ON THE NORMS OF p-stabilized ELLIPTIC NEWFORMS

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1 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS JIM BROWN 1 AND KRZYSZTOF KLOSIN 2, WITH AN APPENDIX BY KEITH CONRAD 3 Abstract. Let f S κ(γ 0(N)) be a Hecke eigenform at with eigenvalue λ f () for a rime N. Let α and β be comlex numbers satisfying α + β = λ f () and α β = κ 1. We calculate the norm of (z) = f(z) β f(z) as well as the norm of U f, both classically and adelically. We use these results along with some convergence roerties of the Euler roduct defining the symmetric square L-function of f to give a local factorization of the Petersson norm of f. f 1. Introduction Let κ 2 and N 1 be integers and an odd rime with N. Let f S κ (Γ 0 (N)) be a newform. It is well-known that the Petersson norm f, f serves as a natural eriod for many L-functions of f [6, 14]. In this aer we focus on related eriods f, f (defined below) for α a Satake arameter of f. When f is ordinary at, the forms f arise naturally in the context of Iwasawa theory as the objects which can be interolated into a Hida family. It is in fact in the context of -adic interolation of some automorhic lifting rocedures (between two algebraic grous, one of them being GL 2 ) that these calculations arise (see [1] for examle); however, our results aly in a more general setu as secified below. Let f S κ (Γ 0 (N)) be an eigenform for the T -oerator with eigenvalue λ f (). Let α and β be the air of comlex numbers satisfying α + β = λ f () and α β = κ 1. We set f (z) = f(z) β f(z). In the case that f is ordinary at, we can choose α and β so that α is a -unit and β is divisible by. In this secial case f is the -stabilized ordinary newform of tame level N attached to f Mathematics Subject Classification. Primary 11F67; Secondary 11F11. Key words and hrases. -stabilized forms, modular eriods, symmetric square L- function, Euler roducts. The first author was artially suorted by the National Security Agency under Grant Number H The United States Government is authorized to reroduce and distribute rerints not-withstanding any coyright notation herein. The second author was artially suorted by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York. 1

2 2 JIM BROWN AND KRZYSZTOF KLOSIN Since f = 1 κ β (U β )f, calculating f, f is in fact equivalent to calculating U f, U f. While comutation of any of these inner roducts does not resent any difficulties (see Section 2), it is an accident resulting from the relative simlicity of the Hecke algebra on GL 2, where the T and the U oerators differ by a single term. It turns out that in the higherrank case it is the calculation of the latter inner roduct that rovides the fastest route to comuting the Petersson norm of various -stabilizations. With these future alications in mind we resent an alternative aroach to calculating U f, U f, this time working adelically (see Sections 3 and 4), as this is the method that generalizes to higher genus most readily (see [1], where this is done for the grou GS 4 ). It is well-known that the Petersson norm f, f is closely related to the value L(κ, Sym 2 f) at κ of the symmetric square L-function of f. The absolutely convergent Euler roduct defining this L-function for Re(s) > κ converges (conditionally) to the value L(s, Sym 2 f) when Re(s) = κ (this and in fact a more general result is roved in the aendix). On the other hand our comutation of f, f shows that this inner roduct differs from f, f by essentially the -Euler factor of L(κ, Sym 2 f). Combining these facts we exhibit a (conditionally convergent) factorization of f, f into local comonents defined via the inner roducts f, f (for details, see Section 5). The authors would like to thank Gergely Harcos and Henryk Iwaniec for helful corresondence. 2. Classical calculation of f, f and U f, U f Let N be a ositive integer. Let Γ 0 (N) SL 2 (Z) denote the subgrou consisting of matrices whose lower-left entry is divisible by N. For [ a holomorhic function f on the comlex uer half-lane h and for γ = ] a b c d GL + 2 (R), where + denotes ositive determinant, and κ Z + we define the slash oerator as ( ) (f κ γ)(z) = det(γ)κ/2 az + b (cz + d) κ f. cz + d If κ is clear from the context we will simly write f γ instead of f κ γ. We will write S κ (Γ 0 (N)) for the C-sace of cus forms of weight κ and level Γ 0 (N) (i.e., functions f as above which satisfy f κ γ = f for all γ Γ 0 (N) and vanish at the cuss - for details see [10]). The sace S κ (Γ 0 (N)) is endowed with a natural inner roduct (the Petersson inner roduct) defined by f, g N = f(z)g(z)y κ 2 dxdy Γ 0 (N)\h for z = x + iy with x, y R and y > 0.

3 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 3 If Γ Γ 0 (N) is a finite index subgrou we also set f, g Γ = f(z)g(z)y κ 2 dxdy. Γ\h From now on let be a rime which does not divide N. Set η = We have the decomosition (2.1) Γ 0 (N) Γ 0 0 (N) = 1 j=0 Γ 0 (N) 1 j Γ 0 0 (N)η. Recall the th Hecke oerator acting on S κ (Γ 0 (N)) is given by 1 T f = κ/2 1 1 j f κ + f 0 κ η and the th Hecke oerator acting on S κ (Γ 0 (N)) is given by 1 U f = κ/2 1 1 j f κ. 0 j=0 j= As we will be viewing f S κ (Γ 0 (N)) as an element of S κ (Γ 0 (N)), we use T and U to distinguish the two Hecke oerators at defined above. Let f S κ (Γ 0 (N)) be an eigenfunction for T with eigenvalue λ f (). There exist (u to ermutation) unique comlex numbers α and β satisfying λ f () = α + β and α β = κ 1. We consider the following two forms: f (z) = f(z) β κ/2 (f κ η)(z), f β (z) = f(z) α κ/2 (f κ η)(z). One immediately obtains that f S κ (Γ 0 (N)) and that f is an eigenfunction for the oerator U with eigenvalue α. Furthermore, if f is also an eigenform for T l for a rime l, then so is f and it has the same T l - eigenvalue as f. The analogous statements for f β hold as well. Note that if f is ordinary at, then one can choose α and β so that ord (α ) = 0 and then f is the -stabilized newform associated to f, see [15] for examle. Theorem 2.1. Let f S κ (Γ 0 (N)) be defined as above, where N. We have U f, U f N = κ 2 + ( 1)λ f () 2 f, f N + 1 and ( ) ( ) f, f N = f, f N α2 κ 1 β2 κ.

4 4 JIM BROWN AND KRZYSZTOF KLOSIN and the fact that U f = T f κ/2 1 f η im- Proof. The definition of f mediately give f, f N = (1 + β 2 κ ) f, f N κ/2 (β f η, f N + β f η, f N ) and U f, U f N = ( κ 2 + λ f () 2 ) f, f N κ/2 1 λ f ()( f η, f N + f η, f N ). Let us now comute f η, f N. Observe that by the definition of T we have 1 T f, g N = κ/2 1 1 j f, g + f η, g 0 N. j=0 N 1 j Using the decomosition (2.1) we can find a j, b j Γ 0 (N) so that a j b 0 j = 0, and a, b Γ 0 0 (N) so that a b =. Using this and the fact that f, g S κ (Γ 0 (N)), we have 1 1 κ/2 1 j T f, g N = f a j, g b 1 0 j + f η, g N j=0 N 1 1 j = f a j b 0 j, g + f η, g N j=0 N = f, g + f η, g 0 N N = f a, g b f η, g N Thus, setting g = f we obtain We can now easily conclude that and f = ( + 1) f η, g N. N f η, f N = 1 κ/2 λ f () + 1 f, f N., f N f, f N = 1 + β 2 κ 1 κ (β + β ) λ f () + 1 U f, U f N = κ 2 + ( 1)λ f () 2. f, f N + 1 Using the fact that T is self-adjoint with resect to the Petersson inner roduct we have α + β = λ f () R. We note by Lemma 4.2 below that

5 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 5 α = β. Thus α 2 = β 2 = α β = κ 1 and β + β = λ f (). This allows us to simlify the formula for f,f N f,f N to f, f N = f, f N λ 1 κ f () Again using that λ f () = α + β we obtain f, f N = 1 ( ) 1 κ ( 2 + β κ 1 ) f, f N + 1 = + 1 Corollary 2.2. We have ( lim is rime 1 α2 κ f ) (, f N β2 κ ). = f, f N. Proof. Using Theorem 2.1 and the fact that f, f N = ( + 1) f, f N, we have for every rime N that ( ) ( ) f, f N + 1 = α2 κ 1 β2 κ f, f N. Since α 2 = β 2 = κ 1, we see that the first three factors on the right tend to 1 as tends to infinity. 3. Relation between the classical and adelic inner roducts While the classical calculations for U f, U f are rather elementary, it is also useful to note that one can erform these calculations adelically. The roblem of calculating U f, U f is one that is local in nature, so it lends itself nicely to such an aroach. Moreover, in a higher genus setting such as when working with Siegel modular forms, it is the adelic aroach that generalizes most readily [1]. In this section we rovide the necessary background relating the adelic and classical inner roducts that is necessary to relate the adelic inner roduct calculated in Section 4 to the calculation given in the revious section. In this and the following sections will denote a rime number and v will denote an arbitrary lace of Q including the Archimedean one, which we will denote by. Let G = GL 2 and fix N 1. By strong aroximation [2, Theorem 3.3.1,. 293] we have (3.1) G(A) = G(Q)G(R) K,

6 6 JIM BROWN AND KRZYSZTOF KLOSIN where K is a comact subgrou of G(Q ) such that det K = Z. One examle would be to take K = K 0 (N), where { } a b K 0 (N) = G(Z c d ) : c 0 mod N. Note that K 0 (N) = G(Z ) if N. We will also set { } a b K 0 (N) := c d G(Ẑ) : c 0 mod N = The decomosition (3.1) imlies [2,. 295] that K 0 (N). (3.2) G(Q) \ G(A) = G + (R) K, where + indicates ositive determinant. This is not a direct roduct. Write K for K and Γ G + (R) for the subgrou corresonding to K, i.e., Γ is the image of the (clearly injective) comosite (G + (R)K) G(Q) G(Q) G(A) G(R). Note that if K = K 0 (N), then Γ = Γ 0 (N). Lemma 3.1. The ma given by (g, k) gk is a bijection. φ : (Γ \ G + (R)) K G(Q) \ G(A) Proof. If γ Γ then φ((γg, k)) = γgk = gk since Γ G(Q), so φ is welldefined. It is surjective by (3.2), so it remains to check injectivity. Suose φ((g, k)) = φ((g, k )). Then there exists γ G(Q) such that g k = γgk, i.e., g 1 γ 1 g = k(k ) 1. Note that the quantity on the right has its infinite comonent equal to 1. Thus, if we denote by γ the infinite comonent of γ and by γ f the finite comonent of γ, then we have g = γ g. On the other hand, the finite comonent of g 1 γ 1 g equals γ 1 f. Hence k(k ) 1 = γ 1 f, so we get γ 1 k(k ) 1 = γ G(Q) (G + (R)K) = Γ. Thus g = γ g = g as an element of Γ \ G + (R). Since φ is well-defined the third equality below is justified: φ((g, k)) = φ((g, k )) = φ((γ g, k )) = φ((g, k )). Reading across, we conclude that there must exist δ G(Q) such that δgk = gk. Note that g commutes with k and k, hence we can transform the above to δkg = k g and conclude that δk = k. Comaring the infinite comonents and using the fact that δ G(Q) we get δ = 1, hence k = k. Let Z G denote the center. We have Z(A) = A = Z + (R) Z

7 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 7 and the bijection in Lemma 3.1 descends to a bijection ψ : (Z + (R)Γ \ G + (R)) (K Z(Q )) \ K Z(A)G(Q) \ G(A). For every there is a unique Haar measure dg on G(Q ) normalized so that the volume of any maximal comact subgrou of G(Q ) is one. This measure descends to a unique ositive regular Borel measure d g on the quotient (K Z(Q )) \ K if we insist that the volumes of K and of (K Z(Q )) \ K in their resective measures coincide, i.e., that dg = d g K (K Z(Q ))\K (see e.g., [2, Proosition 2.1.5]). On the other hand every element g Z + (R) \ G + (R) = SL 2 (R) can be decomosed (uniquely) as [ g = Z + y 1/2 xy (R) 1/2 ] cos(θ) sin(θ) 0 y 1/2, sin(θ) cos(θ) where x, y R, y > 0, θ [0, 2π). Set dg = dxdy dθ. In fact dθ is the y 2 normalized Haar measure on SO(2), a maximal comact subgrou of G + (R), so dg descends to a measure on Z + (R) \ G + (R)/SO(2) = h, which agrees with the measure dxdy on h (for more details see e.g., [2, ]). This, y 2 along with the existence of the bijection ψ, allows us to define a measure on Z(A)G(Q)\G(A) which we will denote by dg and define the adelic analogue of the Petersson inner roduct: φ 1, φ 2 = φ 1 (g)φ 2 (g)dg, Z(A)G(Q)\G(A) where φ 1 and φ 2 lie in L 2 (Z(A)G(Q) \ G(A)) and have the same central character. [ Let] f S κ (Γ 0 (N)). For g = γg k G(A) with γ G(Q), g = a b G c d + (R) and k K 0 (N), set (3.3) φ f (g) = (det g ) κ/2 (ci + d) κ f(g i). Then φ f is an automorhic form on G(A). If f is a newform, then we can write φ f (g) = v φ f,v (g v ), where g = v g v with g v G(Q v ) if v < and g G + (R), and φ f,v : G(Q v ) C for v < and φ f, : G + (R) C. In general if such a factorization holds we have the following relationshi between the adelic and the classical inner roducts. Corollary 3.2. Let K = K, where K is a comact subgrou of G(Ẑ) such that (3.1) holds and let Γ G + (R) be the corresonding subgrou. Let φ 1 and φ 2 be elements of L 2 (Z(A)G(Q) \ G(A)) with the same central character. Assume that φ 1 and φ 2 factor into a roduct of local comonents,

8 8 JIM BROWN AND KRZYSZTOF KLOSIN i.e., φ j = φ j, φ j, for j = 1, 2, and that φ j, is invariant under Γ for j = 1, 2. Then we have φ 1, φ 2 = φ 1,, φ 2, Γ φ 1,, φ 2, K, where and φ 1,, φ 2, K := φ 1, (g )φ 2, (g )dg K φ 1,, φ 2, Γ := Γ\h φ 1, (g)φ 2, (g)dg, with the measure dg on Γ \ h as described before. Note that if φ 1 = φ f1 and φ 2 = φ f2 for some modular forms f 1, f 2, then φ 1,, φ 2, Γ = f 1, f 2 Γ. Furthermore, if f 1 and f 2 are newforms of level Γ (i.e., the corresonding local comonents φ j, are right-k -invariant and φ j, is normalized so that φ j, (1) = 1), then we obtain φ 1, φ 2 = f 1, f 2 Γ vol(k ). Proof. The only thing to exlain is the disaearance of Z(Q ) K on the right-hand side. Since both φ 1, and φ 2, are invariant under the action of the center and the volumes of K and (Z(Q ) K ) \ K coincide in their resective measures, the integrals over K and over (Z(Q ) K )\K clearly coincide as well. 4. Local calculation of U φ f,, U φ f, We will now give a calculation that, when combined with the results of the revious section, rovides a local way to calculate U f, U f. In this section we fix a rime not dividing N. Let φ be a measurable function on G(Q ) that is right invariant under K 0 (N) = K 0 (1) = G(Z ). As noted above, we normalize our Haar measure so that vol(k 0 (1) ) = 1. Define an oerator T as (T φ )(g) = [ φ (gh)dh. K 0 (1) 0 ]K 0 (1) Note that we have the decomositions 0 (4.1) K 0 (1) K 0 (1) = 1 b=0 and 0 (4.2) K 0 () K 0 () = b K 0 (1) 1 b=0 b K 0 (). K 0 0 (1)

9 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 9 Define an oerator U on functions which are right invariant under K 0 (1) by U φ := T φ V φ, where (V φ )(g) = [ φ (gh)dh. 0 ]K 0 (1) This is the adelic counterart of the U -oerator acting on classical modular forms as defined in Section 2, which in this section we will denote by U cl. Indeed, let f S κ (Γ 0 (N)) be a newform with N and let π f = v π v be the associated automorhic reresentation. Let φ = φ f = v φ v v π v be defined as in (3.3). If we set U φ f := (U φ ) v φ v then it follows by the same argument as the one in the roof of [4, Lemma 3.7] that (4.3) φ U cl f = κ/2 1 U φ f. Note that since N, π is an unramified sherical rincial series π (χ 1, χ 2 ). In articular the K 0 (N) -fixed subsace of π (χ 1, χ 2 ) is one-dimensional and clearly contains φ. Moreover, one can describe φ exlicitly as ( ) a φ k = χ 0 b 1 (a)χ 2 (b) ab 1 1/2, where denotes the standard -adic norm ( = 1 ) and k K 0 (N). Lemma 4.1. We have U cl f, U cl f N f, f N = κ 2 U φ, U φ K0 (1). Proof. This follows from Corollary 3.2. Indeed, write θ for φ U cl f and φ f = v φ v. Also, set K 0 (N) := Γ 0 (N). Then we have by (4.3) and Corollary 3.2, θ, θ = κ 2 U φ f, U φ f = κ 2 U φ, U φ K0 (N) φ v, φ v K0 (N) v. Since f is of level N, we get φ, φ = f, f N v vol(k 0(N) v ) and this yields κ 2 θ, θ U φ, U φ K0 (N) = v φ v, φ v K0 (N) v θ, θ = f, f N v vol(k 0(N) v ) φ, φ K0 (N) v θ, θ = f, f N v vol(k 0(N) v ), where the last equality comes from the fact that φ is right-k 0 (N) -invariant, so φ, φ K0 (N) = vol(k 0 (N) ) = 1 since N. Note that U cl f is of level

10 10 JIM BROWN AND KRZYSZTOF KLOSIN N, hence we get (again by Corollary 3.2) θ, θ = U cl f, U cl f N Since f is of level N we also get v vol(k 0 (N) v ) = U cl f, U cl 1 f N + 1 v f, f N = f, f N. vol(k 0 (N) v ). This roves the lemma. Thus we see it is enough to calculate U φ, U φ. As the rest of the section is focused on this calculation, we will from now on write φ for φ and K 0 (1) (res., K 0 ()) for K 0 (1) (res., K 0 () ). Set B := { } b : b {0, 1,..., 1}, B 0 1 := B {}. 0 If g K 0 (1) and β B, there is a ermutation σ g of B and elements k(g, β) K 0 (1) such that gβ = σ g (β)k(g, β). Furthermore, note [ that ] if g K 0 (1) K 0 (), then the corresonding ermutation cannot fix. This 0 imlies that for such a g, there exists β B such that σ g (β) =. Since 0 in the comutation of U φ only matrices in B are used, we are interested in the restriction of σ to [ B. ] As remarked above, if g K 0 (1) K 0 () such a restriction will have in its image. In other words, for such a g there 0 are 1 matrices in the image of σ g which have (, 1) on the diagonal and one that has (1, ) on the diagonal. Set B 1 (g) = {β B : σ g (β) B} and B 2 (g) = {β B : σ g (β) B B}. So for g K 0 (1) K 0 () we have (note that our φ is right-k 0 (1)-invariant and vol(k 0 (1)) = 1) (U φ)(g) = vol(k 0 (1)) β B φ(gβ) = φ(σ g (β)k(g, β)) β B = φ(σ g (β)) + φ(σ g (β)) β B 1 (g) β B 2 (g) = ( 1)χ 1 () 1/2 + χ 2 () 1/2.

11 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 11 If g K 0 (), then the ermutation σ fixes (U φ)(g) = vol(k 0 (1)) β B, hence we obtain 0 φ(gβ) = β B φ(σ(β)) = χ 1 () 1/2 = χ 1 () 1/2. Now let us comute the integral: U φ, U φ K0 (1) = U φ(g)u φ(g)dg + (4.4) We have K 0 () K 0 () U φ(g)u φ(g)dg = and, since vol(k 0 (1) K 0 ()) = /( + 1), (4.5) U φ(g)u φ(g)dg = K 0 (1) K 0 () K 0 (1) K 0 () K 0 () K 0 (1) K 0 () χ 1 () 2 dh = vol(k 0 ()) χ 1 () 2 = χ 1() U φ(g)u φ(g)dg. ( 1) 2 χ 1 () 2 + ( 1)tr (χ 1 ()χ 2 ()) + χ 2 () 2 dg = ( 1) 2 χ 1 () 2 + ( 1)tr (χ 1 ()χ 2 ()) + χ 2 () Putting (4.4) and (4.5) together we get (4.6) U φ, U φ = χ 1 () χ 2() tr (χ 1()χ 2 ()). Lemma 4.2. We have χ j () = s j for j = 1, 2, where s j is a urely imaginary number. In articular, χ j () = 1 for j = 1, 2. Moreover, we have α = β. Proof. The first art follows from [4,. 92] and is a direct consequence of the fact that cus forms on GL 2 satisfy the Ramanujan conjecture. Observe that α = (κ 1)/2 χ 1 () and β = (κ 1)/2 χ 2 (). Using that α β = κ 1, we obtain χ 1 ()χ 2 () = 1. This, combined with the fact that χ j () = s j with s j urely imaginary imlies χ 1 () = χ 2 (). Thus α = β. Using Lemma 4.2 we can simlify (4.6) to U φ, U φ = tr (χ 1()χ 2 ()).

12 12 JIM BROWN AND KRZYSZTOF KLOSIN Moreover using that α = (κ 1)/2 χ 1 (), β = (κ 1)/2 χ 2 () and χ 1 () = χ 2 () we have tr (χ 1 ()χ 2 ()) = (χ 1 () + χ 2 ()) 2 2χ 1 ()χ 2 () = 1 κ λ f () 2 2. Thus we obtain U φ, U φ = ( 1)1 k λ f () = 1 + ( 1)λ f () 2 2 k, + 1 hence we see that by Lemma 4.1 this recovers the classical formula from Theorem Alications to L-values Let f S κ (Γ 0 (N)) be a newform. In this section we aly the results of the revious sections to give a local decomosition of the Petersson norm of f. Recall that the (artial) symmetric square L-function of f is defined by the Euler roduct (5.1) L N (s, Sym 2 f) = 1 L N (s, Sym 2 f), where L (s, Sym 2 f) := ( 1 α2 s ) ( 1 α ) ( β s 1 β2 s The roduct (5.1) converges absolutely for Re s > κ. It is well-known that L N (s, Sym 2 f) admits meromorhic continuation to the entire comlex lane with ossible oles only at s = κ and κ 1, of order at most one [13, Theorem 1]. In our case (since f is assumed to have trivial character), the L-function does not have a ole at s = κ [13, Theorem 2]. We will continue to denote this extended function by L N (s, Sym 2 f). Using that α β = κ 1 we conclude that f, f N (5.2) = 2 1 f, f N 2 1 L (κ, Sym 2 f) = ζ (2) L (κ, Sym 2 f), where ζ (s) = 1/(1 1/ s ). Corollary 5.1. We have Set f, f N = f β, f β N. f, f () N := f, f N = f, f N. f, f N f β, f β N We will now show that f, f () N can in some sense be regarded as a local (at ) eriod for the symmetric square L-function. ).

13 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 13 Theorem 5.2. The value L N (κ, Sym 2 f) given by the meromorhic continuation is equal to the conditionally convergent Euler roduct 1 L (κ, Sym 2 f) N when we order the factors according to increasing. Proof. Let φ f be defined as in (3.3) and let χ 1 () = α / (κ 1)/2 and χ 2 () = β / (κ 1)/2 be its Satake arameters for N as in Section 4. For N define L (s, Sym 2 φ f ) := (1 χ 1() 2 ) (1 1 ) (1 s χ 2() 2 ) and note that L (s, Sym 2 φ f ) = L (s + κ 1, Sym 2 f). roduct L N (s, Sym 2 φ f ) := 1 L N (s, Sym 2 φ f ) s s Thus the Euler converges absolutely for Re s > 1 and inherits all the corresonding roerties (in articular the meromorhic continuation and the lack of a ole at s = 1) from L N (s, Sym 2 f). As before we will continue to denote this extended function by L N (s, Sym 2 φ f ). Let π be the automorhic reresentation of GL 2 (A) associated with φ f. It is known [5, Theorem 9.3] that there exists an automorhic reresentation σ of GL 3 (A) such that the (artial) standard L-function L N (s, σ) coincides with L N (s, Sym 2 π) := L N (s, Sym 2 φ f ). Also L N (s, σ) does not vanish on the line Re s = 1 by a result of Jacquet and Shalika (see [7, Theorem 1]; see also [8]). Finally note that by Lemma 4.2, we have χ 1 () = χ 2 () = 1 if N. Thus we are in a osition to aly Theorem A.1 in the aendix with K = Q and d = 3 to L N (s, Sym 2 φ f ) and the theorem follows. By Theorem 5.2 and (5.2) we have L N (κ, Sym 2 f) N f, f () N = ζ N (2), where the suerscrit means that we omit the Euler factors at rimes dividing N, and the roduct N (here and below) is ordered according to increasing. Using [6, Theorem 5.1] we have L N (κ, Sym 2 f) = (1 λ f () 2 ) 2 2κ π κ+1 κ (κ 1)!δ(N)Nφ(N) f, f N, N where δ(n) = 2 or 1 according as N 2 or not. Using this we obtain the following corollary that can be viewed as a factorization of the global eriod f, f N in terms of the local eriods f, f () N.

14 14 JIM BROWN AND KRZYSZTOF KLOSIN Corollary 5.3. We have f, f N = (κ 1)!δ(N)Nφ(N)ζN (2) 2 2κ π κ+1 N 1 1 λ f () 2 / κ f, f () Aendix A. Convergence of Euler roducts on Re(s) = 1 by Keith Conrad 3 Let K be a number field. A degree d Euler roduct over K is a roduct L(s) = 1 (1 α,1 N s ) (1 α,d N s ), where α,j 1 for all nonzero rime ideals in the integers of K and 1 j d. On the half-lane Re(s) > 1 this converges absolutely and is nonvanishing. Combining factors at rime ideals lying over a common rime number, L(s) is also an Euler roduct over Q of degree d[k : Q]. We want to rove a general theorem about the reresentability of L(s) by its Euler roduct on the line Re(s) = 1. If L(s) is the L-function of a nontrivial Dirichlet character, this is in [3, ], [9, 109], and [11,. 124] if s = 1 and [9, 121] if Re(s) = 1. Theorem A.1. If L(s) is a degree d Euler roduct over K and it admits an analytic continuation to Re(s) = 1 where it is nonvanishing, then L(s) is equal to its Euler roduct on Re(s) = 1 when factors are ordered according to rime ideals of increasing norm: if Re(s) = 1 then L(s) = lim x N x N 1 (1 α,1 N s ) (1 α,d N s ). The roof is based on the following lemma about reresentability of a Dirichlet series on the line Re(s) = 1. Lemma A.2. Suose g(s) = n 1 b nn s has bounded Dirichlet coefficients. If g(s) admits an analytic continuation from Re(s) > 1 to Re(s) 1, then g(s) is still reresented by its Dirichlet series on the line Re(s) = 1. Proof. See [12]. Here is the roof of Theorem A.1. Proof. We will aly Lemma A.2 to a logarithm of L(s), namely the absolutely convergent Dirichlet series (log L)(s) := k 1 α k,1 + + αk,d kn ks, where Re(s) > 1. The coefficient of 1/N ks has absolute value at most d/k d, so if we collect terms and write (log L)(s) as a Dirichlet series indexed by the ositive integers, say n 1 c n/n s, then c n = 0 if n is not N.

15 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 15 a rime ower and c n d[k : Q] if n is a rime ower. Therefore the coefficients of (log L)(s) as a Dirichlet series over Z + are bounded. Since L(s) is assumed to have an analytic continuation to a nonvanishing function on Re(s) 1, (log L)(s) has an analytic continuation to Re(s) 1, so Lemma A.2 imlies that (A.1) (log L)(s) = k α k,1 + + αk,d kn ks for Re(s) = 1, where the terms in the series are collected in order of increasing values of N( k ). Although a rearrangement of terms in a conditionally convergent series can change its value, one articular rearrangement of the series in (A.1) doesn t change the sum: (A.2) k α k,1 + + αk,d kn ks = k 1 α k,1 + + αk,d kn ks when Re(s) = 1, where the sum on the left is in order of increasing values of N( k ) and the outer sum on the right is in order of increasing values of N(). To rove (A.2), we rewrite it as (A.3) N( k ) x α k,1 + + αk,d kn ks = N() x k 1 α k,1 + + αk,d kn ks + o(1) as x, and we will rove (A.3) when Re(s) > 1/2, not just Re(s) = 1. For Re(s) = 1 we can ass to the limit in (A.3) as x and conclude (A.2). The sum on the right in (A.3) the sum on the left in (A.3) is equal to which is equal to N() x k 2 N() k >x α k,1 + + αk,d kn ks, (A.4) x<n() x k 2 α k,1 + + αk,d kn ks + N() x k 3 N() k >x α k,1 + + αk,d kn ks.

16 16 JIM BROWN AND KRZYSZTOF KLOSIN The absolute value of the first sum in (A.4) is bounded above by,1 k + + αk,d d kn ks kn kσ where σ = Re(s) x<n() x x<n() x k 2 N() x k 3 N() k >x < d 2 = d 2 < d 2 k 2 1 N kσ x<n() x k 2 x<n() x 1 N σ (N σ 1) x<n() x 4 N 2σ since N() σ > 2 > 4 3, which tends to 0 as x since 1/N2σ converges. The absolute value of the second sum in (A.4) is bounded above by d kn kσ < d log N (x)n kσ k 3 N() k >x N() x k 3 N() k >x = N() x Letting n be the least integer above log N (x), 1 N kσ = so N() x k 3 N() k >x d kn kσ < d log N() log x k 3 N() k >x 1/N()nσ 1 1/N() σ < 1/xσ 1/4 = 4 x σ, N() x 4d log N() x σ log x 1 N kσ. ( ) x = O x σ, log x which tends to 0 as x since σ > 1/2. Now that we established (A.2), take the exonential of the right side: if Re(s) = 1, then ex k 1 α k,1 + + αk,d kn ks = 1 (1 α,1 N s ) (1 α,d N s ), where the roducts run over in order of increasing norms and the last calculation is justified since α,j /N() s 1/N() < 1. Since L(s) = e (log L)(s) for Re(s) 1, by (A.1) and (A.2) we have L(s) = 1 (1 α,1 N s ) (1 α,d N s ) for Re(s) = 1, where the roduct is in order of increasing values of N.

17 ON THE NORMS OF -STABILIZED ELLIPTIC NEWFORMS 17 Examle A.3. Let L(s) be the L-function of the ellitic curve y 2 = x 3 x over Q. For Re(s) > 3/2 it has an Euler roduct over the odd rimes of the form (A.5) L(s) = 1 1 a s + 2s = 1 (1 α s )(1 β s ), 2 2 where α = and β = for 2. Since y 2 = x 3 x has CM by Z[i], L(s) is also the L-function of a Hecke character χ on Q(i) such that χ((α)) = α = N(α) 1/2 for all nonzero α in Z[i] with odd norm. Therefore L(s) also has an Euler roduct over the nonzero rime ideals of Z[i] of odd norm: for Re(s) > 3/2, (A.6) L(s) = 1 1 χ(π)/n(π) s. (π) (1+i) The function L(s) is entire and is nonvanishing on the line Re(s) = 3/2, so L(s + 1/2) fits the conditions of Theorem A.1 using K = Q and d = 2 for (A.5), and K = Q(i) and d = 1 for (A.6). Therefore (A.5) and (A.6) are both true on the line Re(s) = 3/2. For instance, L(3/2) , the artial Euler roduct for (A.5) at s = 3/2 over rime numbers u to 100,000 is , and the artial Euler roduct for (A.6) at s = 3/2 over nonzero rime ideals in Z[i] with norm u to 100,000 is References 1. J. Brown and K. Klosin. On the action of the U -oerator on Siegel modular forms. rerint. 2. D. Bum. Automorhic Forms and Reresentations, volume 55 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, H. Davenort. Multilicative Number Theory. Graduate Texts in Mathematics. Sringer-Verlag, New York, 3rd edition, S. Gelbart. Automorhic forms on adele grous. Number 83 in Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, S. Gelbart and H. Jacquet. A relation between automorhic reresentations of GL(2) and GL(3). Ann. Sci. École Norm. Su. (4), 11(4): , H. Hida. Congruences of cus forms and secial values of their zeta functions. Invent. Math., 63: , H. Jacquet and J. A. Shalika. A non-vanishing theorem for zeta functions of GL n. Invent. Math., 38(1):1 16, 1976/ W. Kohnen and J. Senguta. Nonvanishing of symmetric square L-functions of cus forms inside the critical stri. Proc. Amer. Math. Soc., 128(6): , E. Landau. Handbuch der Lehre von der Verteilung der Primzahlen. B. G. Teubner, Leizig, T. Miyake. Modular forms. Sringer-Verlag, Berlin, Translated from the Jaanese by Y. Maeda. 11. H. L. Montgomery and R. C. Vaughan. Multilicative number theory. I. Classical theory, volume 97 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, D. J. Newman. Simle analytic roof of the rime number theorem. Amer. Math. Monthly, 87(9): , 1980.

18 18 JIM BROWN AND KRZYSZTOF KLOSIN 13. G. Shimura. On the holomorhy of certain Dirichlet series. Proc. London Math. Soc., 31(1):79 98, G. Shimura. On the eriods of modular forms. Math. Ann., 229: , A. Wiles. On ordinary λ-adic reresentations associated to modular forms. Invent. Math., 94: , Deartment of Mathematical Sciences, Clemson University, Clemson, SC address: jimlb@clemson.edu 2 Deartment of Mathematics, Queens College, City University of New York, Flushing, NY address: kklosin@qc.cuny.edu 3 Deartment of Mathematics, University of Connecticut, Storrs, CT address: kconrad@math.uconn.edu