manuscripta math. 104, 173 – 200 (2001) © Springer-Verlag 2001 Mahdi Asgari · Ralf Schmidt Siegel modular forms and representations Received: 28 March 2000 / Revised version: 25 October 2000 Abstract. This paper explicitly describes the procedure of associating an automorphic rep- resentation of PGSp(2n, A) with a Siegel modular form of degree n for the full modular group n = Sp(2n, Z), generalizing the well-known procedure for n = 1. This will show that the so-called “standard” and “spinor” L-functions associated with such forms are ob- tained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F4, is then used to establish meromorphic continuation for the spinor L-function when n = 3. 1. Introduction Let f be a Siegel modular form of degree n for the full modular group n = Sp(2n, Z).Iff is an eigenfunction for the action of the Hecke algebra, then there are two L-functions associated with f . Let a0,a1,... ,an be the Satake parameters of f , and define the standard L-function n −1 −s −s −1 −s L1(s, f ) = (1 − p ) (1 − aip )(1 − ai p ) , (1) p i=1 and the spinor L-function n −1 L (s, f ) = ( − a a ...a p−s) . 2 1 0 i1 ik (2) p k=0 1≤i1<...<ik≤n One goal of this note is to “explain” the definition of these L-functions within the general framework of automorphic representations. To do so, we first associate an automorphic representation with the classical modular form f . We then identify the above L-functions with certain Langlands L-functions coming from two different representations of the dual group. Langlands’ theory of Euler products will then imply the following (cf. Sect. 4.6). M. Asgari: Department of Mathematics, The University of Michigan, Ann Arbor, MI 48109- 1109, USA. e-mail: [email protected] R. Schmidt: Universität des Saarlandes, Fachrichtung 6.1 Mathematik, Postfach 15 11 50, 66041 Saarbrücken, Germany. e-mail: [email protected] Mathematics Subject Classification (2000): 11F46 174 M. Asgari, R. Schmidt Theorem. The L-functions L1(s, f ) and L2(s, f ) have meromorphic continuation to all of C when n = 3. We should remark that Böcherer has proved stronger results for the standard L-functions (cf. [Bö]). Andrianov also gives all the analytic properties of spinor L-function when n = 2 in [An]. The procedure of associating an automorphic representation to a classical mod- ular form is well known in the case of elliptic modular forms, see [Ge] Chapter 3 or [Bu, Sect. 3.6]. There, one associates with an eigenform f an automorphic representation πf of GL(2, A), where A denotes the adeles of Q.Iff has no char- acter, then πf will have trivial central character, thus descends to a representation of PGL(2, A). In the higher dimensional case we will associate with an eigenform f of degree n an automorphic representation of the group GSp(2n, A) which is isomorphic to GL(2, A) if n = 1. Since f will be assumed to have no character, this is really a representation of PGSp(2n, A). To be more precise, utilizing a strong approximation theorem, we first lift f to a function f on G(A), where G = GSp(2n). We will assume f to be cuspidal, L2(G(Q)\G(A)). so that f lies in the cuspidal subspace 0 (Here we have to trans- late the classical cusp condition into the group theoretic one.) Then let π be the subrepresentation of this space generated by f . This π may not be irreducible, but if f is an eigenform, then all the irreducible components of π will turn out to be isomorphic. This isomorphism class is the automorphic representation πf associated with f . In Theorem 2 we shall describe its local components in terms of Satake parameters (at the finite places) and Harish-Chandra parameters (at the infinite place). We have to go through some Hecke algebra computations to identify the clas- sical Satake parameters of the eigenform f with the Satake parameters of the local components of πf which are spherical representations of the local groups GSp(2n, Qp). The group theoretic Satake parameters can be taken to be in the maximal torus in the dual group of G which is Gˆ = GSpin(2n + 1, C). Since we have a representation with trivial central character, the Satake parameters will in fact be elements of its derived group Spin(2n + 1, C). This latter group has two distinguished finite-dimensional representations, n namely the projection onto SO(2n + 1, C) which we denote by 1, and the 2 - dimensional spin representation 2, the smallest genuine representation (not factor- ing through 1) of this group.We use the weight structure of these finite-dimensional representations to determine the form of the standard Euler factor associated with 1 and 2. It turns out that these Euler factors are precisely the same as the ones in (1) and (2). In other words, the classical L-functions identify with standard Lang- lands L-functions coming from the representations 1 and 2 of the dual group (see Theorem 3). Having established this identification, one can apply results from representation theory to classical modular forms. For modular forms of degree 3, the underlying group GSp(6) can be embedded, as a Levi, in a Chevalley group of type F4. We then make use of Langlands’ method developed in [La] to prove the meromorphic continuation of the spin L-function for n = 3 (cf. Corollary 1). Siegel modular forms and representations 175 2. Notation The right group for handling Siegel modular forms (of degree n) in the context of automorphic representations is t G = GSp(2n) ={g ∈ GL(2n) :∃µ(g) ∈ GL(1)gJg = µ(g)J }, where 1n J = , 1n the n × n identity matrix. −1n AB g = If CD , then the conditions are equivalent to t t t t t t A D − B C = µ(g)1,AB = B A, C D = D C. The function µ is called the multiplier homomorphism. Its kernel is the group Sp(2n) and there is an exact sequence 1 −→ Sp(2n) −→ G −→ GL(1) −→ 1. The center Z of G consists of the scalar matrices, and the standard maximal torus is T ={diag(u1,... ,un,v1,... ,vn) : u1v1 = ...= unvn = 0}. We often write an element t ∈ T in the form t = (u ,... ,u ,u−1u ,... ,u−1u ), u ∈ ( ); diag 1 n 1 0 n 0 i GL 1 (3) then u0 = µ(t). We fix the following characters of the maximal torus T ⊂ G.If t ∈ T is written in the form (3), then let ei(t) = ui,i= 0, 1,... ,n. These characters are a basis for the character lattice of G, X = Ze0 ⊕ Ze1 ⊕ ...⊕ Zen. We also fix the following cocharacters of T : f (u) = ( ,... , ,u,...,u), 0 diag 1 1 n n f (u) = (u, ,... , ,u−1, ,... , ), 1 diag 1 1 1 1 n n . f (u) = ( ,... , ,u, ,... , ,u−1). n diag 1 1 1 1 n n 176 M. Asgari, R. Schmidt Then these elements are a Z-basis for the cocharacter lattice of G, ∨ X = Zf0 ⊕ Zf1 ⊕ ...⊕ Zfn. With the natural pairing , : X × X∨ −→ Z, we have ei,fj =δij . We now choose the following set of simple roots: α (t) = u−1 u , ... ,α (t) = u−1u ,a(t) = u2u−1; 1 n−1 n n−1 1 2 n 1 0 (4) here t is written in the form (3). In other words, α1 = en − en−1, ... , αn−1 = e2 − e1,αn = 2e1 − e0. The numbering of the simple roots is such that the Dynkin diagram is ••··· •••< α1 α2 αn−2 αn−1 αn The corresponding coroots are α∨ = f − f , ... α∨ = f − f ,α∨ = f . 1 n n−1 n−1 2 1 n 1 (5) R ={α ,... ,α }⊂X R∨ ={α∨,... ,α∨}⊂X∨ If we let 1 n , 1 n , then (X,R,X∨,R∨) is the root datum of G = GSp(2n). The Cartan matrix is 2 −1 −12−1 −12−1 α ,α∨= .. .. .. . i j . −12−1 −12−1 −22 With our choice of simple roots, we get the Borel subgroup B = TN, where A 0 1 B N consists of matrices of the form , with A ∈ GL(n) lower 0 tA−1 0 1 triangular unipotent and B symmetric. (Note that some authors use a different set of simple roots that results in A being upper triangular as was pointed out by the referee.) The torus T acts on the Lie-algebra n of N by the adjoint representation Ad. It is easy to compute the modular factor δB (t) = det(Adn(t)).Ift is given in the form (3), then δ (t) = u−n(n+1)/2u2u4 · ...· u2n. B 0 1 2 n (6) Siegel modular forms and representations 177 3. Local representations In Theorem 2 we shall describe the local components of an automorphic repre- sentation associated with a classical Siegel modular form. This is done in terms of Satake parameters at the finite places, and in terms of the weight (or Harish- Chandra parameters, if the weight is large enough) at the archimedean place. In the following sections we shall therefore collect all the required facts about local representations. Notation. In Sects. 3.1 through 3.4, F is a p-adic field, O its ring of integers, ω ∈ O a generator of the maximal ideal, and q = *O/ωO.