A DESCENT CRITERION FOR

ISOBARIC REPRESENTATIONS

Dinakar Ramakrishnan

This will appear as an appendix to the paper Functoriality for the exterior square of GL_{4} and the symmetric fourth of GL_{2} ([K]) by Henry Kim.

The object here is to prove the following extension (from cuspidal) to isobaric automorphic representations of Proposition 3.6.1 of [Ra], which was itself an extension to GL(n) of the Proposition 4.2 (for GL(2)) in [BR]. The argument is essentially the same as in [Ra], but requires some delicate bookkeeping.

Proposition. Fix n, p ∈ ℕ with p prime. Let F a number field, {K_{j} | j ∈ ℕ} an infinite family of cyclic extensions of F with [K_{j} : F] = p, and for each j ∈ ℕ, π_{j} an isobaric automorphic representation of GL(n, 𝔸_{K_{j}}). Suppose that, for all j, r ∈ ℕ, the base changes of π_{j}, π_{r} to the compositum K_{j}K_{r} satisfy

(DC)  (π_{j})_{K_{j}K_{r}} ≃ (π_{r})_{K_{j}K_{r}}.

Then there exists a unique isobaric automorphic representation π of GL(n, 𝔸_{F}) such that

(π)_{K_{j}} ≃ π_{j},

for all but a finite number of j.

Proof. Recall that the set Isob of isobaric automorphic representations of GL(n, 𝔸_{F}) for all n ≥ 1 admits a sum operation ⊞, called the isobaric sum, such that

L(s, π ⊞ π′) = L(s, π)L(s, π′), ∀ π, π′ ∈ Isob.

Moreover, given any isobaric automorphic representation π of GL(n, 𝔸_{F}) there exist cuspidal automorphic representations π^{1}, …, π^{d} of GL(n_{1}, 𝔸_{F}), …, GL(n_{d}, 𝔸_{F}), with n = n_{1} + … + n_{d}, such that

(1)  π ≃ π^{1} ⊞ … ⊞ π^{d}.

Here the cuspidal datum (π^{1}, …, π^{d}) is unique up to (isomorphism and) permutation. We will say that π is of width d. For the basic properties of isobaric representations see [La] and [JS].

Typeset by AMS-TEX

Given any isobaric automorphic representation π of width d in the form (1), and any d-tuple χ := (χ^{1}, …, χ^{d}) of idele clas characters of F, we define the χ-twist of π to be

(2)  π[χ] : = (π^{1} ⊗ χ^{1}) ⊞ … ⊞ (π^{d} ⊗ χ^{d}).

If an isobaric automorphic representation π′ is isomorphic to π[χ] for some χ, we will say that π′ is a twist of π. And if μ is an idele class character of F and m = (m(1), …, m(d)) a d-tuple of integers, we will set

μ^{m} : = (μ^{m(1)}, …, μ^{m(d)}).

Now we need the following

Lemma. Let π = π^{1} ⊞ … ⊞ π^{d} be an isobaric automorphic representation of GL(n, 𝔸_{F}), with π^{1}, …, π^{d} being cuspidal automorphic representations of GL(n_{1}, 𝔸_{F}), …, GL(n_{d}, 𝔸_{F}), n = n_{1} + … + n_{d}. Then there exist at most a finite number of d-tuples χ = (χ^{1}, …, χ^{d}) of idele class characters such that

π ≃ π[χ].

Proof of Lemma. By the uniqueness of the isobaric sum decompostition of π into cuspidals, there must be a permutation σ in S_{d} such that we have, for each i ≤ d, an isomorphism

π^{i} ≃ π^{σ(i)} ⊗ χ_{σ(i)}.

We must necessarily have n_{i} = n_{σ(i)} for each i. So the Lemma is a consequence of the following

Sublemma. Let η, η′ be cuspidal automorphic representation of GL(m, 𝔸_{F}). Then the set X of idele class characters μ such that

η ≃ η′ ⊗ μ

is finite.

Proof of Sublemma. We may assume that X is non-empty, as there is nothing to prove otherwise. Pick, and fix, any member, call it ν, of X. Put

Y = {μν^{−1} | μ ∈ X}.

Since X and Y have the same cardinality, it suffices to prove that Y is finite. We claim that for any χ in Y,

η ≃ η ⊗ χ.

Indeed, if χ = μν^{−1} with μ ∈ X, we have

η ≃ η′ ⊗ μ ≃ (η′ ⊗ ν) ⊗ (μν^{−1}) ≃ η ⊗ χ,

whence the claim.

Now the set Y, which parametrizes the self-twists of η, is finite by Lemma 3.6.2 of [Ra], and hence the Sublemma is proved; so is the Lemma. □

Proof of Proposition (contd.)

For each j, let θ_{j} be a generator of Gal(K_{j}/F), and δ_{j} a character of F cutting out K_{j} (by class field theory). Note that, for each i ≥ 1, the pull back to K_{i} of δ_{j} by the norm map N_{i} from K_{i} to F cuts out the compositum K_{i}K_{j}.

We will write, for each j,

(3)  π_{j} ≃ ⊞_{k=1}^{d(j)}π_{j}^{k},

with each π_{j}^{k} a cuspidal automorphic representation of GL(n_{k}(j), 𝔸_{F}), with n = ∑_{k=1}^{d(j)} n_{k}(j).

We claim that

(4)  π_{j} ∘ θ_{j} ≃ π_{j} (∀j).

For all j, r ≥ 1, let θ_{j,r} denote the automorphism of K_{j}K_{r} such that (i) θ_{j,r}|_{K_{j}} = θ_{j}, and (ii) θ_{j,r}|_{K_{r}} = 1 (where 1 denotes the identity automorphism). It is easy to see that the base change of π_{j} ∘ θ_{j} to K_{j}K_{r} is simply (π_{j})_{K_{j}K_{r}} ∘ θ_{j,r}. (For the basic results on base change, see [AC]; for a quick summary see Prop. 2.3.1 of [Ra].) Applying (DC), we then have

(π_{j} ∘ θ_{j})_{K_{j}K_{r}} ≃ (π_{r})_{K_{j}K_{r}} ∘ θ_{j,r} ≃ (π_{r})_{K_{j}K_{r}} ≃ (π_{j})_{K_{j}K_{r}},

since θ_{j,r} is trivial on K_{r}. Since K_{j}K_{r} is a cyclic extension of K_{j} of prime degree, we must have by Arthur-Clozel,

(5)  π_{j} ∘ θ_{j} ≃ π_{j}[(δ_{r} ∘ N_{j})^{m_{r}}],

for some d(j)-tuple m_{r} = (m_{r}(1), …, m_{r}(d(j))) of integers in {0, 1, …, p − 1}. For every fixed r ≥ 1, and for all k ≠ r, we then have the self-twist identity

π_{j} ≃ π_{j}[(δ_{r} ∘ N_{j})^{m_{r}}][(δ_{k} ∘ N_{j})^{−m_{k}}].

Note that δ_{r} ∘ N_{j} and δ_{k} ∘ N_{j} must be distinct unless their ratio is a power of δ_{j}. So the Lemma above forces m_{r} to be the zero vector for all but a finite number of r. The claimed identity now follows by taking r to be outside this exceptional finite set.

As a result, by applying base change ([AC]; Prop. 2.3.1 of [Ra]) once again, we see that there exists, for each j ≥ 1, an isobaric automorphic representation

π(j) = ⊞_{k=1}^{b(j)} π(j)^{k}

of GL(n, 𝔸_{F}), with each π(j)^{k} a cuspidal automorphic representation of GL(N_{k}(j), 𝔸_{F}) and

n = ∑_{k=1}^{b(j)} N_{k}(j),

such that

(6)  π_{j} ≃ (π(j))_{K_{j}}.

Such a π(j) is of course unique only up to replacing it by π(j)[δ_{j}^{a}] for some d(j)-tuple a = (a_{1}, …, a_{b(j)}) of integers in {0, 1, …, p − 1}. Clearly we have

b(j) ≤ d(j),

but equality need not hold.

It is important to note that, for any r ≠ j, we have the following compatibility for base change in (cyclic) stages:

(7)  ((π(j))_{K_{j}})_{K_{j}K_{r}} ≃ ((π(j))_{K_{r}})_{K_{j}K_{r}}.

We see this as follows. Let v be a finite place of K_{j}K_{r} which is unramified for the data. Denote by u (resp. w, resp. w′) the place of F (resp. K_{j}, resp. K_{r}) below v. If σ_{u} denotes the representation of W′_{F_{u}} associated to π(j)_{u}, then

res_{(K_{j}K_{r})_{v}}^{(K_{j})_{w}}(res_{(K_{j})_{w}}^{F_{u}}(σ_{u})) ≃ res_{(K_{j}K_{r})_{v}}^{(K_{r})_{w′}}(res_{(K_{r})′_{w}}^{F_{u}}(σ_{u})).

Then (2.3.0) of [Ra] implies the local identity (for all such v)

((π(j)_{u})_{(K_{j})_{w}})_{(K_{j}K_{r})_{v}} ≃ ((π(j)_{u})_{(K_{r})′_{w}})_{(K_{j}K_{r})_{v}}.

The global isomorphism (7) follows by the strong multiplicity one theorem for isobaric automorphic representations ([JS]).

We can then rewrite (DC) as saying, for all j, r ≥ 1,

(8)  ((π(j))_{K_{j}})_{K_{j}K_{r}} ≃ ((π(r))_{K_{j}})_{K_{j}K_{r}}.

Consequently we must have, after renumbering, an equality of partitions (∀ (r, j)):

(N_{1}(j), …, N_{b(j)}(j)) = (N_{1}(r), …, N_{b(r)}(r))

of n. In particular, we have

(9)  b : = b(j) = b(r) and N_{k} : = N_{k}(j) = N_{k}(r).

Moreover,

(10)  (π(j))_{K_{j}} ≃ (π(r))_{K_{j}}[(δ_{r} ∘ N_{j})^{m(r,j)}],

for some b-tuple m(r, j) = (m(r, j)_{1}, …, m(r, j)_{b}) of integers. We can replace π(r) by π(r)[δ_{r}^{−m(r,j)}] and get

(11)  (π(j))_{K_{j}} ≃ (π(r))_{K_{j}}.

Then, by replacing π(j) by a twist by δ_{j}^{a} for a b-tuple a of integers, we can arrange to have π(j) and π(r) be isomorphic. In sum, we have produced, for every pair (j, r), a common descent, say π(j, r), of π_{j}, π_{r}, i.e., with

(12)  π(j, r)_{K_{j}} ≃ π_{j} and π(j, r)_{K_{r}} ≃ π_{r}.

Fix non-zero vectors a, c in (ℤ/p)^{b}, and consider the possible isomorphism

(13)  π(j, r) ≃ π(j, r)[δ_{j}^{a}][δ_{r}^{−c}].

We claim that this cannot happen outside a finite set S_{a,c} of pairs (j, r). To see this fix a pair (i, ℓ) and consider the relationship of π(i, ℓ) to π(j, r). Since π(i, ℓ) and π(j, ℓ) have the same base change to K_{ℓ}, they must differ by twisting by a b-tuple power of δ_{ℓ}. Similarly, π(j, ℓ) and π(j, r) differ by a twist as they have the same base change to K_{r}. Put together, this shows that π(i, ℓ) and π(j, r) are twists of each other. Then (13) would imply that

(14)  π(i, ℓ) ≃ π(i, ℓ)[δ_{j}^{a}][δ_{r}^{−c}] ≃ π(i, ℓ)[χ_{a,−c}],

where

χ_{a,−c} = (δ_{j}^{a_{1}}δ_{r}^{−c_{1}}, …, δ_{j}^{a_{b}}δ_{r}^{−c_{b}}).

The claim now follows since, by the Lemma above, π(i, ℓ) admits only a finite number of self-twists, and since the b-tuples χ_{a,−c} are all distinct for distinct pairs (j, r) (as a, c are fixed).

Now choose a pair (j, r) not belonging to S_{a,c} for any pair (a, c) of non-zero vectors in (ℤ/p)^{b}, and set

(15)  π = π(j, r).

We assert that for all but a finite number of indices m,

(16)  π_{K_{m}} ≃ π_{m}.

It suffices to show that, for any large enough m, π = π(j, r) is isomorphic to either π(j, m) or π(m, r). Suppose neither is satisfied. Then there exist non-zero vectors a, c in (ℤ/p)^{b} such that

π(j, m) ≃ π(j, r)[δ_{j}^{a}] and π(m, r) ≃ π(j, r)[δ_{r}^{c}].

We also have π(j, m) ≃ π(m, r)[δ_{m}^{e}], for some vector e in (ℤ/p)^{b}. Putting these together, we get the self-twisting identity

(17)  π(j, r) ≃ π(j, r)[δ_{j}^{a}][δ_{r}^{−c}][δ_{m}^{−e}].

By our choice of (j, r), e cannot be the zero vector. But for each non-zero e, the set of indices m for which such an identity can hold is finite, again by the Lemma. Hence we get a contradiction for large enough m, which implies that a or c should be 0, giving the requisite contradiction. Thus π = π(j, r) must be isomorphic to either π(j, m) or π(m, r) for large enough m. Since we have, by (12),

π(j, m)_{K_{m}} ≃ π_{m} ≃ π(m, r)_{K_{m}},

the Proposition is now proved. □

References

[AC] J. Arthur and L. Clozel, Simple Algebras, Base Change and the Advanced Theory of the Trace Formula, Ann. Math. Studies 120, Princeton, NJ, 1989.

[BR] D. Blasius and D. Ramakrishnan, Maass forms and Galois representations, in Galois groups over ℚ, ed. by Y. Ihara, K. Ribet and J.-P. Serre, Academic Press, NY, 1990, pp. 33-77.

[JS] H. Jacquet and J.A. Shalika, Euler products and the classification of automorphic forms I & II, Amer. J of Math. 103 (1981), 499–558 & 777–815.

[K] H. Kim, Functoriality for the exterior square of GL_{4} and the symmetric fourth of GL_{2}, to appear.

[La] R.P. Langlands, Automorphic representations, Shimura varieties and motives. Ein Märchen, Proc. Symp. Pure Math. 33, ed. by A. Borel and W. Casselman, part 2, AMS, 1979, pp. 205–246.

[Ra] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and Multilicity One for SL(2), Annals of Mathematics 152 (2000), 45–111.

Dinakar Ramakrishnan Department of Mathematics California Institute of Technology Pasadena, CA 91125 dinakar@its.caltech.edu