕Vi is a direct sum of two-dimensional Q`[GK ]-modules Visuch that each Vi is of one of the following two types:(I) Vi is irreducible and det Vi = χ`, where χ` is the cyclotomic`-adic character on GK .
(II) Vi ' V `(Ei), for some CM elliptic curve Ei/K.
Definition: An abelian variety A/K has generalized GL2-type
if there is a finite extension L/K such that(i) End0L(A) = End0Q(A);(ii) V `(A) has restricted GL2-type as a GL-module, ∀`.
Remark: The class (genGL2)K of abelian varieties A/K of generalized GL2-type is closed under products. Moreover, if A ∈(genGL2)K and if B ≤ A, then B, A/B ∈ (genGL2)K .
Lemma 2: If A ∈ (genGL2)K , then there is a decomposition
A ∼ AnCM × ACM such that for any L/K with (i) we havethat(a) ACM ⊗ L ∼ product of CM elliptic curves Ei/L, andV `(ACM ) is a direct sum of 1-dimensional GL-modules;(b) Each GL-irreducible component V of V `(AnCM ) has dimension 2 and is strongly irreducible, i.e. V|U is irreducible, ∀open U ≤ GL. Moreover, V `(AnCM ) has no internal twists,i.e., if Vi are two irreducible submodules of V `(AnCM ), then×

V1 ' V2 ⊗ χ, for some χ ∈ Hom(GL, Q` )⇒χ = 1.

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6. Representation Theory: non-CM Case.
Let: k = Q` and G = GK . Here we study k[G]-modules V with:(4)

V is strongly irreducible of dimension 2.

(Recall: this means that V|U is irreducible, ∀ open U ≤ G.)
Theorem 5 (Irreducibility Criterion): If V, W satisfy (4),then V ⊗ W is irreducible ⇔(5)

V 6' W ⊗ χ,for all χ ∈ Hom(G, k ×).

Remark: By using Schur’s Lemma, this follows easily from a
result of D. Ramakrishnan (2000) on adjoint representations.
Theorem 6 (Cancellation Criterion): If Vi, Wi satisfy (4)for i = 1, 2, and if(6)Vi ⊗ Wj is irreducible, for all i, j ∈ 1, 2,then V1 ⊗ W1 ' V2 ⊗ W2 ⇔ ∃χ ∈ Hom(G, k ×) such that(7)V1 ' V2 ⊗ χ and W1 ' W2 ⊗ χ−1.

Remarks: 1) In view of Lemmas 1 and 2, Theorems 5 and 6
imply Theorem 3 in the non-CM case (i.e, when A ∼ AnCM .)2) The proof of Theorem 6 uses the following identity (whichwas also used in Ramakrishnan’s proof):∧2(V ⊗ W ) ' (S 2V ⊗ ∧2W ) ⊕ (∧2V ⊗ S 2W ).
(As usual, S 2V denotes the symmetric square of V .)

127. Representation Theory: CM Case.
Recall: If E/K is a CM elliptic curve with F := End0K (E) 6= Q,then F ⊂ K and F is an imaginary quadratic field. Moreover,V `(E) ' ψ1 ⊕ ψ2,×with ψi ∈ Hom(GK , Q` ).

In addition, ψ1ψ2 = χ`.
Lemma 3: Let Ei/K be an elliptic curve with CM by Fi ⊂ K,and let V `(Ei) = ψi1 ⊕ ψi2, where i = 1, 2. Assume that
F1 6= F2. If p is a prime which splits completely in K, thenQ(ψ1iψ2j (σP)) ' F1F2,∀i, j = 1, 2,

where σP ∈ GK is a Frobenius element at P | p.
Remarks: 1) Using Lemma 3, it follows easily that Property (2)holds if A = ACM and B = B CM . Since Property (1) is trivial,
we thus see that Theorem 3 holds in this case. Combining thiswith the results of §6, this proves Theorem 3 because it is easy
to verify Properties (1) and (2) for the “mixed terms” Vi ⊗ ψj .
2) By using a more general version of the Irreducibility Criterion (Theorem 5) and the results of Ribet (1980), one can also
show:Theorem 7: If A/Q and B/Q are modular abelian varietieswith HomQ(A, B) = 0, then Property (1) holds, i.e.,V ⊗ W is GQ-irred., if V ⊂ V `(A), W ⊂ V `(B) are GQ-irred.