AN APPROACH TO NONSOLVABLE BASE CHANGE AND DESCENT JAYCE R. GETZ Abstract. Wepresentacollectionofconjecturaltraceidentitiesandexplainwhytheyare equivalent to base change and descent of automorphic representations of GL (A ) along n F 7 nonsolvable extensions (under some simplifying hypotheses). The case n = 2 is treated in 1 0 more detail and applications towards the Artin conjecture for icosahedral Galois represen- 2 tations are given. n a J 6 Contents ] T N 1. Introduction 2 . 1.1. Test functions 3 h t 1.2. Conjectural trace identities 4 a m 1.3. Icosahedral extensions 7 [ 1.4. On the Artin conjecture for icosahedral representations 9 1 2. General notation 11 v 6 2.1. Ad`eles 11 6 7 2.2. Restriction of scalars 11 1 2.3. Characters 11 0 1. 2.4. Harish-Chandra subgroups 11 0 2.5. Local fields 12 7 1 2.6. Field extensions 12 : v 3. Test functions 12 i X 3.1. Unramified Hecke algebras 12 r a 3.2. Base change for unramified Hecke algebras 13 3.3. Transfers 14 4. Limiting forms of the cuspidal spectrum 18 4.1. Rankin-Selberg L-functions and their analytic conductors 19 4.2. Proof of Corollary 4.2 22 5. Restriction and descent of L-parameters 28 5.1. Parameters and base change 29 5.2. Descent of parameters 30 5.3. Primitive parameters and automorphic representations 32 5.4. Restriction of parameters 33 2000 Mathematics Subject Classification. Primary 11F70. 1 2 JAYCE R.GETZ 5.5. The icosahedral group 37 5.6. Motivating conjectures 5.16 and 5.18 40 5.7. Appendix: The representations of some binary groups 41 6. Proofs of the main theorems 43 6.1. Preparation 44 6.2. Functoriality implies the trace identities 47 6.3. The trace identity implies functoriality: first two cases 48 6.4. Artin representations: Theorem 1.8 52 7. Some group theory 57 7.1. Comments on universal perfect central extensions 57 7.2. Generating Gal(E/F) 58 Acknowledgments 59 References 59 1. Introduction Let F be a number field and let v be a nonarchimedian place of F. By the local Langlands correspondence, now a theorem due to Harris and Taylor building on work of Henniart, there is a bijection (1.0.1) ϕ : W GL (C) π(ϕ ) v F′v → n 7−→ v between equivalence classes o(cid:0)f Frobenius semisimp(cid:1)le representations ϕ of the local Weil- v Deligne group W and isomorphism classes of irreducible admissible representations of F′v GL (F ). The bijection is characterized uniquely by certain compatibilities involving ε- n v factors and L-functions which are stated precisely in [HT] (see also [H2]). The correspond- ing statement for v archimedian was proven some time ago by Langlands [La5]. We write ϕ (π ) for any representation attached to π and call it the Langlands parameter or v v v L-parameter of π ; it is unique up to equivalence of representations. v Now let E/F be an extension of number fields, let v be a place of F and let w v be a | place of E. We say that an admissible irreducible representation Π of GL (E ) is a base w n w change of π and write π := Π if v vEw w ϕ(πv) W′ ∼= ϕ(Πw). | Ew In this case we also say that Π descends to π . We say that an isobaric1 automorphic w v representation Π of GL (A ) is a base change (resp. weak base change) of an isobaric n E automorphic representation π of GL (A ) if Π = π for all (resp. almost all) places v of n F w vE F and all places w v of E. If Π is a (weak) base change of π, then we also say Π descends | (weakly) to π. We write π for a weak base change of π, if it exists; it is uniquely determined E 1For generalities on isobaric automorphic representations see [La2] and [JShII]. AN APPROACH TO NONSOLVABLE BASE CHANGE AND DESCENT 3 up to isomorphism by the strong multiplicity one theorem [JShII, Theorem 4.4]. If Π is a weak base change of π, we say that the base change is compatible at a place v of F if Π is w a base change of π for all w v. v | If E/F is a prime degree cyclic extension, then the work of Langlands [La4] for n = 2 and Arthur-Clozel [AC] for n arbitrary implies that a base change always exists. The fibers and image of the base change are also described in these works. Given that any finite degree Galois extension E/F contains a family of subextensions E = E E E = F 0 1 n ≥ ≥ ··· ≥ where E /E is Galois with simple Galois group, to complete the theory of base change it i i+1 is necessary to understand base change and descent with respect to Galois extensions with simple nonabelian Galois group. In this paper we introduce a family of conjectural trace identities that are essentially equivalent to proving base change and descent in this setting. The (conjectural) traceidentity is based oncombining two fundamental paradigms pioneered by Langlands, the second still in its infancy: Comparison of trace formulae, and • Beyond endoscopy. • The point of this paper is to provide some evidence that proving the conjectural trace identities unconditionally is a viable strategy for proving nonsolvable base change. 1.1. Test functions. Fix an integer n 1, a Galois extension of number fields E/F, an ≥ automorphism τ Gal(E/F), and a set of places S of E containing the infinite places and 0 ∈ the places where E/F is ramified. Let w S and let v be the place of F below w. Let 0 6∈ t t 1w 1wτ A = ... and Aτ = ... .     t t  nw  nwτ     We view these as matrices in GL (C[t 1, ,t 1]) For j Z let Symj : GL wv n ±1w ··· ±nw ∈ >0 n → GL be the jth symmetric pow| er representation, where m is the m-choose-j binomial (n+j−1) Q j j coefficient. (cid:0) (cid:1) For a prime power ideal ̟j of define a test function w OE (1.1.1) f(̟j) := 1(tr(Symj(A (Aτ) 1))) C (GL (E F )//GL ( )) w S− ⊗ − ∈ c∞ n ⊗F v n OE ⊗OF OFv where is the Satake isomorphism (see 3.1). We denote by f( ) the characteristic func- S § OEw tion of GL ( ) and regard the f(̟j) as elements of C (GL (A )//GL ( )). n OE ⊗OF OFv w c∞ n ∞E n OE Define f(n) C (GL (A )//GL ( )) in general by declaring that f is multiplicative, ∈ c∞ n ∞E n OE b that is, if n+m = we set E O b f(nm) := f(n) f(m) ∗ where the asterisk denotes convolution in the Hecke algebra. If n is coprime to S , we often 0 view f(n) as an element of C (GL (AS0)//GL ( S0)). c∞ n E n OE b 4 JAYCE R.GETZ Assume that Π is an isobaric automorphic representation of GL (A ) unramified outside n E of S . Define Πτ by Πτ(g) := Π(gτ). The purpose of defining the operators f(m) is the 0 following equality: tr(ΠS0)(f(m)) = LS0(s,Π Πτ) N (m) s × F/Q mXS0 | | ⊂OE This follows from (4.1.9) and the fact that, in the notation of loc. cit., A(Πτ) = A(Π ). w wτ Let φ C (0, ) be nonnegative. Thus φ(1) > 0, where ∈ c∞ ∞ φ(s) := e∞φ(s)xs−1dx Z0 is the Mellin transform of φ. We inetroduce the following test function, a modification of that considered by Sarnak in [S]: (1.1.2) ΣS0(X) := ΣS0(X) := φ(X/ N (m) )f(m). φ | E/Q | mXS0 ⊂OE 1.2. Conjectural trace identities. Assume that E/F is a Galois extension. For conve- nience, let (1.2.1) Π (F) : = Isom. classes of isobaric automorphic representations of GL (A ) n n F { } Π0(F) : = Isom. classes of cuspidal automorphic representations of GL (A ) n { n F } Πprim(E/F) : = Isom. classes of E-primitive automorphic representations of GL (A ) . n { n F } The formal definition of an E-primitive automorphic representation is postponed until 5.3. § If weknew Langlandsfunctoriality wecouldcharacterize themeasily asthose representations that are cuspidal and not automorphically induced from an automorphic representation of a subfield E K > F. We note that there is a natural action of Gal(E/F) on Π (E) that n ≥ preserves Π0(E); we write Π (E)Gal(E/F) for those representations that are isomorphic to n n their Gal(E/F)-conjugates and Π0(E)Gal(E/F) = Π0(E) Π (E)Gal(E/F). n n ∩ n Let S be a finite set of places of F including all infinite places and let S , S be the set ′ 0 of places of F′, E lying above S. Assume that h ∈ Cc∞(GLn(AF′)), Φ ∈ Cc∞(GLn(AF)) are transfers of each other in the sense of 3.3 below and that they are unramified outside of § S and S, that is, invariant under right and left multiplication by GL ( S′) and GL ( S), ′ n OF′ n OF respectively. For the purposes of the following theorems, if G is a finite group let Gab be the b b maximal abelian quotient of G. Assume for the remainder of this subsection that Gal(E/F) is the universal perfect central extension of a finite simple nonabelian group. Let E F F be a subfield such that ′ ≥ ≥ Gal(E/F ) is solvable and H2(Gal(E/F ),C ) = 0. Moreover let τ Gal(E/F) be an ′ ′ × ∈ element such that Gal(E/F) = τ,Gal(E/F ) . ′ h i AN APPROACH TO NONSOLVABLE BASE CHANGE AND DESCENT 5 Remark. In 7.1 we discuss these assumptions, the upshot being that they are no real loss § of generality. Our first main theorem is the following: Theorem 1.1. Consider the following hypotheses: Gal(E/F ) is solvable, H2(Gal(E/F ),C ) = 0 and [E : F ] is coprime to n. ′ ′ × ′ • For all divisors m n there is no irreducible nontrivial representation • | Gal(E/F) GL (C), m −→ The case of Langlands functoriality explicated in Conjectures 5.10 below is true for • E/F, and The case of Langlands functoriality explicated in Conjecture 5.11 is true for E/F . ′ • If these hypotheses are valid and h and Φ are transfers of each other then the limits (1.2.2) Xlim |Gal(E/F′)ab|−1X−1 tr(π′)(h1bE/F′(ΣSφ0(X))) →∞ π′ E-primitive X and (1.2.3) lim X 1 tr(π)(Φ1b (ΣS0(X))) − E/F φ X →∞ π X converge absolutely and are equal. Here the first sum is over a set of representatives for the equivalence classes of E-primitive cuspidal automorphic representations of AGLnF′\GLn(AF′) and the second sum is over a set of representatives for the equivalence classes of cuspidal automorphic representations of A GL (A ). GLnF\ n F Here h1(g) : = h(ag)da ′ ZAGLnF′ Φ1 : = Φ(ag)da ZAGLnF where the da and da are the Haar measures on A and A , respectively, used in the ′ GLnF′ GLnF definition of the transfer. For the definition of A and A we refer to 2.4 and for GLnF′ GLnF § the definition of bE/F and bE/F′ we refer to (3.2.1). Remarks. (1) If we fix a positive integer n, then for all but finitely many finite simple groups G with universal perfect central extensions G any representation G GL will be trivial. This n → follows from [LiSha, Theorem 1.1], for example (the author does not known if it was known e e earlier). Thus the second hypothesis in Theorem 1.1 holds for almost all groups (if we fix n). In particular, if n = 2, then the only finite simple nonabelian group admitting a projective representation of degree 2 is A by a well-known theorem of Klein. Thus when n = 2 and 5 6 JAYCE R.GETZ Gal(E/F) is the universal perfect central extension of a finite simple group other than A 5 the first hypothesis of Theorem 1.1 holds. (2) Conjecture 5.10 and its analogues conjectures 5.14, 5.16 and 5.18 below each amount to a statement that certain (conjectural) functorial transfers of automorphic representations exist andhave certainproperties. Tomotivatethese conjectures, westateandprove theproperties of L-parameters to which they correspond in propositions 5.9, 5.13 and lemmas 5.15, 5.17 respectively. The facts about L-parameters we use are not terribly difficult to prove given basic facts from finite group theory, but they are neither obvious nor well-known, and one of the motivations for this paper is to record them. (3) Conjecture 5.11 is a conjecture characterizing the image and fibers of solvable base change. Rajan [R2] has explained how it can be proved (in principle) using a method of Lapid and Rogawski [LapRo] together with the work of Arthur and Clozel [AC]. It is a theorem when n = 2 [R2, Theorem 1] or when Gal(E/F) is cyclic of prime degree [AC, Chapter 3, Theorems 4.2 and 5.1]. The following weak converse of Theorem 1.1 is true: Theorem 1.2. Assume Conjecture 3.2 on transfers of test functions and assume that F is totally complex. If (1.2.2) and (1.2.3) converge absolutely and are equal for all h unramified outside of S with transfer Φ unramified outside of S, then every cuspidal automorphic rep- ′ resentation Π of GL (A ) satisfying Πσ = Π for all σ Gal(E/F) admits a unique weak n E ∼ ∈ descent to GL (A ). Conversely, if π is a cuspidal automorphic representation of GL (A ) n F n F such that (1.2.4) lim X 1tr(π)(b (ΣS0(X))) = 0 X − E/F φ 6 →∞ then π admits a weak base change to GL (A ). The weak base change of a given cuspidal n E automorphic representation π of GL (A ) is unique. The base change is compatible for the n F infinite places of F and the finite places v of F where E/F and π are unramified or where v π is a twist of the Steinberg representation by a quasi-character. v Here, as usual, Πσ := Π σ is the representation acting on the space of Π via ◦ Πσ(g) := Π(σ(g)). Remark. Conjecture 3.2 is roughly the statement that there are “enough” h and Φ that are transfers of each other. If one assumes that Π (resp. π) and E/F are everywhere unramified, then one can drop the assumption that Conjecture 3.2 is valid. We conjecture that (1.2.4) is always nonzero: Conjecture 1.3. Let π be a cuspidal automorphic representation of A GL (A ); then GLnF\ n F lim X 1tr(π)(b (ΣS0(X))) = 0 X − E/F φ 6 →∞ AN APPROACH TO NONSOLVABLE BASE CHANGE AND DESCENT 7 This conjecture is true for all π that admit a base change to an isobaric automorphic representation of GL (A ) by an application of Rankin-Selberg theory (compare Proposition n E 4.3 and (4.1.2)), however, assuming this would be somewhat circular for our purposes. The author is hopeful that Conjecture 1.3 can be proven independently of the existence of the base change. Indeed, the Chebatarev density theorem is proven despite the fact that the Artin conjecture is still a conjecture. The smoothed sum in Conjecture 1.3 is analogous to some sums that can be evaluated using the Chebatarev density theorem; in some sense the Chebatarev density theorem is the case where π is the trivial representation of GL (A ). 1 F Isolating primitive representations is not a trivial task. For example, the main focus of [V] is the isolation of cuspidal representations that are not primitive when n = 2. Therefore it seems desirable to have a trace identity similar to that of Theorem 1.1 that involves sums over all cuspidal representations. This is readily accomplished under additional assumptions on Gal(E/F) using the following lemma: Lemma 1.4. Let L/K be a Galois extension of number fields. Suppose that there is no proper subgroup H Gal(L/K) such that [Gal(L/K) : H] n. Then ≤ | Πprim(L/K) = Π0(K). n n (cid:3) The proof is immediate from the definition of L-primitive automorphic representations in 5.3. § 1.3. Icosahedral extensions. We now consider the case of the smallest simple nonabelian group A in more detail. We begin by setting notation for specific subsets of Π0(F) and 5 n Π0(E). n Let E/F be a Galois extension, and let ρ : W LGL F′ −→ nF be an L-parameter trivial on W ; thus ρ can essentially be identified with the Galois E′ representation ρ : Gal(E/F) GL (C) obtained by composing ρ with the projection 0 n → LGL GL (C). For every quasi-character χ : F A = (W )ab C we can then form nF → n ×\ ×F ∼ F′ → × the L-parameter ρ χ : W GL (C). ⊗ F′ −→ n We say that a cuspidal automorphic representation π of GL (A ) is associated to ρ χ if n F ⊗ π is the representation attached to the L-parameter (ρ χ) : v v ⊗ π = π((ρ χ) ) v v ⊗ for almost all places v of F (see (1.0.1) above). If π = π((ρ χ) ) for all places v, then we v v ⊗ write π = π(ρ χ). In this case we also say that π and ρ χ are strongly associated. ⊗ ⊗ More generally, if π is a cuspidal automorphic representation of GL (A ) such that π is n F 8 JAYCE R.GETZ associated to ρ χ for some χ we say that π is of ρ-type. If π is associated to ρ χ for ⊗ ⊗ some ρ and χ we say that π is of Galois type. Assumefortheremainder ofthissectionthatGal(E/F) = A , theuniversal perfectcentral ∼ 5 extension of the alternating group A on 5 letters. One can formulate analogues of theorems 5 e 1.1 and 1.2 in this setting. For this purpose, fix an embedding A ֒ A , and let A A 4 5 4 5 → ≤ be the preimage of A under the surjection A A . Thus A is a nonsplit double cover of 4 5 5 4 → e e A . 4 e e e Theorem 1.5. Let n = 2, let F = EA4, and let τ Gal(E/F) be any element of order ′ ∈ 5. Let h ∈ Cc∞(GL2(AF′)) be unramified outside of S′ and have transfer Φ ∈ Cc∞(GL2(AF)) unramified outside of S. Assume the case of Langlands functoriality explicated in Conjecture 5.14 for E/F. Then the limits d3 −1 (1.3.1) 2Xlim ds3(φ(s)Xs)|s=1 |Gal(E/F′)ab|−1 tr(π′)(h1bE/F′(ΣSφ0(X))) →∞(cid:18) (cid:19) π′ X e and d3 −1 (1.3.2) lim (φ(s)Xs) tr(π)(Φ1b (ΣS0(X))) X ds3 |s=1 E/F φ →∞(cid:18) (cid:19) π X e converge absolutely and are equal. Similarly, again assuming Conjecture 5.14 below, the limits (1.3.3) Xlim X−1|Gal(E/F′)ab|−1 tr(π′)(h1bE/F′(ΣSφ0(X))) →∞ π′ not of ρ-type for ρtrivial on W′ X E and (1.3.4) lim X 1 tr(π)(Φ1b (ΣS0(X))) − E/F φ X →∞ π not of ρ-type for ρ trivialon W′ X E converge absolutely and are equal. In both cases the first sum is over a set of representa- tives for the equivalence classes of cuspidal automorphic representations of AGL2F′\GL2(AF′) and the second sum is over a set of representatives for the equivalence classes of cuspidal automorphic representations of A GL (A ). GL2F\ 2 F Again, a converse statement is true: Theorem 1.6. Assume Conjecture 3.2 on transfers of test functions, assume that F is to- tally complex, and assume that the limits (1.3.3) and (1.3.4) converge absolutely for all test functions h unramified outside of S with transfer Φ unramified outside of S. Under these ′ assumptions every cuspidal automorphic representation Π of GL (A ) that is isomorphic to 2 E its Gal(E/F)-conjugates is a weak base change of a unique cuspidal automorphic represen- tation of GL (A ). Conversely, if π is a cuspidal automorphic representation of GL (A ) 2 F 2 F AN APPROACH TO NONSOLVABLE BASE CHANGE AND DESCENT 9 such that lim X 1tr(π)(b (ΣS0(X))) = 0 X − E/F φ 6 →∞ then π admits a unique weak base change to GL (A ). If π is a cuspidal automorphic 2 F representation of GL (A ) that is not of ρ-type for ρ trivial on W , then π is cuspidal. The 2 F F′ E base change is compatible at the infinite places of F and the finite places v of F where E/F and π are unramified or π is a twist of the Steinberg representation by a quasi-character. v v 1.4. On the Artin conjecture for icosahedral representations. As in the last sub- section we assume that Gal(E/F) = A . Fix an embedding Z/2 Z/2 ֒ A and let ∼ 5 5 × → Q ֒ A be the inverse image of Z/2 Z/2 under the quotient A A . For the purposes 5 5 5 → × → e of the following theorem, let S be a subset of the places of F disjoint from S and let S 1 1′ (resp.Se10)bethesetofplacesofF′ (resp.E)aboveS1. MoreoverleethS1′ ∈ Cc∞(GL2(ASF′′)and ΦS1 C (GL (AS1)) be transfers of each other unramified outside of S and S, respectively, ∈ c∞ 2 F ′ and let hS1′ ∈ Cc∞(GL2(FS′1′)//GL2(OFS′1′)). Theorem 1.7. Consider the following hypotheses: One has n = 2 and F = EQ, and the case of Langlands functoriality explicated in ′ • Conjecture 5.16 is true for E/F. e One has n = 3, F = EA4, the case of Langlands functoriality explicated in Conjecture ′ • 5.18 is true for E/F, and Conjecture 5.11 is true for E/F ′ Under these assumptions the limits 1 dn2 1 − (1.4.1) 2Xli→m∞ dsn2−−1(φ(s)Xs) s=1! π′ tr(π′)((hS1′)1hS1′bE/F′(ΣSφ0(X))) (cid:12) X and e (cid:12) 1 dn2 1 − (1.4.2) Xli→m∞ dsn2−−1(φ(s)Xs) s=1! π tr(π)((ΦS1)1bF′/F(hS1′)bE/F(ΣSφ0(X))) (cid:12) X converge absolutely and are eequal. H(cid:12)ere the first sum is over equivalence classes of cuspidal automorphic representations of AGLnF′\GLn(AF′) and the second sum is over equivalence classes of cuspidal automorphic representations of A GL (A ). GLnF\ n F Remarks. (1) One can always find τ Gal(E/F) such that τ,Gal(E/F ) = Gal(E/F) (this follows ′ ∈ h i from Theorem 7.1, for example, or by an elementary argument). (2) The fact that this theorem involves more general test functions than those in theorems 1.1 and 1.5 is important for applications to the Artin conjecture (see Theorem 1.8). Let ρ : W LGL be an irreducible L-parameter trivial on W (i.e. an irreducible 2 F′ → 2F E′ Galois representation ρ : Gal(E/F) GL (C)). Its character takes values in Q(√5) and if 2 2 → 10 JAYCE R.GETZ ξ = Gal(Q(√5)/Q) then ξ ρ is another irreducible L-parameter that is not equivalent 2 h i ◦ to the first (see 5.7). A partial converse of Theorem 1.7 above is the following: § Theorem 1.8. Assume Conjecture 3.2 and that (1.4.1) and (1.4.2) converge and are equal for all test functions as in Theorem 1.7 for n 2,3 . Assume moreover that F is totally ∈ { } complex. Then there is a pair of nonisomorphic cuspidal automorphic representations π ,π 1 2 of GL (A ) such that 2 F π ⊞π = π((ρ ξ ρ )). 1 2 ∼ 2 2 ⊕ ◦ ⊞ Here the denotes the isobaric sum [La2] [JShII]. Remark. It should be true that, upon reindexing if necessary, π = π(ρ ). However, the 1 ∼ 2 author does not know how to prove this at the moment. As a corollary of this theorem and work of Kim and Shahidi, we have the following: Corollary 1.9. Under the hypotheses of Theorem 1.8, if ρ : Gal(E/F) GL (C) is an ir- n → reducible Galois representation of degree strictly greater than 3, then there is an automorphic representation π of GL (A ) such that π = π(ρ). n F The point of the theorems above is that the sums (1.2.2), (1.2.3) and their analogues in the other theorems can be rewritten in terms of orbital integrals using either the trace formula (compare [La6], [FLN]) or the relative trace formula (specifically the Bruggeman- Kuznetsov formula, compare [S], [V])2. One then can hope to compare these limits of orbital integrals and prove nonsolvable base change and descent. The author is actively working on this comparison. He hopes that the idea of comparing limiting forms of trace formulae that underlies theorems 1.2, 1.6, 1.8 will be useful to others working “beyond endoscopy.” Toendtheintroduction weoutlinethesections ofthis paper. Section2states notationand conventions; it can be safely skipped and later referred to if the reader encounters unfamiliar notation. In 3, we review unramified base change, introduce a notion of transfer for test § functions, and prove the existence of the transfer in certain cases. Section 4 introduces the smoothed test functions used in the statement and proof of our main theorems and develops their basic properties using Rankin-Selberg theory. Perhaps the most important result is that the trace of these test functions over the cuspidal spectrum is well-defined and picks out the representations of interest (see Corollary 4.2). The behavior of L-parameters under restriction along an extension of number fields is considered in 5; this is used to motivate § the conjectures appearing in our main theorems above, which are also stated precisely in 5. § Section 6 contains the proofs of the theorems stated above and the proof of Corollary 1.9. Finally, in 7 we explain why the group-theoretic assumptions made in theorems 1.1 and 1.5 § are essentially no loss of generality. 2We note that when applying the relative trace formula the distributions h tr(π′)(h1) will be replaced 7→ byBesseldistributionsdefinedusingWhittakerfunctionals. ThusthedefinitionofΣS0(X)hastobemodified φ to be useful in a relative trace formula approach.