Selberg's Orthogonality Conjecture and Symmetric

SELBERG’S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS

PENG-JIE WONG

Abstract. Let π be a cuspidal representation of GL2pAQq deﬁned by a non-CM holomorphic newform of weight k ě 2. Let K{Q be a totally real Galois extension with Galois group G, and let χ be an irreducible character of G. In this article, we show that Selberg’s orthogonality conjecture implies that the symmetric power L-functions Lps, Symm πq are primitive functions in the Selberg class. Moreover, assuming the automorphy for Symm π, we show that Selberg’s orthogonality conjecture implies that each Lps, Symm π ˆ χq is automorphic. The key new idea is to apply the work of Barnet-Lamb, Geraghty, Harris, and Taylor on the potential automorphy of Symm π.

1. Introduction Thirty years ago, Selberg [21] introduced a class S of L-functions, called the Selberg class nowadays. The class S consists of functions F psq of a complex variable s enjoying the following properties: (Dirichlet series and Euler product). For Repsq ą 1, 8 a pnq 8 F psq “ F “ exp b ppkq{pks ns F n“1 p k“1 ÿ ź ´ ÿ ¯ k kθ 1 with aF p1q “ 1 and bF pp q “ Opp q for some θ ă 2 , where the product is over rational primes p. Moreover, both Dirichlet series and Euler product of F converge absolutely on Repsq ą 1. (Analytic continuation and functional equation). There is a non-negative integer mF such that ps ´ 1qmF F psq extends to an entire function of ﬁnite order. Moreover, there are numbers rF , QF ą 0, αF pjq ą 0, RepγF pjqq ě 0, so that

rF s ΛF psq “ QF ΓpαF pjqs ` γF pjqqF psq j“1 ź satisﬁes ΛF psq “ wF ΛF p1 ´ sq for some complex number wF of absolute value 1. (Ramanujan-Petersson conjecture). For any ﬁxed ą 0, aF pnq “ Opn q. The class S is multiplicatively closed, and the building blocks for S are primitive functions F that the equation F “ F1F2 with F1, F2 P S implies F “ F1 or F “ F2. Indeed, it can be shown that every function F P S factorises into a product of primitive functions in S (see, e.g., [19]). Selberg [21] made the following orthogonality conjecture concerning the primitive functions.

2010 Mathematics Subject Classiﬁcation. 11F66,11M41. Key words and phrases. Selberg’s orthogonality conjecture, symmetric power L-functions. The author was supported by a PIMS postdoctoral fellowship and the University of Lethbridge. 1 2 PENG-JIE WONG

Conjecture 1.1. For any F P S, there exists a positive integer nF such that

|a ppq|2 (1.1) F “ n log log x ` Op1q. p F pďx ÿ Moreover, for any primitive function F , nF “ 1. For any distinct primitive functions F and G, a ppqa ppq (1.2) F G “ Op1q. p pďx ÿ Selberg’s orthogonality conjecture has a strong impact on the theory of L-functions. For instance, Conery and Ghosh [10] showed that under Selberg’s orthogonality conjecture, every F P S has a unique factorisation into primitive functions in S. Moreover, it is known that Selberg’s orthogonality conjecture implies that for any F P S, F psq is non-vanishing on Repsq “ 1. Thus, deﬁning the generalised von Mangoldt function ΛF pnq associated to F by F 1 8 Λ pnq psq “ F , F ns n“1 under Selberg’s orthogonality conjecture, oneÿ has the generalised prime number theorem asserting that

ΛF pnq „ mF x, nďx ÿ as x Ñ 8, where mF is the order of the pole of F at s “ 1. Furthermore, Murty [19] showed that Selberg’s orthogonality conjecture leads to a resolution of Artin’s (holomorphy) conjecture as follows. Theorem 1.2 (Murty). Let K{Q be a Galois extension of number ﬁelds with Galois group G, and let χ be a non-trivial irreducible character of G. Then Conjecture 1.1 implies that the Artin L-function Lps, χ, K{Qq attached to χ is a primitive function in S. In addition, Lps, χ, K{Qq is entire. Suppose, further, that K{Q is solvable Then Conjecture 1.1 implies that the Artin L-function Lps, χ, K{Qq is automorphic. We shall note that although Selberg introduced the class S to study the value distribution of L-functions (for instance, he proved his famous “central limit theorem” for every member in S in [21, Sec. 2]), the study the structural properties of the Selberg class S is also of interest.1 It has been shown by Conery and Ghosh [10] that the only F P S of degree zero is 1 and that there is no function F P S with 0 ă dF ă 1. (Recall that the degree dF of F is the number appearing in the estimate d #tρ | F pρq “ 0, Repρq P r0, 1s, and |Impρq| ď T u “ F T log T ` OpT q π that gives a count of zeros of F .) Moreover, Kaczorowski and Perelli [15] showed that there is no function F P S with 1 ă dF ă 2. In addition, if F P S is of degree one, then 1For instance, the classiﬁcation of the Selberg class may be regarded as an “analytic version” of the Langlands programme (see, e.g., [15]). SELBERG’S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS 3

F is either the Riemann zeta function or a “shifted” Dirichlet L-function Lps ` ir, χq for some r P R, where χ is a primitive Dirichlet character modulo q ą 1 (see [14]). Apart from Riemann zeta function and Dirichlet L-functions, there are many other known members in the Selberg class S as well as L-functions expected to be in S. For instance, the Dedekind zeta functions and Hecke L-functions all belong to S. In this article, we shall emphasise symmetric power L-functions associated to modular forms. Let AQ be the adèlering of Q. Let π be a cuspidal representation πf of GL2pAQq defined by a non-CM holomorphic newform of weight k ě 2, and let Lps, πq be the automorphic L-function of π. From the theory of automorphic L-functions, Lps, πq satisfies the first two axioms of the Selberg class. Moreover, by the celebrated theorem of Deligne, the Ramanujan-Petersson conjecture holds for Lps, πq. In other words, Lps, πq is a member of the Selberg class. Furthermore, via the “symmetric power lifting,” one can construct new L-functions Lps, Symm πq, the symmetric power L-functions of π, from Lps, πq. These L-functions are expected to belong to S, and their analytic properties have many important ap- plications to questions arising from number theory.2 By the work of Barnet-Lamb, Geraghty, Harris, and Taylor [4], it is known that Lps, Symm πq extends holomorphi- m cally to Repsq ě 1 and meromorphically to C. Furthermore, Lps, Sym πq satisfies the Ramanujan-Petersson conjecture and expected functional equation. Thus, under the assumption that Lps, Symm πq extends holomorphically to Repsq ă 1, each Lps, Symm πq would belong to the Selberg class. The primary objective of this article is to show that Selberg’s orthogonality conjecture implies the symmetric power L-functions of π belong to the Selberg class as follows.

Theorem 1.3. Let π be a cuspidal representation of GL2pAQq deﬁned by a non-CM holomorphic newform of weight k ě 2. Let K{Q be a totally real Galois extension with Galois group G. Then Conjecture 1.1 implies that for any m P N and χ P IrrpGq, the Rankin-Selberg L-function Lps, Symm π ˆ χq is a primitive function in S, where IrrpGq is the set of irreducible characters of G. In particular, under Conjecture 1.1, Lps, Symm πq is a member of S. Moreover, we have the following conditional automorphy result for certain Rankin- Selberg L-functions.

Theorem 1.4. Let π be a cuspidal representation of GL2pAQq defined by a non-CM holomorphic newform of weight k ě 2. Let K{Q be a solvable totally real Galois extension with Galois group G. Suppose that Symm π is automorphic. Then subject to m Conjecture 1.1, for any χ P IrrpGq, Lps, Sym π ˆ χq is automorphic over Q. Con- sequently, if m ď 8, then Selberg’s orthogonality conjecture yields the automorphy of Lps, Symm π ˆ χq. Remarks. (i) Our result does not affect the Selberg class S, but verifies expected members of S, i.e. Lps, Symm πq, under Selberg’s orthogonality conjecture. Also, the

2Most famously, they led to the resolution of the Sato-Tate conjecture (now a theorem of Clozel, Harris, Shepherd-Barron, and Taylor [8, 12, 25]). 4 PENG-JIE WONG

(conditional) primitivity of Lps, Symm πq may be seen as a kind of “analytic version” of the conjectural cuspidality of Symm π predicted by the Langlands programme.3 (ii) Assuming the automorphy of the symmetric powers and Artin representations, the theory of Rankin-Selberg L-functions implies that each Lps, Symm π ˆ χq extends to an entire function. Moreover, under the assumption of the functoriality of the tensor product, Lps, Symm π ˆ χq is automorphic. From this viewpoint, our results show the consistency between the Langlands programme and Selberg’s orthogonality conjecture.

2. Preliminaries on representation theory and L-functions In this section, we will recall two induction theorems from character theory and review some theory of automorphic representations and L-functions. 2.1. Induction theorem. We start by recalling Brauer’s induction theorem [6] as well as Solomon’s induction theorem [24], which will play a role in applying the potential automorphy theorem from [2, 3, 4] later. Proposition 2.1 (Brauer’s induction theorem). Let G be a finite group, and let IrrpGq be the set of irreducible characters of G. Then for any χ P IrrpGq, one has χ n IndG ψ , “ i Hi i i ÿ where for each i, ni P Z, Hi is a nilpotent subgroup of G, and ψi is a degree-one character of Hi. Proposition 2.2 (Solomon’s induction theorem). For any finite group G, 1 n IndG 1 , G “ i Hi Hi i ÿ where for each i, ni P Z, Hi is a quasi-elementary subgroup of G, and 1H denotes the trivial character of a group H. (Recall that a group is said to be quasi-elementary if it admits a normal cyclic subgroup with prime-power index.) 2.2. Standard automorphic L-functions. We review in this section some properties of standard automorphic L-functions. To do this, we follow the exposition of [7]. For an (irreducible unitary) cuspidal representation π “ bvπv of GLnpAkq, where, as later, Ak stands for the adèlering of k, the (incomplete) L-function of π is given by

Lps, πq “ Lps, πvq, v ź for Repsq ą 1, where the product is over (finite) primes v of k, for unramified v of π, the local L-function Lps, πvq is defined by n ´s ´1 (2.1) Lps, πvq “ p1 ´ απpv, iqNv q , i“1 ź 3Indeed, the Langlands functoriality conjecture predicts that all the Symm π are automorphic, which implies that Symm π are cuspidal by the work of Ramakrishnan [20, Theorem A(d)] whenever

π is a cuspidal representation of GL2pAQq defined by a non-CM holomorphic newform of weight k ě 2. SELBERG’S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS 5 where απpv, iq are Satake parameters of πv, and Nv is the norm of v. For ramified v of π, one may use the Langlands parameters of πv to define Lps, πvq, which is the reciprocal of a polynomial in Nv´s, of degree at most n and can be written in the form 1{2´pn2`1q´1 of (2.1). Luo-Rudnick-Sarnak [18] showed that for each v, |απpv, iq| ď Nv . The Ramanujan-Petersson conjecture further demands |απpv, iq| “ 1 for unramified v. Moreover, if π is non-trivial and cuspidal, Lps, πq extends to an entire function, which is non-vanishing on Repsq ě 1. Finally, we recall that attached to π is the contragredientπ ˇ, which is also a cuspidal representation of GLnpAkq, such that for every prime v of k,π ˇv is equivalent to the complex conjugation πv. In particular, one has tαπˇpv, iqu “ tαπpv, iqu.

2.3. Rankin-Selberg L-functions. We now brieﬂy review the theory of Rankin- 1 1 Selberg L-functions. Let π “ bvπv and π “ bvπv be cuspidal representations of GLnpAkq and GLn1 pAkq, respectively. The (incomplete) Rankin-Selberg L-function of π and π1 is deﬁned to be

1 1 Lps, π ˆ π q “ Lps, πv ˆ πvq, v ź for Repsq ą 1, as established in [13], where

n n1 1 ´s ´1 Lps, πv ˆ πvq “ p1 ´ απˆπ1 pv, i, jqNv q , i“1 j“1 ź ź 1 where for finite unramified v of either π or π , tαπˆπ1 pv, i, jqu “ tαπpv, iqαπ1 pv, jqu. It has been shown that Lps, π ˆ π1q extends to an meromorphic function, of order 1, over C. Also, assuming the central characters of both π and π1 are trivial, one knows that Lps, π ˆ π1q is entire if and only if π1 fi πˇ. If Lps, π ˆ π1q admits poles, then such poles are simple and at s “ 0, 1. Moreover, Shahidi proved that the Rankin-Selberg L-functions are non-vanishing on Repsq “ 1. Remark. We note that it can be shown that all the automorphic L-functions and Rankin-Selberg L-functions admit analytic continuation and satisfy the functional equations required by the Selberg class. In other words, assuming the Ramanujan- Petersson conjecture, these L-functions are members in the Selberg class. 2.4. Symmetric power L-functions of Hilbert modular forms. In this section, we discuss the Langlands functoriality conjecture for the symmetric powers of Hilbert modular forms. Let k be a totally real number field, and let π be a cuspidal representation corresponding to a non-CM Hilbert modular form of k. (Recall that if π is cuspidal (over k) and with essentially square-integrable π8, then π is associated to a Hilbert modular form of weights ě 2 at all the infinite places of k, and vice versa.) By the work of Blasius [5], the Ramanujan-Petersson conjecture holds for π. The Langlands functoriality conjecture asserts that by writing the local L-function of π at finite unramified v as

´s ´1 ´s ´1 Lps, πvq “ p1 ´ απpvqNv q p1 ´ βπpvqNv q , 6 PENG-JIE WONG

m there exists an automorphic representation Sym π of GLm`1pAkq such that m m m´j j ´s ´1 Lps, pSym πqvq “ p1 ´ απpvq βπpvq Nv q . j“0 ź (We note that the local Langlands correspondence predicts that there are appropriate m local L-functions Lps, pSym πqvq for all primes v.) By the work of Gelbart-Jacquet, Shahidi, and Kim, if m ď 4, this functoriality conjecture is valid. Moreover, Clozel and Thorne [9] derived the following further instances of the automorphy for symmetric powers of π. Theorem 2.3 (Clozel-Thorne). Assume that π corresponding to a non-CM Hilbert modular form of a totally real number field k, and k is linearly disjoint from Qpζ35q, m where ζ35 is a primitive 35th root of unity. Then Sym π is automorphic for m ď 8. Although the Langlands functoriality conjecture remains unproven, from the work of Barnet-Lamb, Gee, Geraghty, and Taylor [2, 3], it follows that that symmetric powers of any cuspidal representation π associated to a non-CM Hilbert modular form are potentially automorphic. More precisely, for every m, there is a totally real number m field L such that L{k is a finite Galois extension, and pSym πq|L is cuspidal over L. Moreover, by [4, Lemma 1.3], if L{F , containing k, is solvable, then it can be shown m that pSym πq|F is cuspidal over F .

2.5. Serre’s equidistribution criterion and the Sato-Tate conjecture for modular forms. Let G be a compact group, and let X be the space of conjugacy classes of G. Let pxpqp be a family of elements in X . Following Serre [22, Ch. I, A.2], we deﬁne the L-function Lps, ρq attached to an irreducible representation ρ of G (with respect to pxpqp) by ´s ´1 Lps, ρq “ detpI ´ ρpxpqp q p ź for Repsq ą 1. Serre [22, I-21–23] proved the following equidistribution criterion. Theorem 2.4 (Serre). Let µ be the push-forward measure of X obtained from the normalised Haar measure of G. Let pxpqp be a family of elements in X . If for any non- trivial irreducible representation ρ of G, Lps, ρq extends to a non-vanishing holomorphic 4 function on Repsq ě 1, then xp are µ-equidistributed. Now, we recall how Serre’s equidistribution criterion plays a role in proving the Sato-Tate conjecture for modular forms. Let G “ SUp2q be the group of 2 ˆ 2 unitary matrices with determinant one, and let X “ XSUp2q be the space of conjugacy classes of SUp2q. Let µ “ µST be the push-forward measure of X obtained from the normalised

4This means that for any complex-valued continuous function f on X , 1 lim fpxpq “ fdµ, xÑ8 π x p q pďx X ÿ ż where πpxq denotes the number of primes p ď x. SELBERG’S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS 7

Haar measure of G. In this case, any element of X contains a unique matrix of the form eiθ 0 M “ θ 0 e´iθ ˆ ˙ 2 2 for some θ P r0, πs. From this, we may identify pX , µST q with pr0, πs, π sin θdθq. 1 On the other hand, let π “ bpπp be a cuspidal representation of GL2pAQq deﬁned by a non-CM holomorphic newform of weight k ě 2 and write ´s ´1 ´s ´1 Lps, πpq “ p1 ´ απppqp q p1 ´ βπppqp q .

Since απppq and βπppq are conjugate and |απppqβπppq| “ 1 for any unramified p of π, iθp íθp there exists θp P r0, πs such that απppq “ e and βπppq “ e . Recall the non-trivial irreducible representations of G “ SUp2q are the mth symmetric powers ρm of the natural representation ρ1 of degree two. Choose xp “ Mθp if p is unramified of π and xp “ Mθ0 otherwise. As for any unramified p of π, m m ´s ´1 ipm´2jqθp ´s ´1 m´j j ´s ´1 detpI ´ ρmpMθp qp q “ p1 ´ e p q “ p1 ´ απpvq βπpvq p q , j“0 j“0 ź ź m we see that Lps, ρmq “ Lps, Sym πq (up to a finite number of terms in their Euler products). Thus, in order to prove the Sato-Tate conjecture for modular forms (i.e., 2 2 θp are equidistributed over r0, πs with respect to π sin θdθ), it suffices to show that each Lps, Symm πq is holomorphic and non-vanishing on Repsq ě 1. This is the main theorem of [4].

3. Consequences of potential automorphy and automorphic induction By virtue of the work of Arthur and Clozel [1] (on the automorphy of accessible characters arising from solvable Galois extensions), assuming the existence of automorphic induction for all Hecke characters, all Artin L-functions are automorphic. Inspired by this, we shall show that the existence of certain automorphic induction implies the automorphy of symmetric powers associated to non-CM Hilbert modular forms. To do this, we require the following lemma. Lemma 3.1. Let π be a cuspidal representation corresponding to a non-CM Hilbert modular form of a totally real number ﬁled k. Then Lps, Symm π ˆ Symm πq extends holomorphically to ts P Czt1u | Repsq ě 1u and has a simple pole at s “ 1. In addition, Lps, Symm π ˆ Symm πq is non-vanishing on Repsq ě 1. Consequently, writing

λ m paq Lps, Symm πq “ Sym π , Nas a‰0 ÿ for Repsq ą 1, where the sum is over non-zero integral ideals a of k, we have

λ m pvqλ m pvq Sym π Sym π “ log log x ` Op1q, Nv Nvďx ÿ where the sum is over primes v of k, and Nv is the norm of v. 8 PENG-JIE WONG

Proof. By the work of Barnet-Lamb, Gee, Geraghty, and Taylor [2, 3], there is a totally m real number ﬁeld L such that pSym πq|L is cuspidal, and L{k is ﬁnite and Galois. Let G “ GalpL{kq. By Solomon’s induction theorem, we have (3.1) 1 n IndG 1 , G “ i Hi Hi i ÿ where for each i, ni P Z and Hi is a quasi-elementary subgroup of G. Recall that m if L{F is solvable, then pSym πq|F is cuspidal. As every quasi-elementary group m is solvable, we conclude that pSym πq|LHi is cuspidal for every i. In addition, the m contragredient of pSym πq|LHi is also cuspidal for every i. Thus, by the theory of m m Rankin-Selberg L-functions, each Lps, pSym πq|LHi ˆ pSym πq|LHi q extends to a non- vanishing holomorphic function on Repsq ě 1 except for a simple pole at s “ 1. (Note that the contragredient of the base change is equivalent to the base change of the contragredient.) Observe that (3.1) implies that

m m m m ni (3.2) Lps, Sym π ˆ Sym πq “ Lps, pSym πq|LHi ˆ pSym πq|LHi q , i ź m m and hence ords“1 Lps, Sym π ˆ Sym πq is equal to m m n ord Lps, pSym πq| H ˆ pSym πq| H q “ n . i s“1 L i L i i i i ÿ ÿ ni Meanwhile, as (3.1) also gives that ζkpsq “ i ζLHi psq , which implies that

1 “ ords“1 ζkpsq “ niśords“1 ζLHi psq “ ni. i i ÿ ÿ (Here ζF psq denotes the Dedekind zeta function of a number ﬁeld F , which has a simple m m pole at s “ 1.) Herein, we conclude that ords“1 Lps, Sym π ˆ Sym πq “ i ni “ 1. m m Furthermore, as each Lps, pSym πq| H ˆ pSym πq| H q extends to a non-vanishing L i L i ř holomorphic function on Repsq ě 1 except for a simple pole at s “ 1, it follows from m m (3.2) that Lps, Sym π ˆ Sym πq extends holomorphically to ts P Czt1u | Repsq ě 1u and is non-vanishing on Repsq ě 1. Finally, by the Wiener-Ikehara Tauberian theorem, Abel’s summation, and the fact that the Ramanujan-Petersson conjecture is known for Symm π, writing

λ m m paq Lps, Symm π ˆ Symm πq “ Sym πˆSym π , Nas a‰0 ÿ for Repsq ą 1, where the sum is over non-zero integral ideals a of k, we have

λ m m pvq Sym πˆSym π “ log log x ` Op1q, Nv Nvďx ÿ We then conclude the proof by recalling that for unramiﬁed primes v of π,

m m λSymm πˆSymm πpvq “ λSym πpvqλSym πpvq and that there are only ﬁnitely many ramiﬁed primes v of π. SELBERG’S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS 9

We remark that assuming the automorphy of Symm π, one can apply the Rankin- Selberg theory to deduce Lemma 3.1 immediately. We are now ready to state and prove the main result of this section. Proposition 3.2. Let π be a cuspidal representation corresponding to a non-CM Hilbert modular form of a totally real number ﬁeld k. Suppose that for any totally F real extension F of k, the automorphic induction AIk pΠq of every pm ` 1q-dimensional cuspidal representation Π of F exists. Then Lps, Symm πq is cuspidal. Proof. In the notation of the proof above, the assumption of the existence of automor- LHi m phic induction yields that each AIk ppSym πq|LHi q exists as an automorphic representation of k, and hence

m m ni ej Lps, Sym πq “ Lps, pSym πq|LHi q “ Lps, πjq , i j ź ź where ej are integers, and πj are distinct cuspidal representations over k. Considering the order of the pole of the Rankin-Selberg L-function of Symm π and Symm π (at 2 s “ 1), the lemma above gives 1 “ j ej . Thus, |e1| “ 1 (say) and the remaining ej are zero. In other words, Lps, Symm πq is either equal to Lps, π q or Lps, π q´1. However, ř 1 1 the latter instance is impossible since it would give “trivial poles” for Lps, π1q. Thus, m Sym π is associated to π1.

4. Proofs of Theorems 1.3 and 1.4 4.1. Remarks on the use of Selberg’s orthogonality conjecture. As remark by Smajlovic, it is apparent that invoking Conjecture 1.1 for the whole Selberg class S to study certain L-functions is not necessary. Indeed, one may instead consider the following weak version of Selberg’s orthogonality conjecture (with respective to a given F P S). Conjecture 4.1 (SOC(F )). Let F P S. Then F itself and every primitive function appearing in a factorisation of F P S satisfies (1.1). Moreover, any pair of primitive functions F1 and F2 appearing in factorisations of F P S satisfies (1.2). As a demonstration of the use of this conjecture, we note that SOC(F ) forces the factorisation of F to be unique (which refines the earlier-mentioned result of Conery and Ghosh). Indeed, assume SOC(F ) and suppose that F admitted two different factorisations into primitive functions

F “ F1 ¨ ¨ ¨ Fr “ G1 ¨ ¨ ¨ Gt.

Without loss of generality, we may assume that no Gj is an F1. Observe that a ppqpa ppq ` ¨ ¨ ¨ ` a ppqq a ppqpa ppq ` ¨ ¨ ¨ ` a ppqq (4.1) F1 F1 Fr “ F1 G1 Gt . p p pďx pďx ÿ ÿ However, under SOC(F ), as x Ñ 8, the sum on the left tends to inﬁnity while the sum on the right is Op1q, which is impossible. 10 PENG-JIE WONG

Furthermore, as shall be seen in the below proofs of Theorems 1.3 and 1.4, instead of invoking the full power of Selberg’s orthogonality conjecture, certain SOC(F ) (for L- functions appearing in the factorisations (4.4) and (4.6)) are suﬃcient for us to proceed with the argument.

4.2. Chebotarev-Sato-Tate distribution and its consequence. Let π and θp be defined as in Section 2.5. Let K{Q be a Galois extension of totally real number fields with Galois group G. Following [26], we consider G “ SUp2qˆG. For such an instance, X “ XSUp2q ˆ XG and µ “ µST ˆ µG, where XG is the space of conjugacy classes of |C| G, and µG is the measure on XG defined by µGpCq “ |G| for any conjugacy class C in G. Let ρ0 and 1G denote the trivial representations of SUp2q and G, respectively. It is clear that any non-trivial irreducible representation of G “ SUp2q ˆ G is of the form ρm b τ ‰ ρ0 b 1G for some m ě 0 and irreducible representation τ of G, where ρm is the mth symmetric power of the natural representation ρ1 of degree two for SUp2q. (Note that ρ0 b 1G is the trivial representation of G “ SUp2q ˆ G.) Let τ be an irreducible representation of G and χ be its character. We choose xp “ pMθp , σpq if p is unramified of both π and K{Q and xp “ pMθ0 , eGq otherwise, where σp is the Artin symbol at p, and eG is the identity of G. We then form the L-function Lps, ρm b τq by

´s ´1 Lps, ρm b τq “ detpI ´ ρmpMθp q b τpσpqp q , p ź for Repsq ą 1. As in Section 2.5, it can be checked that Lps, ρm b τq is the same as Lps, Symm π ˆ χq (up to a ﬁnite number of terms in their Euler products). Moreover, it was shown in [26, Proof of Theorem 1.1] that each Lps, Symm π ˆ χq extends to a non-vanishing holomorphic function on Repsq ě 1 unless m “ 0 and χ “ 1G, where 0 5 Sym π denotes the trivial representation of GL1pAQq. Thus, by Theorem 2.4, xp are µ-equidistributed. 2 Now, let ψ “ trpρm b τq be the character of ρm b τ. As |ψ| is continuous on X and xp “ pMθp , σpq are µ-equidistributed, we have x x x x |ψpx q|2 “ |ψ|2dµ ` o “ ` o , p log x log x log x log x pďx X ÿ ´ ż ¯ ´ ¯ ´ ¯ where the last equality is due to the orthogonality of irreducible characters of compact groups. On the other hand, since

ψ “ trpρm b τq “ trpρmq trpτq “ trpρmqχ, denoting the pth coeﬃcients of Dirichlet series of Lps, Symm πq and Lps, Symm π ˆ χq by λSymm πppq and λSymm πˆχppq, respectively, we obtain

sinpm ` 1qθp m m ψpMθp , σpq “ χpσpq “ λSym πppqχpσpq “ λSym πˆχppq sin θp

5Indeed, the assertion follows from the virtue of (4.3) and the theory of Rankin-Selberg L-functions. SELBERG’S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS 11 for any p unramiﬁed of both π and K{Q. Hence, we have 2 2 |λ m ppq| |ψpMθ , σpq| (4.2) Sym πˆχ “ p ` Op1q “ log log x ` oplog log xq.6 p p pďx pďx ÿ ÿ 4.3. Proof of Theorem 1.3. To prove Theorem 1.3, we require the following lemma concerning the factorisation of F P S and the integer nF deﬁned in Conjecture 1.1 (cf. [19, Proposition 2.5]).

r ej Lemma 4.2. Assume SOC(F ). If F P S and F “ j“1 Fj is a factorisation into primitive functions, where Fj are distinct, then one has ś r 2 nF “ ej . j“1 ÿ In particular, SOC(F ) implies that F P S is primitive if and only if nF “ 1. Proof. It suﬃces to apply SOC(F ) to both sides of

2 r 2 |a ppq| | ejaFj ppq| F “ j“1 . p p pďx pďx ř ÿ ÿ and then compare the asymptotic behaviours. By invoking the Rankin-Selberg theory as in the proof of [19, Lemma 4.2], one may apply Lemma 4.2 to prove the following version of [19, Lemma 4.2].

Lemma 4.3. Let π be a cuspidal automorphic representation of GLnpAQq satisfying the Ramanujan-Petersson conjecture. Then under SOC(π), Lps, πq is primitive. Now, we are in a position to prove Theorem 1.3. Proof of Theorem 1.3. As shown in [2, 3], there is a totally real number ﬁeld L, contain- m m ing K, so that L{Q is Galois and pSym πq|L is automorphic. Moreover, pSym πq|F is cuspidal whenever L{F is a solvable subextension of L{Q. As L{Q is a Galois extension containing K{Q, one may regard χ as a character of G “ GalpL{kq and apply Brauer’s induction theorem to write χ n IndG ψ , “ i Hi i r i ÿ r where for each i, ni P Z, Hi is a nilpotent subgroup of G, and ψi is a character of Hi of degree one. By Artin reciprocity, each ψi can be seen as an id`eleclass character of LHi . We then have the following factorisation r

m m ni (4.3) Lps, Sym π ˆ χq “ Lps, pSym πq|LHi b ψiq . i ź Hi Since nilpotency implies solvability, each Hi is solvable, and so is L{L . Thus, m Hi each pSym πq|LHi is cuspidal over L . Now the functoriality of GLn ˆ GL1 yields m Hi m that pSym πq|LHi b ψi is cuspidal over L . Hence, Lps, pSym πq|LHi b ψiq is an

6 Note that if χ “ 1G, then this estimate follows from Lemma 3.1, immediately. 12 PENG-JIE WONG automorphic L-function, satisfying the Ramanujan-Petersson conjecture, and thus a member in the Selberg class S. Therefore, we can write

` m ej (4.4) Lps, Sym π ˆ χq “ Fjpsq , j“1 ź where for each j, Fj P S and ej P Z. Since every element in S has a factorisation into primitive functions, we may assume all Fj are primitive and distinct. Comparing the pth coeﬃcients of Dirichlet series of both sides of (4.4), one has

m λSym πˆχppq “ ejaFj ppq. j“1 ÿ Herein, we derive

2 ` |λ m ppq| 1 2 (4.5) Sym πˆχ “ e a ppq . p p j Fj pďx pďx ˇ j“1 ˇ ÿ ÿ ˇ ÿ ˇ Assuming SOC(Fj) for all j, we deduce that the asymptoticˇ behaviourˇ of the right-hand side of (4.5) is ` 2 ej log log x ` Op1q. j“1 ´ ÿ ¯ On the other hand, (4.2) tells us that the left-hand side of (4.5) is log log x`oplog log xq. ` 2 Therefore, we deduce that j“1 ej “ 1, from which it follows that e1 “ ˘1 (say). Thus, m by Lemma 4.2, nF “ 1 and Lps, Sym π ˆ χq is primitive. ř 4.4. Proof of Theorem 1.4. Inspired by the proof of Theorem 1.2, we shall show that certain “automorphic factorisation” assumption, combined with Selberg’s orthogonality conjecture, would lead to some instances of the Langlands functoriality conjecture as follows.

Proposition 4.4. Let π be a cuspidal representation of GL2pAQq deﬁned by a non-CM holomorphic newform of weight k ě 2. Let K{Q be a totally real Galois extension with m Galois group G. Suppose that Lps, pSym πq|K q admits a factorisation of the form

m ej (4.6) Lps, pSym πq|K q “ Lps, πjq , j ź where πj are cuspidal representations over Q, and ej P N. If SOC(Fj) holds for each m Fj appearing in (4.4), then SOC(pSym πq|K ), together with SOC(πj), implies that for m every χ P IrrpGq, Lps, Sym π ˆ χq is automorphic over Q, where IrrpGq denotes the set of irreducible characters of G. Proof. Since by the work of Deligne, the Ramanujan-Petersson conjecture holds for π, comparing the local Euler factors on both sides of the decomposition (4.6), we deduce that each Lps, πjq satisﬁes the Ramanujan-Petersson conjecture. Thus, under SELBERG’S ORTHOGONALITY CONJECTURE AND SYMMETRIC POWER L-FUNCTIONS 13

SOC(πj), Lemma 4.3 yields that each Lps, πjq is primitive. On the other hand, we have the classical factorisation m m χp1q Lps, pSym πq|K q “ Lps, Sym π ˆ χq . χPIrrpGalpK{ qq ź Q Note that by Theorem 1.3, each Lps, Symm π ˆ χq is a primitive function in the Sel- m berg class S. As Lps, pSym πq|K q factorises into a product of primitive functions m uniquely (under SOC(pSym πq|K )), and Lps, πjq are primitive (under SOC(πj)), each m Lps, Sym π ˆ χq must be Lps, πjq for some j, which concludes the proof. To end this section, we shall prove Theorem 1.4. m Proof of Theorem 1.4. As Sym π is automorphic (and thus cuspidal) and K{Q is solv- m able, Arthur-Clozel’s base change [1] yields that pSym πq|K exists as an automorphic representation over K. Moreover, the automorphic induction K m AIQ ppSym πq|K q m exists as an automorphic representation over Q. Thus, Lps, pSym πq|K q factorises into a product of cuspidal L-functions over Q, which together with Proposition 4.4 (under m SOC(pSym πq|K ) and SOC(πj) and SOC(Fj) for each πj and Fj appearing in (4.6) and (4.4), respectively) completes the proof. (Note that for m ď 8, the automorphy m assumption of Sym π can be dispensed by invoking Theorem 2.3.)

Acknowledgments The author would like to thank Professors Habiba Kadiri, Wen-Ching Winnie Li, Ram Murty, and Nathan Ng for their kind encouragement and helpful suggestions. He is also grateful to Professor Lejla Smajlovic for making several constructive comments, which help to improve the exposition considerably. Moreover, the author would like to express his sincere gratitude to the referee for the insightful and valuable comments.

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Department of Mathematics and Computer Science, University of Lethbridge, Leth- bridge, Alberta T1K 3M4, Canada. Email address: [email protected]