Maass Cusp Forms with Quadratic Integer Coefficients

Transcription

1 IMRN International Mathematics Research Notices 2003, No. 18 Maass Cus Forms with Quadratic Integer Coefficients Farrell Brumley 1 Introduction Let W be the sace of weight-zero cusidal automorhic forms of level N on GL2) over Q. For each n such that n, N) = 1, denote by Tn) the nth Hecke oerator, and by T the algebra over C which they generate. If B = {f i } is a W-basis of simultaneous eigenvectors of T, then for each f i B let λ fi n) be the eigenvalue of Tn) at f i, normalized so as to lie in the interval [ 2, 2] under the Ramanujan conjecture. A striking result of Sarnak [9] says that if the L-series, Ls, f) = λ f n)n s, 1.1) n=1 of an eigenform f B has integral coefficients, then f must be of Galois tye. The roof utilizes the functorial transfers for the second and third symmetric owers on GL2) established, resectively, by Gelbart and Jacquet in [2] and by Kim and Shahidi in [4]. The condition on the integrality of the coefficients would follow from the stronger suosition that a one-dimensional invariant subsace V of W exists uon which T actsasintegral scalars. Stating the theorem in this way, we might try to extend the imlication to higher-dimensional T-invariant integral subsaces V. TheL-series coefficients of any f in such an n-dimensional subsace V would lie as integers in a number field of degree n over Q, although conversely it is not true that all forms in W having this latter roerty necessarily belong to V. The more general result would again state that the f V are of Galois tye. We accomlish this extension for a two-dimensional V using the techniques of [9] and the remaining fourth symmetric ower transfer established by Kim in [3]. Received 19 June Revision received 1 December Communicated by Peter Sarnak.

2 984 Farrell Brumley Let V be a two-dimensional T-invariant irreducible subsace of W admitting a basis with resect to which the oerators in T can be realized as 2 2 matrices with rational integer entries. If this subsace is irreducible, uon diagonalization these matrices have entries in the integer ring of a quadratic extension K of Q. Thus, if f V is a simultaneous eigenvector of T, its L-series has quadratic integer coefficients. As in [9], we seek to rove that the form f is of Galois tye. In this aer, we show that in fact this can be done, rovided the quadratic field K is not Q 5). This single excetion can be exlained by observing that, if f is attached through the Artin conjecture to an even icosahedral Galois reresentation with trivial central character, its coefficients lie as integers in Q 5). Without higher symmetric ower transfers at our disosal however, it is imossible for our method of roof to fully distinguish the analytic roerties of its L-function. Henceforth, the subsace V will be such that its Hecke eigenvalues do not lie in the field Q 5). Theorem 1.1. Let V W be as above. Then V comrises forms associated to a Galois reresentation of either dihedral, tetrahedral, or octahedral tye. In articular, V has Lalacian eigenvalue 1/4 and satisfies the Ramanujan conjecture at every finite lace. It should be remarked that with the strength of full functoriality, from which follow the conjectures of Ramanujan and Sato-Tate, this result becomes transarent, for a finite number of eigenvalues in an interval cannot be continuously equidistributed. The advantage of theorems of this tye is that they give artial rogress toward Ramanujan in an altogether different manner than imroving local bounds on the Satake arameters. Indeed, a finite number of functorial lifts, along with their cusidality conditions, suffice for roving that those automorhic forms which satisfy a certain integrality condition on their coefficients must also satisfy the Ramanujan conjecture. 2 Preliminaries 2.1 Let ρ be an irreducible two-dimensional comlex Galois reresentation of a number field F over Q. For each rime, unramified in F, let Frob denote the Frobenius class at. Then the artial Artin L-function is defined on a suitable right half-lane as Ls, ρ) = det 1 ρ ) Frob s ) 1 = 1 λρ ) s + ω ρ ) 2s) 1, 2.1)

3 Maass Cus Forms with Quadratic Integer Coefficients 985 the roduct being taken over all unramified rimes. If the numbers λ ρ ) lie in a totally real or urely imaginary number field, then from the unicity of ω ρ, it follows that ω 2 ρ = 1. The image of ρ must therefore lie in GL m) 2 C) := { g GL 2 C) det g) m = 1 }, for m = 1 or ) By exlicitly constructing all subgrous of GL m) 2 C) having quadratic integer traces, we discover in this section which trace fields can actually occur in the setting of our roblem. Klein, in [6], has classified all finite subgrous of PGL 2 C). We lift them to subgrous of GL m) 2 C), requiring additionally that the traces be quadratic integers. For dihedral subgrous of PGL 2 C), the condition of a quadratic determinate already ensures rational integrality of the traces. These lifts are { [ ] [ ] [ ] [ ] } 1 0 i 0 0 i 0 1 U 2 = ±, ±, ±, ± i i { [ ] [ ] [ ] [ ] } V 2 = ±, ±, ±, ± GL 1) 2 C), GL 2) 2 C), 2.3) both of which have image in PGL 2 C) equal to the Klein four grou D 2 ; and { [ ] [ ] [ ] [ ] [ ] [ ]} i i i i 0 U 3 = ±, ±, ±, ±, ±, ±, 0 1 i 0 0 i i i { [ ] [ ] [ ] [ ] [ ] [ ]} 2.4) V 3 = ±, ±, ±, ±, ±, ± in GL 1) 2 C) and GL2) 2 C) resectively, both having an image in PGL 2C), the order-6 dihedral grou D 3.Nowletζ 8 = 1 be a rimitive 8th root of unity. The grou { [ 1 ζ r U 4 = 2 ζ s ζ s ζ r ] } r, s {1, 3, 5, 7} U 2 GL 1) 2 C) 2.5) has image in PGL 2 isomorhic to A 4, the grou of tetrahedral rotations. The trace field for both the dihedral and tetrahedral grous is evidently Q.

4 986 Farrell Brumley For brevity, we give the generating matrices for the remaining grous. Let ζ again denote a rimitive 8th root of unity. The matrices ζ ζ ) ζ ζ, ζ 0 ) 0 ζ, 0 1 ) ) generate an octahedral subgrou of PGL 2 C). This grou lifts to a finite subgrou of GL 1) 2 C) and GL2) 2 C) with traces lying as integers in the field Q 2).Finally, we let ɛ C satisfy ɛ 5 = 1 and set = ɛ 4 ɛ)/ 5, q = ɛ 2 ɛ 3 )/ 5. Then the following matrices generate an icosahedral subgrou of PGL 2 C): ɛ 3 0 ) 0 ɛ 2, 0 1 ) 1 0, q q ). 2.7) One could look at [1, age 73] for the derivation of this fact. This grou lifts to a finite subgrou of GL 1) 2 C) with traces lying as integers in the field Q 5). 2.2 Retaining the notation of the introduction, we let {f, g} be a basis of V of simultaneous eigenvectors for T with eigenvalues λ f n) and λ g n) for every n rime to N. Though a riori the field of definition of λ f n) and λ g n) may deend on n, it can easily be seen that one quadratic field K contains them all. For, with resect to the integral basis, T embeds in the matrix algebra M 2 Z) as a commutative subalgebra. As such, it is isomorhic either to Z or Z d). In the first case, the eigenforms have rational integer coefficients, and by the work of Sarnak [9], they must be of either dihedral or tetrahedral tye. We therefore restrict our attention to the latter case, when f g, and denote by K = Q d) the smallest field containing the coefficients. The integrality of T) imlies that λ g ) = λ f ), the Galois conjugate within K. We therefore write f in lace of g. Now, associated to f res., f ) is a weight-zero irreducible cusidal automorhic reresentation π res., π ) on GL 2 A), with A the adele ring of Q. The artial L-function of π on the comlex right half-lane Res) >1is given by Ls, π) = det 1 A s) 1 = 1 λπ ) s + ω π ) 2s) 1, 2.8) N N where A is the matrix of Satake arameters at the rime, and ω π is the central character of π. By the same arguments given in Section 2.1 alied to Galois reresentations, the central character ω π of π is quadratic.

5 Maass Cus Forms with Quadratic Integer Coefficients We roceed as in [9], demonstrating successively that noncusidality of each symmetric ower lift, together with quadratic integrality of its L-series coefficients, imlies the theorem. The arguments in [9] for the second and third symmetric owers when the coefficients were rational integers aly here, making all necessary changes, when the coefficients are quadratic integers. Thus, when sym 2 π is noncusidal, π is monomial, by [2, Theorem 3.3.7].Using[7], we show that π corresonds to a dihedral reresentation of the Weil grou, which, since it has quadratic integer coefficients, must have a finite order. When sym 3 π is noncusidal, Kim and Shahidi in [5, Proosition 3.3.8] show that π corresonds to a tetrahedral reresentation ρ of the Weil grou. By the integrality assumtion on its coefficients, it follows that ρ descends to a Galois reresentation. Finally, if sym 4 π is noncusidal, π corresonds to an octahedral reresentation ρ of the Weil grou, [5, Proosition 3.3.8]. Sinceρ has a finite image in GL 2, it is roerly a Galois reresentation, and again the theorem follows in this case. 2.4 What remains, and what we intend to disrove, is the case when all three symmetric ower lifts, sym 2 π, sym 3 π, and sym 4 π, are cusidal. The cusidality of these lifts will be the working hyothesis for the remainder of this aer. The artial L-function Ls, sym k π) is defined on a suitable right half-lane as L s, sym k π ) = N det 1 sym k A ) s ) ) Let χ be the character of the reresentation sym k on GL 2.Thenwehave d ds logl s, sym k π ) = χ A n ) log ) ns. 2.10) n 1 We will write the above equation as d ds logl s, sym k π ) = λ π )log ) s + R k) s), 2.11) where R k) s) = n 2 χ A n ) log ) ns. 2.12)

6 988 Farrell Brumley All the above sums over rimes are, and henceforth will be, imlicitly taken over those not dividing the level. Lemma 2.1. On Res) 1, d ds log L s, sym k π ) = λ symk π)log ) s + O1) 1 k 8). 2.13) Proof. We first demonstrate that the inner sum in R k) s) converges. Accordingly, set W k) n X) = X χ A n ) log ) ns. 2.14) Taking the absolute values, we obtain W k) n X) < X χ A n ) log ) nσ, 2.15) where σ = Res). Now, the coefficient χa n ) can be written as an integral olynomial in both λ π ) and ω π ), with λ π ) aearing in the leading term to the ower nk. Thus χ A n ) λπ ) nk, 2.16) from which we get W k) n X) ɛ λπ ) nk nσ+ɛ. 2.17) X A summation by arts affords λπ ) nk nσ+ɛ ɛ S nk ) nσ 1+ɛ, 2.18) X where we have denoted X S nk X) = λπ m) nk. 2.19) m X From the general theory of Rankin-Selberg integrals alied to the function Ls, sym 4 π sym 4 π), both factors being cusidal by assumtion, we deduce a bound for the eighth moment λπ n) 8 CX, 2.20) n X

7 Maass Cus Forms with Quadratic Integer Coefficients 989 for some ositive constant C. Combining this with the bound on the Hecke eigenvalues λ π ) 2 1/9, obtained by Kim and Shahidi in [4], we have S nk X) max λπ m) nk 8 λπ m) 8 ɛ X nk+1)/9+ɛ. 2.21) m X m X From this it follows that, as long as k 8, the sum in 2.18) converges on the region σ 1. With these restrictions on k and s, the inner sum of R k) s) converges, and we write, switching the order of summation, 1/9 ) n = 10/9 1 ) ) R k) s) = 8/9 n 2 This sum converges and the lemma is roved. Directly from our cusidality assumtion, we know that for integers 1 k 4, the function Ls, sym k π) is invertible at s = 1. In fact, we can establish, through Rankin- Selberg factorization cf. [4]) that Ls, sym k π) is invertible at s = 1 for the extended range of 1 k 8.This, along with Lemma 2.1, imlies the following identity: lim M 1 λ M symk π) log = 0 1 k 8). 2.23) M 2.5 We would like to obtain a result for Rankin-Selberg roducts of automorhic forms similar to that of Lemma 2.1. Throughout this section, we will denote by Π any one of the following tensor roduct reresentations on GL m+1)n+1) : sym m π sym n π, sym m π sym n π, sym m π sym n π, 2.24) when 1 m, n 4. IfA and B, for N, are the resective diagonal matrices of Satake arameters at the rime for the factors in Π, then the artial L-function Ls, Π) is defined on Res) >1by Ls, Π) = det 1 Π ) A B s ) ) N If we let χ denote the character of the reresentation sym m sym n, then uon taking the logarithmic derivative we get d ds logls, Π) = A ) ) j χ B log ) js. 2.26) j=1

8 990 Farrell Brumley We write the above equation as d ds logls, Π) = λ Π )log ) s + R m,n) s), 2.27) where R m,n) s) = j 2 A ) ) j χ B log ) js. 2.28) The roof of the following lemma mimics exactly that of Lemma 2.1. Lemma 2.2. On Res) 1, if Π is a reresentation in 2.24) with 1 m, n 4, then d ds log Ls, Π) = λ Π )log ) s + O1). 2.29) When the two factors in the Rankin-Selberg roduct are contragredient, the L- function has a simle ole at s = 1, and is invertible there otherwise. We conclude from Lemma 2.2 that for any reresentation Π in 2.24), with 1 m, n 4, lim M 1 1, if the factors in Π are contragredient, λ Π ) log = M 0, otherwise. M 2.30) 2.6 To make use of these quantities, we calculate some linearity relations. Denote by P n x) the olynomial sending the trace of a matrix in GL 2) 2 C), with determinant ω π, to the trace of its nth symmetric ower. In articular, λ symn π) = P n λπ ) ), λ symm π sym n π ) = P m λπ ) ) P n λπ ) ). The coefficients of each P n lie in Z[ω π ], where ω π is a symbol satisfying ω 2 π following relations hold: 2.31) = 1. The P 1 X) = X, P 2 X) = X 2 ω π, P 3 X) = X 3 2ω π X, P 4 X) = X 4 3ω π X 2 + 1, P 5 X) = X 5 4ω π X 3 + 3X, P 6 X) = X 6 5ω π X 4 + 6X 2 ω π, P 7 X) = X 7 6ω π X X 3 4ω π X, 2.32) P 8 X) = X 8 7ω π X X 4 10ω π X

9 Maass Cus Forms with Quadratic Integer Coefficients 991 From identity 2.23), we have lim M 1 M P m λπ ) ) log = 0 1 m 8), 2.33) M and from identity 2.30), assuming 1 m, n 4, lim M 1 M P m λπ ) ) P n λπ ) ) 1, if π π, log = 0, otherwise. M 2.34) The obvious equalities hold for Rankin-Selberg roducts sym m π sym n π and sym m π sym n π. We now have the tools with which to eliminate, case by case, the otential trace fields of the automorhic π afforded by the hyothesis of the theorem. 3 K-imaginary quadratic 3.1 We first treat the case when K is an imaginary quadratic extension of Q, say K = Q d), for some square-free ositive integer d. The comlex conjugate reresentation π = π of π is an irreducible cusidal automorhic reresentation of GL 2 over Q, which is given locally by the comlex conjugate of the Satake arameters for π. As the central character ω π is unitary, we have π = π.ongl 2, there is an isomorhism π π ω 1 π, and in the resent context, ω 1 π = ω π. For those rimes at which ω π ) = 1, the Satake arameters, and thus the Hecke eigenvalues, are real, being equal to their comlex conjugate. Likewise, for at which ω π ) = 1, the Hecke eigenvalues are urely imaginary, being the negative of their comlex conjugate. In the comlex lane, the coefficients of π lie in R ir. We are fortunate in this setting to have a convenient descrition of the reresentation π. The field K is imaginary; the Galois action is therefore comlex conjugation, and we have π π. We also observe that from the condition that π is not equal to π, the reresentation π is not self-dual, and the central character is therefore nontrivial. Throughout the remainder of this section, we will write π in lace of π.thefollowing relationshis will be used in later calculations: i) π and sym 3 π are not self-dual; ii) sym 2 π and sym 4 π are self-dual; iii) ω π is nontrivial.

10 992 Farrell Brumley 3.2 Let S be the subsace of R[x] generated by monomials whose degrees are even and no greater than 8. Elements in S can be considered as functions from R ir to R. Define a linear form I on S by 1 IT) = lim M M T λ π ) ) log, 3.1) M for T S. We calculate I on the standard basis of S by alying linearity to the relations between owers of the trace of a matrix and the traces of its symmetric owers. The olynomials P n defined in Section 2.6, the nontriviality ω π, and identity 2.33), rovide the following values for I: I1) = 1, I x 2) = 0, I x 4) = 2, I x 6) = 0, I x 8) = ) If K = Qi), consider the olynomial Tx) = x 2 1)x 2 4)x 2 + 1)x 2 + 4). As a real-valued function, T is negative at x = 0, zero at all other K-integral oints in [ 2, 2] [ 2i, 2i], and negative elsewhere in R ir. The mean trace IT) accordingly should be nonositive. Using the identities of 3.2), however, we comute that ITx)) = I x x 4 16) = 4. This contradiction eliminates the ossibility that K = Qi). If K = Q d), for d 3, then set Tx) = x 2 x 2 1)x 2 4)x 2 +3). The olynomial T vanishes at all integer oints in [ 2, 2], and is negative outside of this interval. On the K-integral oints in [ 2i, 2i], the value of T is zero when d = 3, and negative when d 5. Furthermore, T is negative outside of this interval in ir. The mean trace IT) should therefore be nonositive. Once again, however, we obtain a contradiction by comuting with the identities of 3.2) the value ITx)) = I x 8 + 2x x 4 12x 2 ) = Now, when K = Q 2), the above setting for the linear form I does not roduce a contradiction. The critical distance d on the imaginary axis beyond which a olynomial Tx) = x 2 x 2 1)x 2 4)x 2 +d) must have a root, in order to have a ositive mean, is d = 11/5 a distance small enough to eliminate all fields Q d), d 3, but too great to eliminate Q 2). Similarly, the critical distance d within which a olynomial Tx) = x 2 1)x 2 4)x 2 + d)x 2 + 4d) must have a root, in order to have a ositive mean, is aroximately a distance small enough only for Qi). ForQ 2) then, it is necessary to make use of the Galois structure afforded by the hyothesis of the theorem. In a two-variable

11 Maass Cus Forms with Quadratic Integer Coefficients 993 setting that registers π, as well as π, all quadratic imaginary fields can be eliminated at once, as we will see. Let K = Q d) for d 1. For each rime N, let Tr) = λ π ) + λ π ) and Nm) = λ π )λ π ) be the trace and norm in Z of the coefficients, as elements in K. Then, for Tx, y) in S = R[x, y], set 1 IT) = lim M M T Tr), Nm) ) log, 3.3) M for T S. The linear form I is evaluated at oints x, y) in the standard Z 2 lattice in R 2 such that x = Tr) and y = Nm). Droing the deendence on, if λ π = a + b d, then when x = Tr = 2a, we have y = Nm = a 2 + db 2 a 2 = 1/4)x 2. Only those lattice oints on or above the arabola y = 1/4)x 2 can ossibly be reresented as trace norm of the coefficients. Set Tx, y) = y 14 x2 ) y 1) ) The function T vanishes on the arabola y = 1/4)x 2.Withinit, T is zero on the line y = 1, and strictly ositive above it a region which for every quadratic imaginary field includes all oints Tr), Nm)). Accordingly, T should always have a nonnegative mean. We comute however, using the values Iy) = 1, I y 2) = 2, I x 2) = 2, I x 2 y ) = 4, I y 3) = 4, I x 2 y 2) = 8, I y 4) = 20, I x 2 y 3) = 48, 3.5) that ITx, y)) = 3/2 < 0, which gives the desired contradiction. 4 K-real quadratic 4.1 Next, we consider the case when K is a real quadratic extension of Q. LetS be the real vector sace generated by olynomials in two variables x and y of the form x i, i 8; y j, j 8; x i y j, 1 i, j )

12 994 Farrell Brumley Define a linear form I on S by 1 IT) = lim M M T λ π ),λ π ) ) log, 4.2) M for T S. We assume for the moment that sym 2 π sym 2 π, and then evaluate I at any olynomial T S using identity 2.34). This assumtion will be in effect until Section 4.3.For brevity, in the list that follows, we give only the nonzero values of I for those monomials ins needed in the analysis: I1) = 1, I x 2) = I y 2) = 1, I x 2 y 2) = 2, I x 4) = I y 4) = 2, I x 6) = 5, I x 8) = ) From sym 2 π sym 2 π, we deduce that λ π ) = ±λ π ) for every. As one eigenvalue is the Galois conjugate of the other, either λ π ) Z or dλ π ) Z. Thus in the x, y-lane, all coefficient airs λ π ),λ π )) lie on the line y = x, or y = x. Set T ɛ x, y) = y x) 2 x 2 + y 2 4 ɛ ), 4.4) for ɛ 0. Whend 3, the olynomial T ɛ is nonnegative on all ossible coefficient airs as long as ɛ 2. Indeed, when K = Q 3), the coefficient airs off the line y = x lie at distance at least 6 from the origin. This is the closest any off-diagonal coefficient air can be to the origin, for in the case that K = Q 5), any K-integer that is lus or minus its Galois conjugate has trivial denominator.) Thus, having as a factor in T ɛ, the equation for a circle of radius at most 6 ensures nonnegativity for the value of T ɛ at all ossible coefficient airs. Accordingly, T ɛ should have zero mean-trace. We comute, however, using the identities of 4.3), that for ɛ>0, IT ɛ ) < 2ɛ. Thus for d 3, we have obtained a contradiction. 4.2 Now, when d = 2, the above arguments do not roduce a contradiction, for then ɛ = 0 gives IT ɛ )=0.Thisiswellso, for as the exlicit matrix realization of Section 2.1 showed, Q 2) is recisely the case which occurs for automorhic π coming from octahedral Galois reresentations. In this case, we construct the subsace V of the theorem, along with its integral basis, as follows. Let ρ be an even octahedral reresentation of the absolute Galois grou of Q. The image of ρ in GL 2) 2 C) is given in Section 2.1. By comosing ρ with

13 Maass Cus Forms with Quadratic Integer Coefficients 995 the nontrivial Galois automorhism of GalQ 2)/Q), which acts entrywise on the image of ρ, we obtain the Galois conjugate octahedral reresentation ρ. The results of Langlands [8] and Tunnell [10] roduce corresonding cusidal automorhic reresentations π, H π ) and π,h π ) of GL 2 over Q reserving L-functions. Moreover, π is weight-zero since ρ is even. If f H π and f H π are eigenfunctions of the Hecke subalgebra T, then we take V = C{f, f }. A basis of V with resect to which T acts by integral matrices is {f + f )/2, f f )/2 2}. The symmetric ower L-functions of an octahedral π are invertible at s = 1 for all owers u to the seventh, but have a ole there at the eighth. In order to rule out the existence of a nonoctahedral π, the olynomial roducing the contradiction must be able to distinguish between the two by icking u the behavior of Ls, sym 8 π) at s = 1. The degree of the olynomial should accordingly be at least 8. We set Tx, y) = x 2 x 2 1 ) x 2 2 ) x 2 4 ). 4.5) The seven vertical lines in the vanishing locus of T coincide with the coefficient airs which satisfy the Ramanujan conjecture. The assumtion that sym 2 π sym 2 π ensures that all other coefficient airs lie to the left or right of these vertical lines. In those two regions, the olynomial T is ositive. With these considerations, it should follow that the mean trace of T is nonnegative. We comute, however, using the identities of 4.3), that IT) = 1. This contradiction eliminates the ossibility that the π of our hyothesis can be non-galois. Finally, we check that if π corresonded to an octahedral Galois reresentation, the value Ix 8 ) would be 15, IT) would be 0, and no contradiction then occurs. 4.3 We now go through similar arguments under the comlementary assumtion that sym 2 π sym 2 π. This changes the value of Ix 2 y 2 ) listed in 4.3) from 2 to 1. We take K = Q d) where d 5. If the Ramanujan conjecture holds, the only oints λ π ),λ π )) which contribute to the value of I are those for which both coordinates are bounded in absolute value by 2. Those that are also K-integral lie either on the diagonal y = x, as with 0, 0), 1, 1), 1, 1), 2, 2), and 2, 2); or off the diagonal and at distance from the origin no less than 2, as with 2, 2) and 2, 2), when K = Q 2). Set Tx, y) = x 2 + y 2 4 ) y x) )

14 996 Farrell Brumley The olynomial T is nonnegative in the region bounded away from the circle centered at the origin of radius 2, and also on all Ramanujan oints enumerated above. The mean IT) should therefore be nonnegative. However, using the identities of 4.3), we comute that IT) = 2, a contradiction. 5 The excetional field Q 5) U to this oint, we have roven the theorem when the coefficient field is any quadratic extension of Q excet Q 5). This case must remain outstanding for reasons we now set forth. If ρ is an even icosahedral reresentation of the absolute Galois grou of Q, with trivial central character, then its range in GL 2 C) is described in Section 2.1 and we can form the Galois conjugate ρ by comosing ρ with the nontrivial element in GalQ 5)/ Q).Toρ and ρ are associated, through a conjectural Artin corresondence, cusidal automorhic reresentations π, H π ) and π,h π ) both of which are irreducible constituents of W if f H π and f H π are eigenfunctions of the Hecke algebra T, the subsace V = C{f, f } admits the basis {f + f )/2, f f )/2 5} with resect to which the Hecke oerators act as integral matrices. The symmetric ower L-functions of f, being equal to those of ρ, are invertible at s = 1 for all owers u to the eleventh, and have a ole there at the twelfth. This roerty serves to test whether a given automorhic form is Galois, for any Maass cus form whose coefficients are integers in Q 5) could not ossibly corresond to an icosahedral Galois reresentation if its symmetric ower L-functions were invertible at s = 1 for all owers u to and including the twelfth. This latter roerty would resumably hold, via Rankin-Selberg factorization, if there existed symmetric ower functorial lifts u through the sixth, and if under every one of these lifts the given form was actually cusidal. As we have at resent only functorial lifts u to the fourth ower, and from this, knowledge of the form s symmetric ower L-functions only u to the eighth, those Maass forms which, conjecturally at least, are of icosahedral Galois rovenance are effectively indistinguishable from those which are not. This exlains why in the theorem no claims are made about forms with quadratic integer coefficients inside Q 5). Acknowledgments I would like to thank Peter Sarnak for suggesting this roblem and for his invaluable hel in realizing the solution. I extend thanks to the referee as well for his useful editorial comments.

15 Maass Cus Forms with Quadratic Integer Coefficients 997 References [1] H. F. Blichfeldt, Finite Collineation Grous, University of Chicago Press, Illinois, [2] S. Gelbart and H. Jacquet, A relation between automorhic reresentations of GL2) and GL3), Ann. Sci. École Norm. Su. 4) ), no. 4, [3] H. H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2, J. Amer. Math. Soc ), no. 1, , Aendix 1 by D. Ramakrishnan, Aendix 2 by H. H. Kim and P. Sarnak. [4] H. H. Kim and F. Shahidi, Functorial roducts for GL 2 GL 3 and functorial symmetric cube for GL 2, C. R. Acad. Sci. Paris Sér. I Math ), no. 8, [5], Cusidality of symmetric owers with alications, Duke Math. J ), no. 1, [6] F. Klein, Lectures on the Icosahedron, Kegan Paul, London, [7] J.-P. Labesse and R. P. Langlands, L-indistinguishability for SL2), Canad. J. Math ), no. 4, [8] R. P. Langlands, Base Change for GL2), Annals of Mathematics Studies, vol. 96, Princeton University Press, New Jersey, [9] P. Sarnak, Maass cus forms with integer coefficients, A Panorama of Number Theory or the View from Baker s Garden G. Wustholz, ed.), Cambridge University Press, Cambridge, [10] J. Tunnell, Artin s conjecture for reresentations of octahedral tye, Bull. Amer. Math. Soc. N.S.) ), no. 2, Farrell Brumley: Deartment of Mathematics, Princeton University, Princeton, NJ 08544, USA address: fbrumley@math.rinceton.edu