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Selberg's Orthogonality Conjecture and Symmetric

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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 ¥ 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 ¡ 1, 8 a pnq 8 F psq “ F “ exp b ppkq{pks ns F n“1 p k“1 ¸ ¹ ´ ¸ ¯ 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 ¡ 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 ¡ 0, αF pjq ¡ 0, RepγF pjqq ¥ 0, so that rF s ΛF psq “ QF ΓpαF pjqs ` γF pjqqF psq j“1 ¹ satisfies ΛF psq “ wF ΛF p1 ´ sq for some complex number wF of absolute value 1. (Ramanujan-Petersson conjecture). For any fixed ¡ 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 p¤x ¸ 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 p¤x ¸ 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 conjec- ture, 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, one¸ has the generalised prime number theo- rem asserting that ΛF pnq „ mF x; n¤x ¸ 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 distri- bution 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| ¤ 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 ¡ 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 mod- ular forms. Let AQ be the ad`elering of Q. Let π be a cuspidal representation πf of GL2pAQq defined by a non-CM holomorphic newform of weight k ¥ 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 ¥ 1 and meromorphically to C. Furthermore, Lps; Sym πq satisfies the Ramanujan-Petersson conjecture and expected functional equation. Thus, un- der 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 ar- ticle 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 ¥ 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 ¥ 2. Let K{Q be a solvable totally real Galois ex- tension 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 ¤ 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.
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