Compositio Math. 142 (2006) 541–550
doi:10.1112/S0010437X06001904

Generic transfer from GSp(4) to GL(4)Mahdi Asgari and Freydoon Shahidi
Abstract
We establish the Langlands functoriality conjecture for the transfer from the generic spectrum of GSp(4) to GL(4) and give a criterion for the cuspidality of its image. We apply thisto prove results toward the generalized Ramanujan conjecture for generic representationsof GSp(4).
1. Introduction
Let k be a number ﬁeld and let G denote the group GSp(4, Ak ). The (connected component of the)L-group of G is GSp(4, C), which has a natural embedding into GL(4, C). Langlands functoriality
predicts that associated to this embedding there should be a transfer of automorphic representationsof G to those of GL(4, Ak ) (see [Art04]). Langlands’ theory of Eisenstein series reduces the proofof this to unitary cuspidal automorphic representations. We establish functoriality for the generic
spectrum of GSp(4, Ak ). More precisely (cf. Theorem 2.4), we prove the following.
Let π be a unitary cuspidal representation of GSp(4, Ak ), which we assume to be globally
generic. Then π has a unique transfer to an automorphic representation Π of GL(4, Ak ).
 ⊗ ωπ . Here,The transfer is generic (globally and locally) and satisﬁes ωΠ = ωπ2 and Π  Π
ωπ and ωΠ denote the central characters of π and Π, respectively.
Moreover, we give a cuspidality criterion for Π and prove that, when Π is not cuspidal, it is an
isobaric sum of two unitary cuspidal representations of GL(2, Ak ) (cf. Proposition 2.2).
We give a number of applications of this result. The ﬁrst is Theorem 3.3 which gives estimates
toward the generalized Ramanujan conjecture for generic representations of the group GSp(4) (cf.
§ 3.1). We also prove in Theorem 3.7 that any generic unitary cuspidal representation of GSp(4, Ak )is weakly Ramanujan (cf. § 3.2 for the deﬁnition). In § 3.3 we use our main result to give an immediate
proof that the spin L-function of a generic unitary cuspidal representation of GSp(4, A) is entire.
This is due to the fact that the spin L-function of GSp(4) now becomes a standard L-function (orproduct of two such L-functions) of general linear groups. This fact has also been proved recently fora certain class of globally generic representations of GSp(4, Ak ) by Takloo-Bighash using diﬀerent
methods from ours.
We should note here that the transfer from GSp(4) to GL(4) has been expected by experts in theﬁeld for a long time. It is our understanding that Jacquet, Piatetski-Shapiro and Shalika knew how
to prove this result, at least in principle, but, as far as we know, their result was never published.
We should also point out that their proof is based on methods that are fairly disjoint from ours.
Our method of proof is to start with our earlier, more general but weaker, result on generictransfer from GSpin groups to GL (cf. [AS06]). This gives us the existence of Π. We then use the
Received 5 January 2005, accepted in final form 15 August 2005.
2000 Mathematics Subject Classification 11F70, 11R42, 22E55.
Keywords: Langlands functoriality, automorphic representation, L-function.
First author partially supported by the NSF grant DMS–0111298 at IAS; second author partially supported by the
NSF grant DMS–0200325.
c Foundation Compositio Mathematica 2006.
This journal is 

M. Asgari and F. Shahidi
results of Piatetski-Shapiro and Soudry on analytic properties of L-functions of GSp(4) twisted byGL(1) and GL(2) to get more information about the representation Π. It is exactly the lack of such
results in the general case of GSpin groups that prevents us from carrying out our analysis for the
more general case for now. However, as we pointed out in [AS06], there are currently two ways toovercome this problem. One is to prove an analogue of descent theory for these groups as was done
for classical groups by Ginzburg, Rallis and Soudry [GRS01, Sou05]. The other is to use techniquesalong the lines of [Kim01, Kim02].
The case of the transfer for all automorphic representations of GSp(4) (whether generic or not)requires Arthur’s trace formula. The process and the expected issues in this case are outlined in[Art04].
2. Main result
Let k be a number ﬁeld and let A = Ak denote its ring of adeles. We deﬁne the similitude symplectic
group of degree four via
GSp(4) = {g ∈ GL(4) : t gJg = µ(g)J},
where



J =

−1

1
1
−1






and µ(g) ∈ GL(1) is the similitude character. We ﬁx the following parametrization of the elementsof the maximal torus T in GSp(4):



a
a
a


0
1
2





a
a
0
1
 .
T = t = t(a0 , a1 , a2 ) = 


a0 a2





a0
The above agrees with our previous more general notation for the group GSpin(2n + 1) in [AS06].
Recall that the group GSp(4) is identiﬁed with GSpin(5).
Let π =  v πv be a globally ψ-generic unitary cuspidal automorphic representation of GSp(4, A).
Here, ψ = v ψv is a non-trivial additive character of k\A deﬁning a character of the unipotent
radical of the standard upper-triangular Borel subgroup in the usual way. We ﬁx ψ now and do notrepeat it in the rest of this paper. Let S be any non-empty ﬁnite set of non-archimedean places v,
which includes those
v with πv or ψv ramiﬁed. We proved in [AS06] that there exists an automorphic
representation Π = v Πv of GL(4, A) such that Πv is a local transfer of πv for v outside of S.
To be more explicit, assume that v ∈ S. If v is archimedean, then πv is given by a parameter
φv : Wv −→ GSp(4, C), where Wv is the Weil group of kv (cf. [Lan89]). Let Φv : Wv −→ GL(4, A)
be given by Φv = ι ◦ φv , where ι : GSp(4, C) −→ GL(4, C) is the natural embedding. Then Φv is theparameter of Πv .
If v ∈ S is non-archimedean, then πv is the unique unramiﬁed subquotient of the representationinduced from an unramiﬁed character χ of T(kv ) to GSp(4, kv ). Writing
χ(t(a0 , a1 , a2 )) = χ0 (a0 )χ1 (a1 )χ2 (a2 ),(1)

where χi are unramiﬁed characters of kv× and ai ∈ kv× , the representation Πv is then the unique
irreducible unramiﬁed subquotient of the representation of GL(4, kv ) parabolically induced from the542

Generic transfer from GSp(4) to GL(4)character−1χ1 ⊗ χ2 ⊗ χ−1
2 χ0 ⊗ χ1 χ0(2)

of T(kv ).
Moreover, we proved that ωΠ = ω 2 , where ω = ωπ and ωΠ denote the central characters of π
 v ⊗ ωπv , i.e. Π is nearly equivalent to Π ⊗ ω.
and Π, respectively, and for v ∈ S we have Πv  Π
The representation Π is equivalent to a subquotient of some representation
Ind(|det|r1 σ1 ⊗ · · · ⊗ |det|rt σt ),(3)where induction is from GL(n1 ) × · · · × GL(nt ) with n1 + · · · + nt = 4 to GL(4) and σi are theunitary cuspidal automorphic representation of GL(ni , A) and ri ∈ R.
Without loss of generality we may assume that r1  r2  · · ·  rt . Moreover, as Π is unitary wehave n1 r1 + · · · + nt rt = 0, which implies that rt  0. Let T = S ∪ {v : v|∞} and considert ) = LT (s, Π × σt ) =LT (s, π × σtLT (s + ri , σi × σ
t ).

(4)

i=1

Here, LT denotes the product over v ∈ T of the local L-factors.
If nt = 1, then the left-hand side is entire by a result of Piatetski-Shapiro (cf. [Pia97, p. 274]).
Now consider the right-hand side at s0 = 1 − rt  1. The last term in the product has a pole at s0 ,whereas all of the others are non-zero there as (s0 + ri ) = 1 + ri − rt  1. This is a contradiction.
Now assume that nt = 3, i.e. t = 2 with n1 = 1 and n2 = 3. Replacing π and Π by theircontragredients will change ri to −ri and takes us back to the above situation, which gives acontradiction again.
Therefore, nt = 2. In this case, the left-hand side of (4) may have a pole at s = 1 (cf. [PS84,
Theorem 1.3] and the beginning of its proof), and if so, arguing as above, we conclude that rt = 0.
This means that we either have t = 2 with n1 = n2 = 2 or t = 3 with n1 = n2 = 1 and n2 = 2.
However, we can rule out the latter as follows.
Assume that t = 3 with n1 = n2 = 1 and n3 = 2. Then, it follows from the fact that r3 = 0
and the conditions r1  r2  r3 and r1 + r2 + 2r3 = 0 that all of the ri would be zero in this case.
This implies that if we consider the L-function of π twisted by σ
1 , we have1 ) = LT (s, σ1 × σ
1 )LT (s, σ2 × σ
1 )LT (s, σ3 × σ1 ).
LT (s, π × σ(5)

Now the left-hand side is again entire by Piatetski-Shapiro’s result mentioned above and the righthand side has a pole at s = 1, which is a contradiction.
Therefore, the only possibilities are t = 1 (i.e. Π unitary cuspidal) or t = 2 and n1 = n2 = 2 with
r2 = 0. In the latter case, we also get r1 = 0, as r1 + r2 = 0 by unitarity of the central character.
Moreover, in this case we have σ1  σ2 as, otherwise,1 ) = LT (s, σ1 × σ
1 )LT (s, σ2 × σ1 )LT (s, π × σ(6)must have a double pole at s = 1 while any possible pole of the left-hand side at s = 1 is simple(cf. the proof of [PS84, Theorem 1.3]).
Therefore, we have proved the following.
Proposition 2.1. Let π be a globally generic unitary cuspidal automorphic representation ofGSp(4, A) and let Π be any transfer of π to GL(4, A). Then Π is a subquotient of an automorphic representation as in (3) with either t = 1, n1 = 4 and r1 = 0 (i.e. Π is unitary cuspidal) ort = 2, n1 = n2 = 2 and r1 = r2 = 0. In the latter case, we have σ1  σ2 .
In fact, we can get more precise information.
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M. Asgari and F. Shahidi
Proposition 2.2. Let π be a globally generic unitary cuspidal automorphic representation of ⊗ω
GSp(4, A) with ω = ωπ its central character and let Π be any transfer as above. Then, Π  Π(not just nearly equivalent). Moreover:(a) the representation Π is cuspidal if and only if π is not obtained as a Weil lifting from GSO(4, A);(b) if Π is not cuspidal, then it is the isobaric sum of two representations Π = Π1  Π2 , wherei ⊗ ω
each Πi is a unitary cuspidal automorphic representation of GL(2, A) satisfying Πi  Πand Π1  Π2 .
Proof. By Proposition 2.1, Π is not cuspidal if and only if it is a subquotient ofGL(4,A)Σ = IndGL(2,A)×GL(2,A) (σ1 ⊗ σ2 ),(7)

where σi are unitary cuspidal automorphic representation of GL(2, A).
On the other hand, by [PS84, Theorem 1.3] mentioned above, the representation π is obtained
as a Weil lifting from GSO(4, A) if and only if there exists an automorphic representation τ ofGL(2, A) such that LT (s, π × τ ) has a pole and in that case τ can be normalized so that the poleoccurs at s = 1.
Now assume that Π is cuspidal. Then for any τ as above we have
LT (s, π × τ ) = LT (s, Π × τ ),which is entire. Therefore, π is not obtained as a Weil lifting from GSO(4, A). Moreover, as Π is

cuspidal, so is Π⊗ωand they are nearly equivalent, therefore, by the strong multiplicity one theorem ⊗ ω.
[JS81a, JS81b, Pia79] we have Π  Π
Next, assume that Π is not cuspidal and, hence, is given as a subquotient of Σ above. We claimthat each Σv = Ind(σ1,v ⊗ σ2,v ) is irreducible. To see this note that each σi,v is generic unitaryand is either a tempered representation of GL(2, kv ) or a complementary series I(χ| |α , χ| |−α ) withα ∈ (0, 12 ) and χ a unitary character. If both of the σi,v are tempered, then irreducibility of Σv is clear.
If both are complementary series of the form I(χ1 | |α , χ1 | |−α ) and I(χ2 | |β , χ2 | |−β ) with α, β ∈ (0, 12 )and χi unitary characters, then for Σv to be reducible we should have α ± β = ±1, which is notpossible. Finally, if one of σi,v is tempered and the other is complementary series, then we have Σv =Ind(χ| |−α , χ1 , χ2 , χ| |α ), Σv = Ind(χ| |−α , Q(χ1 | |−1/2 , χ1 | |1/2 ), χ| |α ) or Σv = Ind(χ| |−α , η, χ| |α ).
Here, χ1 , χ2 and χ are unitary characters, α ∈ (0, 12 ), η is a unitary supercuspidal representationof GL(2, kv ), and Q(χ1 | |−1/2 , χ1 | |1/2 ) denotes the Steinberg representation twisted by the unitarycharacter χ1 . Again, in all these cases the representation Σv is irreducible as α ∈ (0, 12 ).
Therefore, at every place v the representation Πv is the same as the irreducible Σv and, hence,
it is the Langlands quotient. This means that Π is an isobaric representation, i.e. the isobaric sumof σ1 and σ2 . Again by the strong multiplicity one theorem, which remains valid for isobaric repre ⊗ ω. Now, just take Πi to be σi .
sentations [JPS83], we conclude that Π  ΠFinally, by [AS06, Proposition 7.4], which was based on classiﬁcation theorems of Jacquet and i ⊗ω for i = 1, 2 or we have Π1  Π
 2 ⊗ω (or equivalently,
Shalika, we know that we either have Πi  ΠΠ2  Π1 ⊗ ω). However, the latter case will not occur when π is cuspidal and generic as, otherwise,
π will be nearly equivalent to an Eisenstein series representation, i.e. π will be a CAP representationof GSp(4, A). This is impossible by [PS87, Theorem 1.1]. This completes the proof.
Remark 2.3. Note that any other transfer Π of π is also a subquotient of Σ in (7), which isirreducible. Therefore, π has a unique transfer to GL(4, A), which we continue to denote by Π.
In particular, this implies that we have not lost any information at the places where we did nothave a natural candidate for the local transfer.
Moreover, as Π is either a unitary cuspidal representation of a general linear group or an isobaric
sum of two such, every local representation Πv is full induced and generic.
544

Generic transfer from GSp(4) to GL(4) 2 ) has no pole at s = 1.
Furthermore, if Π is not cuspidal, then Π1  Π2 implies that L(s, Π1 × ΠThis implies that the Fourier coeﬃcient of Π along the unipotent radical of our ﬁxed Borel subgroupis non-vanishing, i.e. Π is globally generic [Sha81].
We collect the above results in the following theorem, which is our main result.
Theorem 2.4. Let π be a globally generic unitary cuspidal automorphic representation of GSp(4, A) ⊗ω
with central character ω. Then π has a unique transfer Π to GL(4, A), which satisﬁes Π  Π
2and its central character is ω . Moreover, Π is either unitary cuspidal or an isobaric sum Π1  Π2 of i ⊗ ω.
two inequivalent unitary cuspidal automorphic representations of GL(2, A) satisfying Πi  Π
The latter is the case if and only if π is obtained as a Weil lifting from GSO(4, A). Furthermore, Πis globally generic, i.e. it has a non-vanishing Fourier coeﬃcient along the unipotent radical of our
ﬁxed Borel subgroup.
In fact, we can get more precise information about the local representations at places v ∈ S.
Proposition 2.5. Fix v ∈ S and let
πv  Ind(π1,v |det|b1,v ⊗ · · · ⊗ πt,v |det|bt,v ⊗ π0,v )(8)

be an irreducible generic representation of GSp(4, kv ), where each πi,v is a tempered representation ofsome GL(ni , kv ), b1,v > · · · > bt,v , and π0,v is a tempered generic representation of some GSp(2m, kv ).
(Note that there are very few possibilities for ni and m as n1 + · · · + nt + m = 2. We are allowing
m = 0 and, by convention, GSp(0) = GL(1).) Let ωv denote the central character of πv . Assume
that πv is the local component of the globally generic unitary cuspidal representation π of GSp(4, A)
and let Π be its transfer to GL(4, A). Then the local component Πv of Π at v is generic and of theform
πt,v ⊗ωv )|det|−bt,v ⊗· · ·⊗(
π1,v ⊗ωv )|det|−b1,v ), (9)Πv  Ind(π1,v |det|b1,v ⊗· · ·⊗πt,v |det|bt,v ⊗Π0,v ⊗(
where Π0,v is a tempered generic representation of GL(2m, kv ) if m > 0.
Proof. Let us remark that, as in [CKPS04, § 7], one could deﬁne the notion of local transfer andobtain complete information about such transfers for a general irreducible admissible generic representation, whether a local component of a global representation or not. (In fact, the representation
Π0,v would then be the local transfer of π0,v .) However, we do not need the full extent of such resultsin this paper.
Recall that we already proved (cf. Remark 2.3) that each Πv is generic and is full induced. Letv ∈ S and consider πv and Πv as in the proposition. We ﬁrst show that if ρv is any supercuspidal
representation of GL(r, kv ), thenγ(s, πv × ρv , ψv ) = γ(s, Πv × ρv , ψv ).

(10)

The key here is the fact that there exists a unitary cuspidal representation ρ of GL(r, A) suchthat its local component at v is ρv and at all other ﬁnite places w = v the local component ρw
is unramiﬁed (cf. [Sha90, Proposition 5.1]). Now applying the converse theorem of Cogdell andPiatetski-Shapiro with S  = S − {v} will give the result exactly as in the proof of [CKPS04,
Proposition 7.2]. Moreover, by multiplicativity of γ-factors, we conclude that (10) also holds if ρv
is a discrete series representation of GL(r, kv ).
Next, we claim that if πv is tempered, then so is Πv . Here, again the main tool is multiplicativityof γ-factors and the proof is exactly as in [CKPS04, Lemma 7.1]. This proves the proposition forthe case m = 2. If m = 1, then the group GSp(2m) = GSp(2) is the same as GL(2) and we set Π0,v
to be π0,v itself. For m = 0 we need no choice of Π0,v . Now, let T = {w0 } consist of a single ﬁnite
place w0 = v at which πv is unramiﬁed and consider the representation Π of GL(4, A) whose local545

M. Asgari and F. Shahidi
components are the same as Π outside of S and are the irreducible induced representations on
the right-hand side of (9) when v ∈ S. We can now apply the converse theorem again to Π and
T = {w0 } to conclude that Π is a transfer of π. The key here is that the induced representations
on the right-hand side of (9) have the right L-functions. Therefore, by uniqueness of the transfer
we proved earlier, we have Πv  Πv for v ∈ S. This completes the proof.
3. Applications
We ﬁrst recall that the current formulation of the Ramanujan conjecture for generic cuspidal representations states that for any quasi-split group and any globally generic unitary cuspidal automorphic representation π = v πv , the local components πv are tempered for all places v. As anapplication of our main theorem, we can prove two types of results in this direction: estimates toward
this conjecture for the group GSp(4) as well as a weaker version of it for generic representations of
this group.
3.1 Estimates toward Ramanujan

Following [CKPS04], we introduce the following notation in order to prove estimates. Let Π = v Πv
be a unitary cuspidal automorphic representation of GL(m, Ak ). For every place v the representation
Πv is unitary generic and can be written as a full induced representation
Πv  Ind(Π1,v |det|a1,v ⊗ · · · ⊗ Πt,v |det|at,v )(11)

with a1,v > · · · > at,v and each Πi,v tempered [Vog78, Zel80].
Definition 3.1. We say Π satisﬁes H(θm ) with θm  0 if for all places v we have −θm  ai,v  θm .
The classiﬁcation of generic unitary dual of GL(m) (see [Tad86, Vog78]) trivially gives H( 12 ).
The best result currently known for a general number ﬁeld k says that any unitary cuspidal representation of GL(m, A) satisﬁes H( 12 − 1/(m2 + 1)) (see [LRS99]). When k = Q and m  4, it isH( 12 −1/(1 + m(m + 1)/2)). The same bound is also available for m > 4 for k = Q provided that one
knows that the symmetric square L-function of Π is absolutely convergent for (s) > 1 (cf. [KS03]),
but this is only currently available for m  4. When m = 2 we have the better bounds of H( 19 ) for a7) for k = Q (see [KS03]). The Ramanujan conjecture
general number ﬁeld k (see [KS02]) and H( 64demands H(0).

Similarly, if π = v πv is a unitary generic cuspidal automorphic representation of GSp(2n, Ak ),then by [Mui01] and [Vog78], each πv can be written as a full induced representation
πv  Ind(π1,v |det|b1,v ⊗ · · · ⊗ πt,v |det|bt,v ⊗ τv ),(12)

where each πi,v is a tempered representation of some GL(ni , kv ) and τv is a tempered genericrepresentation of some GSp(2m, kv ) with n1 + · · · + nt + m = n.
Definition 3.2. We say π satisﬁes H(θn ) with θn  0 if for all places v we have −θn  bi,v  θn .
The classiﬁcation of generic unitary dual of GSp(4) (cf. [LMT04], for example) trivially gives
the estimate H(1). The Ramanujan conjecture demands H(0) again. For a survey of results in thisdirection and their applications we refer to [Sar03, Sha04].
Theorem 3.3. Let k be a number ﬁeld and assume that all unitary cuspidal representations ofGL(4, Ak ) (respectively, GL(2, Ak )) satisfy H(θ4 ) (respectively, H(θ2 )) and θ2  θ4 . Then any
globally generic unitary cuspidal representation π of GSp(4, Ak ) satisﬁes H(θ4 ), and even the betterbound of H(θ2 ) if π transfers to a non-cuspidal representation of GL(4, Ak ) (cf. Theorem 2.4).
546

Generic transfer from GSp(4) to GL(4)Proof. Let Π be the functorial transfer of π to GL(4, Ak ). If v is an archimedean place of k, then thetheorem is clear as in this case local functoriality is well understood through Langlands’ parametrization (cf. the proof of [AS06, Theorem 6.1], for example, for more details).
Let v be a non-archimedean place of k at which πv is unramiﬁed. Then it follows from (2) that
πv is given by its Frobenius–Hecke (Satake) parameter, which is of the form−1diag(χ1 (), χ2 (), χ−12 ()χ0 (), χ1 ()χ0 ()),(13)

where  denotes a uniformizer of kv . If Π is cuspidal, then for i = 1, 2 we have by assumption
qv−θ4  |χi ()|  qvθ4 .
If Π is not cuspidal, then we have similar inequalities with θ4 replaced by the even better estimateof θ2 . As πv is unitary, we have |χ0 ()| = 1. Therefore, Frobenius–Hecke parameters of πv alsosatisfy similar inequalities.
Next, assume that v is a place of k in S. Then, by Proposition 2.5, a similar argument as above
works again.
Corollary 3.4. Let π be a globally generic unitary cuspidal representation of GSp(4, Ak ). Then π1satisﬁes H( 1534 ). If π transfers to a non-cuspidal representation of GL(4, Ak ), then it satisﬁes H( 9 ).
97
If k = Q, then we have the better estimates of H( 22 ) and H( 64 ), respectively.
Proof. The proof is immediate if we combine Theorem 3.3 with the known results on estimates forthe general linear groups mentioned above.
Corollary 3.5. The Ramanujan conjecture for unitary cuspidal representations of GL(4) andGL(2) imply the Ramanujan conjecture for the generic spectrum of GSp(4).

3.2 The weak Ramanujan conjecture
Following [CP95, Ram97, Kim03] we recall the following deﬁnition.
Definition 3.6. Let G be a split reductive group over the number ﬁeld k. Let π =πv be anautomorphic representation of G(Ak ). We say that π is weakly Ramanujan if, given  > 0, there
exists a set T of places of k containing the archimedean places and the non-archimedean ones withπv ramiﬁed such that T has density zero and for v ∈ T the Frobenius–Hecke parameter diag(λv,i )of πv satisﬁes
max{|λv,i |, |λ−1v,i |}  qv .
iHere, qv denotes the cardinality of the residue ﬁeld.
We are concerned with the cases of G = GL(m) or G = GSp(4) in this paper. We recall that
(unitary) cuspidal representations of GL(m) for m  4 are weakly Ramanujan (cf. [Ram97] and[Kim03, Propositions 3.7 and 6.3]).
Let π be a globally generic unitary cuspidal representation of GSp(4, Ak ). For any v ∈ T as
above, letdiag(a0,v , a1,v , a2,v )(14)

be the Frobenius–Hecke parameter of πv (cf. (1)). Then, as in (13), the parameter of the local
transfer Πv is given by−1diag(a1,v , a2,v , a−1
2,v a0,v , a1,v a0,v ).

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M. Asgari and F. Shahidi
Moreover, |a0,v | = 1 as πv is unitary. Therefore, the above results about weak Ramanujan propertyof unitary cuspidal representations of GL(m) immediately imply the following.
Theorem 3.7. Let π be a globally generic unitary cuspidal representation of GSp(4, Ak ). Then π
is weakly Ramanujan.
3.3 Spinor L-function for GSp(4)As another application we get the following immediate corollary of our main result, Theorem 2.4.
Proposition 3.8. Let π be a globally generic unitary cuspidal representation of GSp(4, Ak ). Thenthe spinor L-function L(s, π, spin) is entire.
Proof. Let Π be the transfer of π to GL(4, A). If Π is unitary cuspidal, then L(s, π, spin) = L(s, Π)and if Π = Π1  Π2 is the isobaric sum of two unitary cuspidal representations of GL(2, A), thenwe have L(s, π, spin) = L(s, Π1 )L(s, Π2 ). In either case the L-functions on the right-hand side
are standard L-functions of unitary cuspidal representations of the general linear group and are
entire.
Remark 3.9. This result has also been proved for a certain class of globally generic representations π
by Takloo-Bighash in [Tak]. His methods are diﬀerent from ours and are based on using an integral
representation. T. Moriyama has proved certain cases of this result as well [Mor04, IM05].
Acknowledgements
We would like to thank James Cogdell for many helpful discussions and Peter Sarnak for his interestin this work. The ﬁrst author would also like to thank Steve Gelbart, Dihua Jiang, Robert Langlands,
Brooks Roberts and Ramin Takloo-Bighash for many helpful discussions during the course of thiswork.
References
Art04

AS06
CKPS04
CP95
GRS01
IM05
JPS83
JS81a
JS81b
Kim01

J. Arthur, Automorphic representations of GSp(4), in Contributions to automorphic forms,
geometry, and number theory (Shalika volume), eds H. Hida, D. Ramakrishnan and F. Shahidi(Johns Hopkins University Press, Baltimore, MD, 2004), 65–81.
M. Asgari and F. Shahidi, Generic transfer for general spin groups, Duke Math. J. 132 (2006),
137–190.
J. Cogdell, H. Kim, I. Piatetski-Shapiro and F. Shahidi, Functoriality for the classical groups,
Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163–233.
J. Cogdell and I. Piatetski-Shapiro, Unitarity and functoriality, Geom. Funct. Anal. 5 (1995),
164–173.
D. Ginzburg, S. Rallis and D. Soudry, Generic automorphic forms on SO(2n + 1): functorial lift
to GL(2n), endoscopy, and base change, Internat. Math. Res. Notices 14 (2001), 729–764.
T. Ishi and T. Moriyama, Spinor L-functions for generic cusp forms on GSp(2) belonging toprincipal series representations, Preprint (2005).
H. Jacquet, I. Piatetskii-Shapiro and J. Shalika, Rankin–Selberg convolutions, Amer. J. Math.
105 (1983), 367–464.
H. Jacquet and J. Shalika, On Euler products and the classiﬁcation of automorphic representations. I, Amer. J. Math. 103 (1981), 499–558.
H. Jacquet and J. Shalika, On Euler products and the classiﬁcation of automorphic forms. II,
Amer. J. Math. 103 (1981), 777–815.
H. Kim, Residual spectrum of odd orthogonal groups, Internat. Math. Res. Notices 17 (2001),
873–906.

548

Generic transfer from GSp(4) to GL(4)
Kim02
Kim03

KS02
KS03
Lan89

LMT04
LRS99

Mor04
Mui01
Pia79

Pia97
PS84
PS87
Ram97
Sar03

Sha81
Sha90
Sha04
Sou05
Tad86
Tak
Vog78
Zel80

H. Kim, Applications of Langlands’ functorial lift of odd orthogonal groups, Trans. Amer. Math.
Soc. 354 (2002), 2775–2796.
H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 (withAppendix 1 by D. Ramakrishnan and Appendix 2 by H. Kim and P. Sarnak), J. Amer. Math.
Soc. 16 (2003), 139–183.
H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112(2002), 177–197.
H. Kim and P. Sarnak, Reﬁned estimates towards the Ramanujan and Selberg conjectures, J. Amer.
Math. Soc. 16 (2003). Appendix 2 to [Kim03].
R. Langlands, On the classiﬁcation of irreducible representations of real algebraic groups,
in Representation theory and harmonic analysis on semisimple Lie groups, Mathematical
Surveys and Monographs, vol. 31 (American Mathematical Society, Providence, RI, 1989),
101–170.
E. Lapid, G. Muić and M. Tadić, On the generic unitary dual of quasisplit classical groups,
Internat. Math. Res. Notices 26 (2004), 1335–1354.
W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for GL(n), in
Automorphic forms, automorphic representations, and arithmetic, Fort Worth, TX, 1996, Proceedings of Symposia in Pure Mathematics, vol. 66 (American Mathematical Society, Providence,
RI, 1999), 301–310.
T. Moriyama, Entireness of the spinor L-functions for certain generic cusp forms on GSp(2),Amer. J. Math. 126 (2004), 899–920.
G. Muić, A proof of Casselman-Shahidi’s conjecture for quasi-split classical groups, Canad. Math.
Bull. 44 (2001), 298–312.
I. Piatetski-Shapiro, Multiplicity one theorems, in Automorphic forms, representations andL-functions, Oregon State University, Corvallis, OR, 1977, part 1, Proceedings of Symposia in
Pure Mathematics, vol. 33 (American Mathematical Society, Providence, RI, 1979), 209–212.
I. Piatetski-Shapiro, L-functions for GSp4 . Olga Taussky-Todd: in memoriam, Paciﬁc J. Math.
181 Special Issue (1997), 259–275.
I. Piatetski-Shapiro and D. Soudry, L and ε functions for GSp(4) × GL(2), Proc. Natl Acad. Sci.
USA 81 (1984), 3924–3927.
I. Piatetski-Shapiro and D. Soudry, On a correspondence of automorphic forms on orthogonalgroups of order ﬁve, J. Math. Pures Appl. (9) 66 (1987), 407–436.
D. Ramakrishnan, On the coeﬃcients of cusp forms, Math. Res. Lett. 4 (1997), 295–307.
P. Sarnak, Notes on the generalized Ramanujan conjectures, Fields Institute Lectures, 2003, inHarmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings,
vol. 4 (American Mathematical Society, Providence, RI, 2005), 659–685.
F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297–355.
F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for
p-adic groups, Ann. of Math. (2) 132 (1990), 273–330.
F. Shahidi, On the Ramanujan conjecture for quasi-split groups, Asian J. Math. 8 (2004), 813–836.
D. Soudry, On Langlands functoriality from classical groups to GLn , Astérisque 298 (2005),
335–390.
M. Tadić, Classiﬁcation of unitary representations in irreducible representations of general linear
group (non-Archimedean case), Ann. Sci. École Norm. Sup. (4) 19 (1986), 335–382.
R. Takloo-Bighash, Spinor L-functions, theta correspondence, and Bessel coeﬃcients, Forum
Math., to appear.
D. Vogan, Gelfand–Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978),75–98.
A. Zelevinsky, Induced representations of reductive p-adic groups II. On irreducible representationsof GL(n), Ann. Sci. École Norm. Sup. (4) 13 (1980), 165–210.

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Generic transfer from GSp(4) to GL(4)Mahdi Asgari asgari@math.okstate.edu
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08450, USA
Current address: Department of Mathematics, Oklahoma State University, Stillwater,
OK 74078-1058, USA
Freydoon Shahidi shahidi@math.purdue.edu
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

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