ON THE DEGREE OF CERTAIN LOCAL L-FUNCTIONS

Transcription

1 ON THE DEGREE OF CERTAIN LOCAL L-FUNCTIONS U K ANANDAVARDHANAN AND AMIYA KUMAR MONDAL Abstract Lt π b an irrducibl suprcuspidal rprsntation of GL n (F ), whr F is a p-adic fild By a rsult of Bushnll and Kutzko, th group of unramifid slf-twists of π has cardinality n/ whr is th o F -priod of th principal o F -ordr in M n (F ) attachd to π This is th dgr of th local Rankin-Slbrg L-function L(s, π π ) In this papr, w comput th dgr of th Asai, symmtric squar and xtrior squar L-functions associatd to π As an application, assuming p is odd, w comput th conductor of th Asai lift of a suprcuspidal rprsntation, whr w also mak us of th conductor formula for pairs of suprcuspidal rprsntations du to Bushnll, Hnniart, and Kutzko [BHK98] 1 Introduction Lt F b a p-adic fild Lt o F dnot its ring of intgrs and lt p F b th uniqu maximal idal of o F Lt q dnot th cardinality of th rsidu fild o F /p F Lt W F dnot th Wil-Dlign group of F For a rductiv algbraic group G dfind ovr F, lt L G b its Langlands dual Givn a Langlands paramtr ρ : W F L G and a finit dimnsional rprsntation r : L G GL(V ), w hav an L-function L(s, ρ, r) dfind as follows If N is th nilpotnt ndomorphism of V associatd to r ρ, thn 1 L(s, ρ, r) = ) dt (1 (r ρ)(frob) q s (Kr N)I whr Frob is th gomtric Frobnius and I is th inrtia subgroup of th Wil group of F Thus, L(s, ρ, r) = P (q s ) 1 for som polynomial P (X) with P (0) = 1, and by th dgr of L(s, ρ, r) w man th dgr of P (X) If π = π(ρ) dnots th L-packt of irrducibl admissibl rprsntations of G(F ) corrsponding to ρ undr th conjctural Langlands corrspondnc, thn its Langlands L-function, dnotd by L(s, π, r), is xpctd to coincid with L(s, ρ, r) In many cass, candidats for L(s, π, r) can also b obtaind ithr via th Rankin-Slbrg mthod of intgral rprsntations or by th Langlands-Shahidi mthod, and in svral instancs it is known that all ths approachs lad to th sam L-function [Sha84, Sha90, AR05, Hn10, Mat11, KR12] If G = GL(n 1 ) GL(n 2 ), π i an irrducibl admissibl rprsntation of GL ni (F ) (i = 1, 2), and if r is th tnsor product rprsntation of L G = GL n1 (C) GL n2 (C) on C n C n givn by r((a, b)) (x y) = ax by, thn th rsulting L-function is th Rankin-Slbrg L-function L(s, π 1 π 2 ) [JPSS83, Sha84] If w assum that both π 1 and π 2 ar suprcuspidal rprsntations, thn on knows that L(s, π 1 π 2 ) 1 unlss n 1 = n 2 and π2 = π 1 χ dt for an unramifid charactr χ of F Hr, 1991 Mathmatics Subjct Classification Primary 22E50; Scondary 11F33, 11F70, 11F85 1

2 2 U K ANANDAVARDHANAN AND AMIYA KUMAR MONDAL π dnots th rprsntation contragrdint to π Morovr, in th lattr cas, th dgr of L(s, π 1 π 2 ) is qual to th dgr of L(s, π 1 π 1 ) which in turn quals th cardinality of th group {η : F C π 1 η dt = π 1, η unramifid} Th rsult of Bushnll and Kutzko mntiond in th abstract computs th cardinality of th abov group of unramifid slf-twists of π = π 1 [BK93, Lmma 625] In ordr to stat th rsult, lt [A, m, 0, β] b th simpl stratum dfining a maximal simpl typ occurring in th irrducibl suprcuspidal rprsntation π Hr, A is a principal o F -ordr in M n (F ), m 0 is an intgr calld th lvl of π, and β M n (F ) is such that F [β] is a fild with F [β] normalizing A Lt = (A o F ) b th o F -priod of A; this quantity in fact quals th ramification indx (F [β]/f ) of F [β]/f Thn, divids n, and th cardinality of th group of unramifid slf-twists of π is n/ W mntion in passing that th lvl m of π is rlatd to th conductor f(π) of π by f(π) = n ( ) 1 + m Th aim of th prsnt work is to analogously comput th dgr of som othr local L-functions in th suprcuspidal cas Invstigating th suprcuspidal cas would suffic as th L-function of any irrducibl admissibl rprsntation can b usually built out of L-functions associatd to suprcuspidal rprsntations Th L-functions that w study in this papr ar: th Asai L-function, th symmtric squar L-function, and th xtrior squar L-function For th Asai L-function, tak G = Rs E/F GL(n), th Wil rstriction of GL(n), whr E is a quadratic xtnsion of F Thus, G(F ) = GL n (E) In this cas, th dual group is L G = GL n (C) GL n (C) Gal(E/F ), whr th nontrivial lmnt of th Galois group Gal(E/F ), say σ, acts by σ (a, b) = (b, a) Th rprsntation r is th Asai rprsntation, also known as th twistd tnsor rprsntation, of L G on C n C n givn by r((a, b)) (x y) = ax by and r(σ) (x y) = y x Th Asai L-function can b studid both by th Rankin-Slbrg mthod (cf [Fli93, Appndix],[Kab04]) and by th Langlands-Shahidi mthod [Sha90] It is also known that all th thr dfinitions match [AR05, Hn10, Mat11] For th symmtric squar L-function (rsp th xtrior squar L-function), tak G = GL(n) and r to b th symmtric squar (rsp th xtrior squar) of th standard rprsntation of L G = GL n (C) Th Langlands-Shahidi thory of ths L-functions is satisfactorily undrstood [Sha90, Sha92] and this dfinition is known to match with th on via th Langlands formalism [Hn10] For th Rankin-Slbrg thory of ths L-functions, w rfr to [JS90, BG92, KR12] Ths L-functions ar ubiquitous in numbr thory and th dgr of L(s, π, r) oftn has svral maningful and important intrprtations For instanc, ths L- functions dtct functorial lifts from classical groups In particular, by th work of Shahidi [Sha92] and Goldbrg [Gol94], for an irrducibl suprcuspidal rprsntation π, th dgr of L(s, π, r) is ithr th numbr of unramifid twists or half th numbr of unramifid twists of π which ar functorial lifts from classical groups (cf [Sha92, Thorm 77] and [Gol94, Thorms 51 and 52]) W rfr to 2 for som mor dtails in this rgard Sinc rducibility of parabolic induction is undrstood in trms of pols of ths L-functions, th dgr of L(s, π, r), whn π is slf-dual

3 (rsp conjugat slf-dual) if r = Sym 2 or 2 (rsp if r = Asai) counts th numbr of unramifid twists or half th numbr of unramifid twists of π such that th parabolically inducd rprsntation to th rlvant classical group is irrducibl (cf [Sha92, Thorm 76] and [Gol94, Thorm 65]) Ths L-functions ar also rlatd to th thory of distinguishd rprsntations If π is a suprcuspidal rprsntation of GL n (E), thn th dgr of its Asai L-function is th numbr of unramifid charactrs µ of F for which π is µ-distinguishd with rspct to GL n (F ) (cf [AKT04, Corollary 15]) Similarly, if π is a suprcuspidal rprsntation of GL n (F ), th dgr of its xtrior squar L-function is half th numbr of unramifid charactrs µ of F such that π µ dt admits a Shalika functional (cf [JNQ08, Thorm 55]) Our main thorm computs th dgr of L(s, π, r), whn π is a suprcuspidal rprsntation, in trms of th simpl stratum [A, m, 0, β] dfining a maximal simpl typ occurring in th irrducibl suprcuspidal rprsntation π Not that π is a suprcuspidal rprsntation of GL n (E), with E/F a quadratic xtnsion, in th Asai cas, whras othrwis it is a suprcuspidal rprsntation of GL n (F ) As bfor, lt dnot th o-priod of A whr o = o E in th Asai cas and o = o F othrwis Lt ω = ω E/F b th quadratic charactr of F associatd to th xtnsion E/F and lt κ b an xtnsion of ω to E For th purposs of this papr, lt us say that a suprcuspidal rprsntation, and mor gnrally a discrt sris rprsntation, π of GL n (E) is distinguishd (rsp ω-distinguishd) if its Asai L-function L(s, π, r) (rsp L(s, π κ, r)) has a pol at s = 0; strictly spaking this is not how distinction is usually dfind but th proprty abov dos charactriz distinction for th pair (GL n (E), GL n (F )) (cf [AKT04, Corollary 15]) It follows that a suprcuspidal rprsntation, and mor gnrally a discrt sris rprsntation, cannot b both distinguishd and ω-distinguishd bcaus of th idntity L(s, π π σ ) = L(s, π, r)l(s, π κ, r) Hr, σ is th nontrivial lmnt of th Galois group Gal(E/F ) Rcall also that a suprcuspidal rprsntation π, and mor gnrally a discrt sris rprsntation, of GL n (F ) which is slf-dual, i, π = π, is said to b orthogonal (rsp symplctic) if its symmtric squar L-function L(s, π, Sym 2 ) (rsp its xtrior squar L-function L(s, π, 2 )) has a pol at s = 0 Thus, a suprcuspidal rprsntation, and mor gnrally a discrt sris rprsntation, cannot b both orthogonal and symplctic, sinc w hav th factorization L(s, π π) = L(s, π, Sym 2 )L(s, π, 2 ) Thanks to th abov factorizations, if π is a suprcuspidal rprsntation of GL n (E), w can conclud that dg L(s, π, r) + dg L(s, π κ, r) = { 2n n if E/F is unramifid if E/F is ramifid and similarly if π is a suprcuspidal rprsntation of GL n (F ), w can conclud that dg L(s, π, Sym 2 ) + dg L(s, π, 2 ) = n, 3

4 4 U K ANANDAVARDHANAN AND AMIYA KUMAR MONDAL by th rsult of Bushnll and Kutzko mntiond arlir Our main rsults assrt that if both th dgrs on th lft hand sid of th abov idntitis ar non-zro, thn thy ar qual To stat th rsult mor prcisly, w introduc th following notion Lt [π] dnot th inrtial quivalnc class of π; thus [π] consists of all th unramifid twists of π W say that [π] is µ-distinguishd (rsp orthogonal, symplctic) if thr is an unramifid twist of π which is µ-distinguishd (rsp orthogonal, symplctic) W hav: Thorm 11 Lt π b a suprcuspidal rprsntation of GL n (E), with E/F a quadratic xtnsion Lt b th o E -priod of th principal o E -ordr in M n (E) attachd to π Lt L(s, π, r) b th Asai L-function of π (1) Suppos E/F is unramifid Thn th dgr of L(s, π, r) is { 0 if [π] is not distinguishd d(asai) = if [π] is distinguishd n (2) Suppos E/F is ramifid Thn th dgr of L(s, π, r) is 0 if [π] is not distinguishd n d(asai) = if [π] is both distinguishd and ω-distinguishd 2 if [π] is distinguishd but not ω-distinguishd n Thorm 12 Lt π b a suprcuspidal rprsntation of GL n (F ) Lt b th o F -priod of th principal o F -ordr in M n (F ) attachd to π Thn th dgr of its symmtric squar L-function L(s, π, Sym 2 ) is 0 if [π] is not orthogonal d(sym 2 n ) = if [π] is both orthogonal and symplctic 2 n if [π] is orthogonal but not symplctic Thorm 13 Lt π b a suprcuspidal rprsntation of GL n (F ) Lt b th o F -priod of th principal o F -ordr in M n (F ) attachd to π Thn th dgr of its xtrior squar L-function L(s, π, 2 ) is 0 if [π] is not symplctic d( 2 n ) = if [π] is both symplctic and orthogonal 2 n if [π] is symplctic but not orthogonal Rmark As mntiond arlir, a consqunc of Thorm 12 and Thorm 13 is that dg L(s, π, Sym 2 ) = dg L(s, π, 2 ) if both ths L-functions ar not idntically 1 In this contxt, w also rfr to th rmark following Thorm 21 in 2, which placs th abov obsrvation in th framwork of th work of Shahidi [Sha92] Finally, in Sction 6, w prov th following thorm W strss that th assumption of odd rsidu charactristic is ssntial in its proof

5 Thorm 14 Lt E/F b a quadratic xtnsion of p-adic filds If it is ramifid, assum also that p 2 Lt κ b a charactr of E which rstricts to th quadratic charactr ω E/F of F associatd to E/F Lt π b an irrducibl suprcuspidal rprsntation of GL n (E) and lt r(π) b its Asai lift to GL n 2(F ) Thn, f(r(π)) + dg L(s, π, r) = f(r(π) ω E/F ) + dg L(s, π κ, r) Rmark Th conductor formula of Bushnll, Hnniart, and Kutzko [BHK98, Thorm 65] givs an xplicit formula for f(π π σ ) (cf 5) Thus, togthr with Thorm 11, and this xplicit conductor formula for pairs of suprcuspidal rprsntations of gnral linar groups of [BHK98], Thorm 14 in fact producs an xplicit conductor formula for th Asai lift Sinc th statmnt of such an xplicit formula involvs introducing furthr notations, w lav th prcis formula to 6 (cf Thorm 61) 2 Rsults of Shahidi and Goldbrg W rcall th rsults of Shahidi and Goldbrg [Sha92, Gol94] to plac our thorms - Thorm 11, Thorm 12, and Thorm 13 - in contxt W first stat [Sha92, Thorm 77]) For th unxplaind dfinitions in th following, w rfr to [Sha92, Dfinition 74, Dfinition 75] Thorm 21 (Shahidi) Lt π b an irrducibl suprcuspidal rprsntation of GL n (F ) Thn, (1) Th L-function L(s, π, 2 ) 1 unlss som unramifid twist of π is slf-dual Assum π is slf-dual Lt S b th st (possibly mpty) of all th unramifid charactrs η, no two of which hav qual squars, for which π η dt coms from SO n+1 (F ) Thn, L(s, π, 2 ) = η S (1 η 2 (ϖ)q s ) 1 (2) Th L-function L(s, π, Sym 2 ) 1 unlss som unramifid twist of π is slfdual Assum π is slf-dual If π coms from Sp n 1 (F ), thn L(s, π, Sym 2 ) = (1 q rs ) 1, whr r is th numbr of unramifid slf-twists of π Othrwis, lt S b th st (possibly mpty) of all th unramifid charactrs η, no two of which hav qual squars, for which π η dt coms from SO n(f ) Thn, L(s, π, Sym 2 ) = η S (1 η 2 (ϖ)q s ) 1 5 Rmark A consqunc of Thorm 12 and Thorm 13 is that S and S hav th sam cardinality if both ths sts ar non-mpty Nxt w stat [Gol94, Thorms 51 and 52] Hr, E/F is a quadratic xtnsion of p-adic filds and σ dnots th non-trivial lmnt of Gal(E/F ) For an irrducibl admissibl rprsntation of GL n (E), lt L(s, π, r) dnot its Asai L-function In th following, q = q F is th rsidu cardinality of F For th unxplaind dfinitions in th following two thorms, w rfr to [Gol94, Dfinition 111, Dfinition 112]

6 6 U K ANANDAVARDHANAN AND AMIYA KUMAR MONDAL Thorm 22 Lt n b odd Suppos that π is an irrducibl suprcuspidal rprsntation of GL n (E) such that π = π σ Lt S b th st of all unramifid charactrs η of E, no two of which hav qual squars, such that π η dt is a stabl lift from U(n, E/F ) (1) Suppos E/F is ramifid Thn L(s, π, r) = η S(1 η(ϖ F )q s ) 1 (2) If E/F is unramifid, thn, L(s, π, r) = η S(1 η 2 (ϖ F )q s ) 1 Thorm 23 Lt n b vn Suppos that π is an irrducibl suprcuspidal rprsntation of GL n (E) such that π = π σ Lt S b th st of all unramifid charactrs η of E, no two of which hav qual valu at ϖ F, such that π η dt is an unstabl lift from U(n, E/F ) Thn, L(s, π, r) = η S(1 η(ϖ F )q s ) 1 Rmark Thorm 11 computs xplicitly th cardinality of S in Thorm 22 and Thorm 23 3 Th Asai lift W collct togthr various rsults on th Asai rprsntation in this sction Lt H b a subgroup of indx two in a group G Lt ρ b a finit dimnsional rprsntation of H of dimnsion n Its Asai lift, which w do not dfin hr, is a rprsntation of G of dimnsion n 2 Lt r(ρ) dnot th Asai lift of ρ to G Th following proposition summarizs th ky proprtis of th Asai lift (cf [Pra99, MP00]) Proposition 31 Th Asai lift satisfis: (1) r(ρ 1 ρ 2 ) = r(ρ 1 ) r(ρ 2 ) (2) r(ρ) = r(ρ ) (3) r(χ) for a charactr χ is χ tr, whr tr: G/[G : G] H/[H : H] is th transfr map (4) r(ρ σ ) = r(ρ), whr σ is th nontrivial lmnt of G/H (5) r(ρ) H = ρ ρ σ (6) For a rprsntation τ of G, r(τ H ) = Sym 2 τ ω G/H 2 τ, whr ω G/H is th nontrivial charactr of G/H (7) Lt Ind G Hρ dnot th rprsntation of G inducd from ρ Thn, (a) Sym ( 2 Ind G Hρ ) = Ind G H Sym 2 ρ r(ρ) (b) ( 2 Ind G Hρ ) = Ind G H 2 ρ r(ρ) ω G/H Rmark W hav assumd [G : H] = 2 sinc that is th cas of intrst to us Th Asai lift can b mor gnrally dfind whn H is of any finit indx in G

7 4 Proofs of Thorms W now prov Thorms 11,12, and 13 W first prov (1) of Thorm 11, us this to prov Thorm 12 and Thorm 13, and finally prov (2) of Thorm 11 W will hav to appal to th rsult of Bushnll and Kutzko [BK93, Lmma 625], mntiond alrady in 1, which w formally stat now in ordr to mak it asy to rfr to it latr Thorm 41 (Bushnll-Kutzko) Lt π b an irrducibl suprcuspidal rprsntation of GL n (E) Lt [A, m, 0, β] b th simpl stratum dfining a maximal simpl typ occurring in π, whr A is a principal o E -ordr in M n (E), m 0 is th lvl of π, and β M n (E) is such that E[β] is a fild with E[β] normalizing A Lt = (A o E ) b th o E -priod of A (which is th sam as th ramification indx (E[β]/E) of E[β]/E) Thn, divids n, and th cardinality of th group of unramifid slf-twists of π is n/ 41 Proof of Thorm 11 (1) Lt E/F b quadratic unramifid Lt π b a suprcuspidal rprsntation of GL n (E) Lt ρ π : GL n (C) b its Langlands paramtr W assum that its Asai lift r(ρ π ) : W F GL n 2(C) contains th trivial charactr of W F, which in particular implis that ρ σ π = ρ π Sinc ω = ω E/F is unramifid, clarly th numbr of unramifid charactrs in r(ρ π ) and r(ρ π ) ω is th sam Sinc, dg L(s, π, r) + dg L(s, π κ, r) = dg L(s, π π ) = 2n, by Thorm 41, (1) of Thorm 11 is immdiat 42 Proof of Thorms 12 and 13 Lt π b a suprcuspidal rprsntation of GL n (F ) Lt ρ π : W F GL n (C) b its Langlands paramtr W assum that r(ρ π ) contains th trivial charactr of W F, which in particular implis that ρ π = ρ π Hr, r is ithr th symmtric squar rprsntation or th xtrior squar rprsntation of GL n (C) Thus, th dimnsion of r(ρ π ) is ithr n(n+1) or n(n 1) W hav th 2 2 idntity L(s, π π) = L(s, π, Sym 2 )L(s, π, 2 ), and w know that th lft hand sid L-function has dgr n/ by Thorm 41 If n/ = 1, thn th trivial charactr of W F is th only unramifid charactr apparing in ρ π ρ π and hnc in r(ρ π ) Thus, in this cas thr is nothing to prov Othrwis, thr is a nontrivial unramifid charactr χ : W F C such that ρ π χ = ρ π Thus, ρ π = Ind W F W τ, F for som irrducibl rprsntation τ of W F, whr F /F is th unramifid xtnsion of dgr n/ Lt σ dnot a gnrator of Gal(F /F ) W know that ρ π ρ π = Ind W F W F (τ τ) IndW F W F (τ τ σ ) Ind W F W F (τ τ σ( n 1)) If n/ is an odd intgr, thn obsrv that ach summand othr than th vry first on on th right hand sid of th abov idntity appars twic This is indd th 7

8 8 U K ANANDAVARDHANAN AND AMIYA KUMAR MONDAL cas sinc, Ind W F W F (τ τ σa ) = Ind W F W F (τ τ σ( n a)), for vry 1 a n/ Sinc th trivial charactr of W F appars xactly onc on th lft hand sid, it follows that 1 Ind W F W F (τ τ), whn n/ is odd Thrfor, prcisly on of Sym 2 τ or 2 τ contains th trivial charactr of W F and hnc prcisly on of Ind W F W F (Sym2 τ) or Ind W F W F ( 2 τ) contains all th unramifid slf-twists of ρ π Thus, Thorm 12 and Thorm 13 follow in th cas whn n/ is an odd intgr If n/ is an vn intgr, w procd by induction on dim ρ π W start by writing ρ π = Ind W F τ for an irrducibl rprsntation τ of whr E is th quadratic unramifid xtnsion of F This can always b don bcaus an unramifid xtnsion of vn dgr ncssarily has th quadratic unramifid subxtnsion By (7) of Proposition 31, w hav: r (ρ π ) = { Ind W F Sym 2 τ Asai(τ) if r = Sym 2 Ind W F 2 τ Asai(τ) ω E/F if r = 2 Now ithr τ = τ or τ σ = τ but not both, sinc ρ π is an irrducibl rprsntation of W F Hr, σ is th lmnt of ordr two in Gal(E/F ) W claim that Asai(τ) (rsp Asai(τ) ω E/F ) contains an unramifid charactr of W F only if Sym 2 τ (rsp 2 τ) dos not contain an unramifid charactr of W F Indd, if Asai(τ) contains an unramifid charactr of W F, th total numbr of unramifid charactrs in Asai(τ) Asai(τ) ω E/F is n + n = n, by applying part (1) of Thorm 11 to th rprsntation τ which 2 2 has dimnsion n/2, and by obsrving that ω E/F is unramifid Not also that = (ρ π ) = (τ), sinc th xtnsion E/F is unramifid Sinc this numbr quals th numbr of unramifid charactrs containd in ρ π ρ π = Sym 2 ρ π 2 ρ π, th claim follows Thrfor, if Asai(τ) contains an unramifid charactr, th proof is complt by appaling to part (1) of Thorm 11 Othrwis, sinc dim τ = 1 2 dim ρ π, th proof is complt by appaling to th induction hypothsis Not that th bas cas of induction is asily vrifid sinc thr ar at most two unramifid charactrs to considr whn dim ρ π = 2; i, whn dim τ = 1 43 Proof of Thorm 11 (2) Now lt E/F b a ramifid quadratic xtnsion, and lt π b a suprcuspidal rprsntation of GL n (E) Lt ρ π : GL n (C) b its Langlands paramtr W may assum that r(ρ π ) 1, whr r dnots th Asai lift from to W F Not that this implis that r(ρ π ) dos not contain ω E/F, th non-trivial charactr of W F / In what follows, w us this assumption many tims to rduc th numbr of cass that w nd to analyz

9 Now considr th 2n-dimnsional rprsntation Ind W F ρ π of W F W hav: (1) Sym 2 (Ind W F ρ π ) = Ind W F Sym 2 ρ π r(ρ π ) and (2) 2 (Ind W F ρ π ) = Ind W F 2 ρ π r(ρ π ) ω E/F W divid th proof into two cass First, w assum that π = π σ so that Ind W F ρ π is irrducibl Lt Ind F Eπ dnot th corrsponding suprcuspidal rprsntation of GL 2n (F ) Not that by our assumption that r(ρ π ) 1, Ind F Eπ is orthogonal and not symplctic, by (1) Thrfor, it follows from Thorm 12 and Thorm 13 that x = dg L(s, Ind F Eπ, Sym 2 ) dg L(s, Ind F Eπ, 2 ) is givn by { dg L(s, Ind F (3) x = Eπ, Sym 2 ) if [ Ind F Eπ ] is orthogonal but not symplctic, 0 if [ Ind F Eπ ] is orthogonal and symplctic Sinc th xtnsion E/F is ramifid, th priod associatd to Ind F Eπ may b or 2, and thus th dgr of L(s, Ind F Eπ, Sym 2 ) is ithr 2n or n On th othr hand, th diffrnc (4) y = dg L(s, π, Sym 2 ) dg L(s, π, 2 ) could b, a priory, n or 0 or n Now w do a cas by cas analysis to list all th possibl candidats for th pair (x, y) To this nd, not that: (i) in (1) and (2), possibl valus for th dgr of th first summand on th right hand sid ar 0, n, and n (by Thorms 12 and 13) 2 (ii) in (1) and (2), th scond summand on th right hand sid cannot hav dgr mor than n (sinc dg L(s, π, r)+ dg L(s, π κ, r) = n) (iii) in addition, in (1), th scond summand on th right hand sid has non-zro dgr (by th assumption that r(ρ π ) 1) W hav alrady obsrvd, using (3), that whn x 0, it is ithr 2n or n, and th dgr of L(s, Ind F Eπ, 2 ) is 0 In particular, whn x 0, all th trms in (2) hav dgr 0 Whn x = 2n, both th summands in (1) hav dgr n by (i) and (ii), and thus y = n Whn x = n, w claim that th dgr of th first summand is 0 (and that of th scond summand is n ) in (1), and thus y = 0 Indd, if th first summand has a non-zro dgr it has to b ithr n or n by (i), but it cannot 2 b n by (iii), and it cannot b n sinc this would imply that th scond summand 2 in (2) has dgr n as wll, but w know that it is 0 Whn x = 0, th dgr of 2 th lft hand sid in both (1) and (2) ar qual by (3), and is ithr n or n Whn 2 this dgr is n, th dgr of th first summand in (1) is ithr 0 or n by making 2 us of (iii) Not that th dgr of th first summand in (2) would thn b n or n rspctivly, and thus y = n or 0 rspctivly In th prcding argumnt, w 2 hav mad us of th idntity 9 (5) dg L(s, π, r) + dg L(s, π κ, r) = n

10 10 U K ANANDAVARDHANAN AND AMIYA KUMAR MONDAL Whn th dgr of th lft hand sid in both (1) and (2) is n, th dgr of th 2 first summand is 0 in both (1) and (2), and thus y = 0, onc again by arguing with (i), (iii), and (5) Obsrv that th numbr of unramifid charactrs in Ind W F Sym 2 ρ π (rsp Ind W F 2 ρ π ) is sam as th numbr of unramifid charactrs in Sym 2 ρ π (rsp 2 ρ π ), sinc E/F is ramifid It follows that dg L(s, π, r) dg L(s, π κ, r) = x y is ithr n or 0 This provs (2) of Thorm 11 in this cas Nxt, suppos that π = π σ = π Sinc π = π σ, it follows that ρ π = τ WE for an irrducibl rprsntation τ of W F In this cas, Thus, w gt: Ind W F ρ π = τ τ ωe/f (6) Sym 2 τ Sym 2 τ τ τ ω E/F = Sym 2 (Ind W F ρ π ) = Ind W F Sym 2 ρ π r(ρ π ) and (7) 2 τ 2 τ τ τ ω E/F = 2 (Ind W F ρ π ) = Ind W F 2 ρ π r(ρ π ) ω E/F By our assumption that r(ρ π ) 1, w conclud that th irrducibl rprsntation τ is not symplctic This is bcaus if 2 τ 1, thn th lft hand sid of (7) contains th trivial charactr at last twic whras th right hand sid can contain th trivial charactr at most onc sinc r(ρ π ) ω E/F 1 As bfor, w now do a cas by cas analysis to list all possibl pairs (a, b) whr (8) a = dg L(s, τ, Sym 2 ) dg L(s, τ, 2 ), and (9) b = dg L(s, ρ π, Sym 2 ) dg L(s, ρ π, 2 ), and vrify that dg L(s, ρ π, r) dg L(s, ρ π κ, r) = 2a b is ithr n or 0 Sinc w hav obsrvd that th irrducibl rprsntation τ is not symplctic, a 0 and it is ithr n or 0 by Thorm 12 and Thorm 13 Now th possibl valus for b could b, a priori, n or 0 or n Whn a = n, considring th sum of (6) and (7), w can conclud that all th trms in (7) ar of dgr 0 Also, not that both th trms on th right hand sid of (6) will hav dgr n, and in particular b = n Whn a = 0, th lft hand sids of both (6) and (7) ar ach of total dgr n Sinc r(ρ π) 1, th dgr of L(s, ρ π, Sym 2 ) is ithr 0 or n It follows that th valu of b is n or 0 rspctivly 2 Thus, in all th cass 2a b is n or 0, and th rsult follows

11 5 Conductor formula of Bushnll, Hnniart, and Kutzko W stat th xplicit conductor formula for pairs of suprcuspidal rprsntation du to Bushnll, Hnniart, and Kutzko This sction closly follows [BHK98, 6] Lt π b a suprcuspidal rprsntation of GL n (F ) Following [BK93], lt [A, m, 0, β] b a simpl stratum of a maximal simpl typ occurring in π Hr, A is a principal o F -ordr in M n (F ), m is th lvl of π, and β M n (F ) is such that E = F [β] is a fild with E normalizing A If dnots th o F -priod of A, thn th numbr of unramifid slf-twists of π is n/ by Thorm 41 As mntiond in th introduction, th conductor f(π) of π is givn by ( f(π) = n 1 + m ) Lt π i b two suprcuspidal rprsntations of GL ni (F ) for i = 1, 2 Thr ar thr distinct possibilitis: (i) π 1 and π 2 ar unramifid twists of ach othr, (ii) π 1 and π 2 ar compltly distinct, and (iii) π 1 and π 2 admit a common approximation W do not gt into dfining ths notions and rfr to [BHK98, 6] instad Suffic to say that whn π 1 and π 2 admit a common approximation, thr is a bst common approximation and this is an objct of th form ([Λ, m, 0, γ], l, ϑ), whr th stratum [Λ, m, 0, γ] is dtrmind by π 1 and π 2, 0 l < m is an intgr, and ϑ is a charactr of a compact group attachd to th data coming from π 1 and π 2 Anothr ingrdint in th conductor formula is an intgr c(β) associatd to β This coms from th gnralizd discriminant, say C(β), associatd to th xact squnc 0 E End F (E) a β End F (E) s β E 0, whr s β is a tam corstriction rlativ to E/F [BK93, 13] and a β is th adjoint map x βx xβ Th constant c(β) is dfind such that C(β) = q c(β) Now w stat th conductor formula of [BHK98] Thorm 51 (Bushnll, Hnniart, and Kutzko) For i = 1, 2, lt π i b an irrducibl suprcuspidal rprsntation of GL ni (F ) Dfin quantitis m i, i, β i as abov Lt = lcm( 1, 2 ) and m/ = max(m 1 / 1, m 2 / 2 ) (1) Suppos that n 1 = n 2 = n and π 1 and π 2 ar unramifid twists of ach othr Lt β = β 1 and d = [F [β] : F ] Thn, ( f(π1 π 2 ) = n c(β) ) dg L(s, π d 2 1 π 2 ) (2) Suppos that th π 1 and π 2 ar compltly distinct Thn, ( f(π1 π 2 ) = n 1 n m ) (3) Suppos that π 2 is not quivalnt to an unramifid twist of π 1 but that th π i ar not compltly distinct Lt ([Λ, m, 0, γ], l, ϑ) b a bst common approximation to th π i, and assum that th stratum [Λ, m, l, γ] is simpl Put 11

12 12 U K ANANDAVARDHANAN AND AMIYA KUMAR MONDAL d = [F [γ] : F ] Thn, ( f(π1 π 2 ) = n 1 n c(γ) + l ) d 2 d Rmark Obsrv that in (2) and (3), dg L(s, π 1 π 2 ) = 0 6 Conductor of th Asai lift Lt E/F b a quadratic xtnsion of p-adic filds Lt π b a suprcuspidal rprsntation of GL n (E) Lt ρ π : GL n (C) b its Langlands paramtr Lt r(ρ π ) : W F GL n 2(C) b th Asai lift of ρ π In this sction, w comput th Artin conductor of r(ρ π ) Throughout this sction, w assum that p is odd For a rprsntation τ of th Wil-Dlign group, lt f(τ) dnot its Artin conductor Our formula for th Asai lift is a consqunc of th conductor formula for pairs of suprcuspidal rprsntations du to Bushnll, Hnniart, and Kutzko [BHK98, Thorm 65] Sinc r(ρ π ) WE = ρπ ρ σ π, it follows that { f(ρ π ρ σ π) = f(r(ρ π )) f(r(ρ π )) + f(r(ρ π ) ω E/F ) n 2 if E/F is unramifid if E/F is ramifid In th abov, in th scond cas, w hav mad us of th fact that E/F is tamly ramifid, which is tru sinc p is odd by our assumption Sinc th formula of Bushnll, Hnniart, and Kutzko [BHK98] computs th lft hand sid, in ordr to driv a formula for th Asai lift, it suffics to comput f(r(ρ π )) f(r(ρ π ) ω E/F ), which is what w ar going to do Lt r(ρ π ) = ρ i i b th dirct sum dcomposition of r(ρ π ) into irrducibl rprsntations Now, r(ρ π ) ω E/F = ρ i ω E/F, and sinc th Artin conductor is additiv, it follows that f(r(ρ π )) f(r(ρ π ) ω E/F ) = [f(ρ i ) f(ρ i ω E/F )] i W know that f(ρ χ) max{f(ρ), dim ρ f(χ)}, with quality in th abov idntity if f(ρ) dim ρ f(χ) Thus, f(ρ i ω E/F ) = f(ρ i ) unlss ρ i is a on dimnsional charactr with Artin conductor on, in which cas f(ρ i ω E/F ) can b 0 or 1 Obsrv that th contribution to f(r(ρ π )) f(r(ρ π ) ω E/F ) i

13 13 from tamly ramifid charactrs ρ i in r(ρ π ) such that ρ i ω E/F is unramifid is dg L(s, π κ, r), whras th contribution from unramifid charactrs ρ i in r(ρ π ) such that ρ i ω E/F is tamly ramifid is Thrfor, it follows that dg L(s, π, r) f(r(ρ π )) f(r(ρ π ) ω E/F ) = dg L(s, π κ, r) dg L(s, π, r) Now making us of Thorm 51, w gt th following conductor formula for th Asai lift Thorm 61 Lt E/F b a quadratic xtnsion of p-adic filds, whr p is odd, with ramification indx (E/F ) Lt σ dnot th non-trivial lmnt of Gal(E/F ) Lt π b a suprcuspidal rprsntation of GL n (E) Lt b th o E -priod of th principal o E -ordr in M n (E) attachd to π Lt r(π) b its Asai lift to GL n 2(F ) and lt L(s, π, r) b th Asai L-function attachd to π (1) Suppos π and π σ ar unramifid twists of ach othr Thn, ( f(r(π)) = n c(β) ) dg L(s, π, r) (E/F )d 2 (2) Suppos π and π σ ar compltly distinct Thn, ) f(r(π)) = n (1 2 m + (E/F ) (3) Suppos that π is not quivalnt to an unramifid twist of π σ and that thy ar not compltly distinct Lt ([Λ, m, 0, γ], l, ϑ) b a bst common approximation to (π, π σ ), and assum that th stratum [Λ, m, l, γ] is simpl Put d = [F [γ] : F ] Thn, f(r(π)) = n 2 (1 + c(γ) (E/F )d 2 + ) l (E/F )d Rmark Togthr with Thorm 11, Thorm 61 givs an xplicit conductor formula for th Asai lift As in th cas of Thorm 51, dg L(s, π, r) = 0 in cass (2) and (3) Acknowldgmnts Th authors would lik to thank Dipndra Prasad for svral usful discussions Thy thank th anonymous rfr for carfully rading th manuscript and for making svral usful suggstions Th scond namd author would lik to acknowldg a grant (07IR001) from Industrial Rsarch and Consultancy Cntr, Indian Institut of Tchnology Bombay, during th priod of th prsnt work

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