The following theorem is proven in [, p. 39] for a quadratic extension E=F of global elds, such that each archimedean place of F splits in E. e prove

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1 On poles of twisted tensor L-functions Yuval. Flicker and Dmitrii inoviev bstract It is shown that the only possible pole of the twisted tensor L-functions in Re(s) is located at s = for all quadratic extensions of global elds.. Introduction. Let E be a quadratic separable eld extension of a global eld F. Denote by E, F the corresponding rings of adeles. Put G n for GL n and n for its center. Then n ( E ) is the group E of ideles of E. Fix a cuspidal representation of the adele group G n ( E ). ithout lost of generality, we may assume that the central character of is trivial on the split component of E. This is the multiplicative group R of the eld of real numbers embedded in E via x! (x; :::; x; ; :::) (xin the archimedean, in the nite components). Let S be a nite set of places of F (depending on ), including the places where E=F ramify, and the archimedean places, such that for each place v of E above a place v outside S the component v of is unramied. Following [], let r be the twisted tensor representation of bg = [GL(n; ) GL(n; )] Gal(E=F) on n n. It acts by r((a; b))(x y) =ax by and r()(x y) = y x ( 2 Gal(E=F); = ). Let q v be the cardinality of the residue eld R v = v R v of the ring R v of integers in F v. e dene the twisted tensor L-function to be the Euler product L(s; r(); S)= Y v2=s det [, q,s v r(t v)], : The representation is called distinguished if its central character is trivial on F and there is an automorphic form 2 in L 2 (G n (E) n ( F )ng n ( E )), such that R (g)dg =. The integral is taken over the closed subspace G n (F ) n ( F )ng n ( F )ofg n (E) n ( F )ng n ( E ). The rst named author wishes to express his very deep gratitude to Professor Toshio Oshima for inviting him to visit the University oftokyo in March 995, and to him and Professor Takayuki Oda for their warm hospitality. Department of Mathematics, The Ohio State University, 23. 8th ve., olumbus, OH 32-.

2 The following theorem is proven in [, p. 39] for a quadratic extension E=F of global elds, such that each archimedean place of F splits in E. e prove it for any quadratic extension of global elds, i.e. also for number elds with completions E v =F v = =R. Theorem. The product L(s; r(); S) converges absolutely, uniformly in compact subsets, in some right half-plane. It has analytic continuation as a meromorphic function to the right half plane Re(s) >,, for some small >. The only possible pole of L(s; r(); S)in Re(s) >, is simple, located at s =. The function L(s; r(); S) has a pole at s =if and only if is distinguished. Proof. The proof of this theorem is the same as that of the Theorem of [, x], pp On lines and 8 of page 3 of [], we use the proposition below. It holds in the non-split archimedean case too. Hence the restriction put in [] on the extension E=F can be removed. For the functional equation satised by L(s; r(); S), see []. For the local L-factors at all non-archimedean places of F, see [2]. The non-vanishing of this L-function on the edge Re(s) = of the critical strip has been shown by Shahidi []. Twisted tensor L-functions are used in the study (see Kon-no [5]) of the residual spectrum of unitary groups.. Local computations. From now on, we consider the local case only. Let E=F be a quadratic extension of local elds. Thus in the archimedean case E=F = =R. Denote by x! x the non-trivial automorphism of E over F. Let = be an element of E, such that =,. Put G n for GL n. The groups of F and E-points are denoted by G n (F ) and G n (E). Denote by N n the unipotent radical of the upper triangular subgroup of G n, and by n the diagonal subgroup. Let be a non trivial additive character of F. For example, if F = R then (x) =e 2ix. Let be the (non-trivial) character (z) = ((z, z)=) of E. It is trivial on F. For u 2 N n (E), set (u) = ( P n, u i= i;i+). Fix an irreducible admissible representation of G n (E) on a complex vector space V. The representation is called generic if there exists a non-zero linear form on V, such that ((u)v) = (u)(v) for all v in V and u in N n (E). The dimension of the space of such is bounded by one. Let (; ) be the space of functions on G n (E) of the form (g) = ((g)v), where v 2 V. e have (ug) = (u) (g) (g 2 G n (E), u 2 N n (E)). Denote by (; ) those functions in (; ) whose corresponding vectors v are in the space of K-nite vectors, where K=K n (E) is the standard maximal compact subgroup of G n (E). 2

3 For 2 S(F n ), dene the integral (s; ;)= Nn(F)nGn(F) where n =(;; :::; ; ) is a row vector of size n. (g)( n g)jdet gj s dg; Proposition. (i) There exists some small constant, >, such that the integral (s; ; ) converges absolutely, uniformly in compact subsets, for Re(s) >, ; (ii) There exists in (; ) and in S(F n ), such that (s; ;)=. Proof. hen E=F is an extension of non-archimedean local elds, (i) and (ii) are treated in the Proposition of [], x, p. 38. e prove (i) in general, including the case (E; F) =(;R), following Jacquet and Shalika [3], pp Using the Iwasawa decomposition G n (F )=N n (F) n (F )K n (F ), and the associated measure decomposition, we need to show the convergence of the integral n(f )Kn(F ) Here a =diag(a ; a 2 ; :::;a n, ; ). Recall that j (ak)jjdet aj s, n;f (a)dadk: n;f (a) = n,;f (a)jdet aj = jdet aj Y i<jn, ja i j ja j j ; and (see e.g. [], p. 3) that n;e (a) =n;f 2 (a). y Proposition 3 of Jacquet and Shalika [3, x] there is a nite set X of nite functions in n, variables such that j (ak)j is bounded by a nite sum of expressions of the form =2 n,;e(a) a a 2 ; a 2 a 3 ; :::; a n, : Here is the absolute value of some element of X and is in S(F n, ). Thus, it suces to show that the integral obtained by replacing by this estimate is convergent. Using that n,;e(a) =2, n;f (a) = n,;f (a), n;f (a) =jdet aj, ; we arrive atthenite sum of integrals a a 2 ; a 2 a 3 ; :::; a n, jdet aj s, da: The change of variables a = t :::t n, ; a 2 = t 2 :::t n, ; :::; a n, = t n, ; has the Jacobian t 2 t 2 3 :::tn,3 n,2. e obtain a sum of expressions of the form Y n,2 (t ; t 2 ; :::; t n, ) 3 j=2 t j, j n, Y j= t j(s,) j dt:

4 gain, by Proposition 3 of Jacquet and Shalika [3, x] the set X is such that any in it is the product of () a polynomial in the logarithms of the absolute values of the variables, and (2)acharacter of the form (t ) 2 (t 2 )::: n, (t n, ); with Re( i ) >, for each i. It follows that the aboveintegral converges uniformly in compact subsets of Re(s) >,, for some small >. This completes the proof of (i). For (ii) we will follow the proof of Proposition.3 of Jacquet and Shalika [3]. ssume that (s; ;) = for all choices of in (; ) and in S(F n ). e will show that (e) = for all, a contradiction which will imply (ii) of the lemma. Since is arbitrary, it follows that for all we have Dene Nn,(F )ngn,(f ) I k ( )= Nk(F)nGk(F) "!# jdet gj s dg =: " n,k!# jdet gj s dg: e claim that I k ( ) is zero for all and all k with k n,. The lemma would then follow, since (e) = I ( ). e will show this claim by descending induction on k. e have just seen that I n, ( )=. So x k n, with I k ( )=for all. e proceed to show that I k, ( ) = for all. e apply the fact that I k ( )=tothefunction dened by (g) = 2 F k g k u 3 5 (u)du: Here u is a column of size k, 2 S(F k ) and 2 (; ). Proposition 2. of Jacquet and Shalika [; II], p. 8, and the remark following it (top of p. 8), assure us that this function is in the space (; ). Note that "!# "!# = b(k g); n,k n,k where (y) b = R F k(u) (y u)du denotes the Fourier transform of 2 S(F k ). Indeed X k g kj u j du: b( k g)= F k(u) ( k g u)du = F k (u) j=

5 Further, since k u = k gu ; we have 2 = = k gu X k j= k gu g kj u j X 5 = k Now substituting for in I k ( )=,weobtain Nk(F )ngk(f ) j= g kj u j 2 " n,k!# b(k g)jdet gj s dg = for all 2 S(F k ) and all 2 (; ). In this integral b can be replaced by any element of S(F k ). Hence I k, ( ) = for all and we are done. 3 5 : References [] Y. Flicker, \Twisted Tensors and Euler Products", ull. Soc. Math. France, 988, 295{33. [2] Y. Flicker, \On the local twisted tensor L-function", ppendix to: \On zeroes of the twisted tensor L-function", Math. nn. 29, 993, 99{29. [3] H. Jacquet and J. Shalika, \Exterior square L-functions", in utomorphic Forms, Shimura Varieties, and L-functions, 99, 3{22 (ed. L. lozel, S. Milne). [] H. Jacquet and J. Shalika, \On Euler products and the classication of automorphic representations", mer. J. Math. 3, 98, I, 9{558; II, {85. 5

6 [5] T. Kon-no, \The residual spectrum of U(2; 2), preprint. [] F. Shahidi, \On certain L-functions", mer. J. Math. 3, 98, 29{355.