On the archimedean Euler factors for spin L functions

Transcription

1 On the archimedean Euler factors for spin L functions Ralf Schmidt Abstract. The archimedean Euler factor in the completed spin L function of a Siegel modular form is computed. Several formulas are obtained, relating this factor to the recursively defined factors of Andrianov and to symmetric power L factors for GL. The archimedean ε factor is also computed. Finally, the critical points of certain motives in the sense of Deligne are determined. Introduction Let f be a classical holomorphic Siegel modular form of weight k and degree n, assumed to be an eigenform of the Hecke algebra. Andrianov [An] has associated to f an L function Z f s as an Euler product over all finite primes, where the Euler factor at p is a polynomial in p s of degree n. This is called the spin L function of f since it is attached to the n dimensional spin representation of the L group Spinn + 1, C of the underlying group PGSpn, see [AS]. A serious problem is that of obtaining the analytic continuation and a functional equation for Z f s. The case n = 1 is the classical Hecke theory, the case n = was done by Andrianov in [An]. Beyond that, very little is presently known. To obtain smooth functional equations, the partial L function Z f s has to be completed with an Euler factor at the archimedean prime. For example, for n = 1, the function Ls, f = π s ΓsZ f s has the functional equation Ls, f = 1 k/ Lk s, f. For n =, the definition Ls, f = π s ΓsΓs k + Z f s leads to the functional equation Ls, f = 1 k Lk s proved in [An]. The purpose of this paper is to give formulas for the archimedean Euler factor in any degree. Automorphic representation theory provides the recipe to compute this factor. Let Π k be the archimedean component of the automorphic representation of PGSpn, A attached to the eigenform f see [AS]. By the local Langlands correspondence over R, there is an associated local parameter ϕ : W R Spinn + 1, C, where W R is the real Weil group. Let ρ be the spin representation. Then the factor we are looking for is Ls, Π k, ρ = Ls, ρ ϕ, where on the right we have the L factor attached to a finite-dimensional representation of the Weil group. The formulas we will thus obtain coincide for n = 1 and n = with the Γ factors from above, except for some constants. Note however that we are working with the automorphic normalization that is designed to yield a functional equation relating s and 1 s. To compare with the classical formulas, we have to make a shift in the argument s. At the end of the paper [An] Andrianov gives another definition of an archimedean Euler factor by a recursion formula. It turns out that for n 3 this definition leads to a factor that is different from ours.

2 However, the difference is not too serious. One factor can be obtained from the other by multiplying with a quotient of reflection polynomials, where we call a polynomial δ a reflection polynomial if its zeros are integers or half-integers and if it has a functional equation δs = ±δ1 s. Therefore, for questions of meromorphic continuation and functional equation, either factor is suitable. The difference becomes however very relevant if one is interested in controlling poles in a global L function. If the degree n is even we shall establish a formula relating Ls, Π k, ρ to symmetric power L factors for GL. To be precise, we shall prove for n = n that Ls, Π k+n, ρ = δs L s r/, Dk 1, Sym n j βr,j,n, 1 n r Z where βr, j, n are certain combinatorially defined exponents, Dk 1 is the discrete series representation of PGL, R of lowest weight k, and δs is a reflection polynomial. The existence of such a formula is already a hint that there is a lifting from elliptic modular forms of weight k to Siegel modular forms of weight k + n and degree n. This is indeed the case, as proved by Ikeda [Ik]. The above formula is needed to express the completed spin L function of an Ikeda lift in terms of symmetric power L functions, see [Sch]. As part of a global L function, the reflection polynomial δs in formula 1 should make a contribution of δ = δs/δ1 s to the global ε factor. This is indeed the case; we shall prove the analogous formula εs, Π k+n, ρ, ψ = δ n r Z ε s r/, Dk 1, Sym n j, ψ βr,j,n. We shall also compute the ε factor in odd degree. In each case except n = 1 it is just a sign. It is important to know this sign because it appears in the global functional equation. Deligne has defined critical points of motives and made conjectures about the values of the L function of the motive at these points. The critical points do only depend on the archimedean Euler factor. Having these factors for spin L functions explicitly at hand, we will compute the critical points of motives corresponding to Siegel modular forms in the final part of this paper. The first section of this paper contains background material on Weil group representations and their L factors. In the second section we shall compute the archimedean L factors for symmetric powers on GL, both for later use and as an illustration of our method. The next section contains the basic formula for Ls, Π k, ρ. We shall continue by establishing a recursion formula that will enable us to compare our factors with Andrianov s. This is followed by a section in which the formula 1 is established. In the next section we use similar methods to compute the archimedean ε factor. The final section is devoted to critical points. I would like to thank R. Schulze Pillot for various helpful discussions on the topics of this paper.

3 3 1 Preparations 1.1 Representations of the Weil group We recall some basic facts about representations of the real Weil group. References are [Ta] and [Kn]. The real Weil group W R is a semidirect product W R = C j, where j is an element with j = 1 which acts on C by jzj 1 = z. A representation of W R in some finite-dimensional complex vector space is called semisimple if its image consists entirely of semisimple elements. Every such representation is fully reducible. An irreducible semisimple representation is either one- or two-dimensional. Here is a complete list. One-dimensional representations: τ +,t : z z t, j 1, 3 τ,t : z z t, j 1. 4 Here we have t C and is the usual absolute value on C not its square. Two-dimensional representations: τ l,t : re iθ r t e ilθ r t e ilθ, j 1 l 1. 5 Here the parameters are a positive integer l and some t C. An L factor is attached to a semisimple representation of W R. For the irreducible representations the L factors are given as follows see [Kn] 3.6. s + t Ls, τ +,t = π s+t/ Γ, 6 s + t + 1 Ls, τ,t = π s+t+1/ Γ, 7 Ls, τ l,t = π s+t+l/ Γ s + t + l. 8 For an arbitrary semisimple representation the associated L factor is the product of the L factors of its irreducible components. The local Langlands correspondence is a parametrization of the infinitesimal equivalence classes of irreducible admissible representations of a real reductive group G = GR by admissible homomorphisms W R L G into the L group of G. If G is split over R, then L G may be replaced by its identity component, which is a complex Lie group. If π is an irreducible, admissible representation of G and ρ is a finite-dimensional representation of L G, then an L factor Ls, π, ρ is defined as follows. Let ϕ : W R L G be the local parameter attached to π. Define a semisimple representation of W R by τ := ρ ϕ. Then the L factor associated to π and ρ is Ls, π, ρ := Ls, τ.

4 4 1 PREPARATIONS In this paper we shall be concerned with the following situation. G is the group PGSpn, R rank n and π is a certain holomorphic discrete series representation corresponding to a Siegel modular form of weight k. The identity component of the L group is Spinn + 1. As ρ we take the smallest genuine representation of this group, the n dimensional spin representation. The resulting factor Ls, π, ρ is of interest because it is the correct archimedean Euler factor of the spin L function of a Siegel modular form, see [AS]. There are also ε factors attached to semisimple representations of W R. In the archimedean case they do not depend on the complex variable s. They do however depend on the choice of an additive character ψ of R. Throughout we will fix the character ψx = e πix. Then the ε factors are given on irreducible representations as follows. εs, τ +,t, ψ = 1, εs, τ,t, ψ = i, εs, τ l,t, ψ = i l The ε factor of an arbitrary semisimple representation is again the product of the factors of the irreducible components. Via the local Langlands correspondence we have ε factors attached to irreducible, admissible representations of a reductive group G. 1. Archimedean factors for symmetric power L functions Let l be a positive integer, and let Dl be the discrete series representation of SL ±, R = {g GL, R : detg {±1}} with a lowest weight vector of weight k = l + 1 and a highest weight vector of weight l 1. For t C we consider the representation Dl, t = Dl det t of GL, R which coincides with Dl on SL ±, R and with a t on matrices a a, a > 0. The connected component of the L group of GL, R is GL, C, and, by [Kn], the local parameter W R GL, C attached to Dl, t is the representation τ l,t defined in 5. For a positive integer n let Sym n 1 be the n dimensional irreducible representation of GL, C on the space of homogeneous polynomials in two variables of degree n 1 given by Sym n 1 gp x, y = P x, yg, g GL, C. The L functions Ls, π, Sym n 1, for a global representation π of GL, A, are called symmetric power L functions, cf. [Sh]. It is easy to compute their archimedean Euler factors, and we shall do so for the representations Dl, t of GL, R Lemma. The archimedean symmetric power L factors for the representation Dl, t of GL, R are given as follows. i If n is even, then Ls, Dl, t, Sym n 1 = n/ π sn/ π tnn 1/ ln /8

5 1. Archimedean factors for symmetric power L functions 5 ii If n is odd, then where n/ j=1 Γ s + tn ln + 1 lj. Ls, Dl, t, Sym n 1 = n 1/ π sn 1/ π tn 1 / ln 1/8 π s+ε/ tn 1 n 1/ Γ s + tn s + ε ln + 1 lj Γ + tn 1, ε = { 0 if n 1 mod 4, 1 if n 3 mod 4. j=1 Proof: We have to decompose the composition τ : W R GL, C GLn, C into irreducibles, where the first map is τ l,t and the second one is Sym n 1. If we take the natural basis x n 1 j y j, j {0,..., n 1}, for the space of Sym n 1, then this representation is explicitly given by re iθ r tn 1 j 1 1 e iln 1θ e iln 3θ.... e iln 3θ e iln 1θ From this the irreducible components are obvious. If n is even, we have n/ τ = τ ln+1 j,tn 1, j=1 with τ l,t as in 5, and if n is odd, then τ = n 1/ j=1 τ ln+1 j,tn 1 τ,tn 1, where means + if n 1 mod 4 and if n 3 mod 4 see 3 and 4. Using the definitions 6 to 8, the assertion is now easily obtained.,

6 6 THE EULER FACTOR FOR THE SPIN REPRESENTATION For later use we shall rewrite the formulas in the lemma slightly, assuming that t = 0. If n is odd, then Ls, Dl, Sym n = n 1 step Ls, τ ln j,0 = π s n+1/ π ln+1 /8 n 1 step Γ s + 1 ln 1 lj. 1 If n is even, then Ls, Dl, Sym n = n step Ls, τ ln j,0 Ls, τ,0 = π s n n/ π lnn+/8 π s+ε/ step Γ s + 1 ln 1 s + ε lj Γ 13 where ε = { 0 if n 0 mod 4, 1 if n mod 4, = { + if n 0 mod 4, if n mod 4. The Euler factor for the spin representation.1 Basic computation Let X resp. P, Q denote the character lattice resp. weight lattice, root lattice and X resp. P, Q the cocharacter lattice resp. coweight lattice, coroot lattice of the group PGSpn. We can choose a basis e 1,..., e n of X Z Q and a dual basis f 1,..., f n of X Z Q such that these lattices are given as follows: P = e 1,..., e n P = { c i f i : c i Z i or c i Z + 1 i} X = { c i e i : c i Z} X Q = e 1 e,..., e n 1 e n, e n Q = f 1 f,..., f n 1 f n, f n The root datum of Spinn + 1 is dual to this one, meaning that characters cocharacters become cocharacters characters. Thus the e 1,..., e n identify with cocharacters of Spinn+1 and f 1,..., f n identify with characters of Spinn + 1. See [Sp] for basic facts about root data. Let us choose the standard set e 1 e,..., e n 1 e n, e n of simple roots of PGSpn. The element ν = k 1e k ne n P

7 .1 Basic computation 7 then lies in the fundamental Weyl chamber provided k > n. It is the Harish-Chandra parameter for a certain discrete series representation Π k of PGSpn, R. This representation has a scalar minimal K type with highest weight ke ke n. It appears as the archimedean component of automorphic representations of PGSpn, A Q attached to classical Siegel modular forms of weight k, see [AS]. If k = n then we get a representation Π k which is a limit of discrete series..1.1 Proposition. Let n and k be positive integers with k n, and let Π k be the limit of discrete series representation of PGSpn, R considered above. Let ρ be the n dimensional spin representation of the dual group Spinn + 1, C. Then Ls, Π k, ρ = εi>0 εi=0, iε i>0 π s π k ε i iε i / Γ s + k ε i iε i π s π iε i/ Γ s + 1 N/. iεi π Γs s 14 Here N = #{ε 1,..., ε n {±1} n : ommitted for odd n. εi = iε i = 0}. The second line of the formula can be Proof: As before let e 1,..., e n span the character lattice and f 1,..., f n span the cocharacter lattice of PGSpn, in a way such that e i f i z = z and e i f j z = 1 for i j. The e 1,..., e n identify with cocharacters of Spinn + 1, and the f 1,..., f n identify with characters. Let ν = k 1e k ne n be the Harish-Chandra parameter of our representation Π k. The local parameter ϕ : W R Spinn+ 1, C attached to Π k is then given by ϕz = z ν z ν z C, ϕj = w, where w is a representative for the longest Weyl group element sending e i to e i for each i, see [Bo] We have to consider the decomposition of τ := ρ ϕ into irreducibles. The weights of the spin representation ρ are well known; they are ε 1 f ε n f n, ε i {±1}. Each weight space is one-dimensional, so that ρ is a n dimensional representation. Let v ε1,...,ε n vectors spanning the one-dimensional weight spaces. Then, writing z = re iθ, be τzv ε1,...,ε n = ρz ν z ν v ε1,...,ε n = ρe iθ ν v ε1,...,ε n = ε 1 f ε n f n e iθ ν v ε1,...,ε n = e iε1k εnk nθ v ε1,...,ε n = e ik ε i iε iθ v ε1,...,ε n. 15

8 8 THE EULER FACTOR FOR THE SPIN REPRESENTATION Since w represents the longest Weyl group element, we have that τjv ε1,...,ε n is a multiple of v ε1,..., ε n It follows that the two-dimensional spaces v ε1,...,ε n v ε1,..., ε n are invariant for the action of W R. Let τ ε1,...,ε n be the representation on this two-dimensional space. If l := k ε i iε i 0, then obviously τ ε1,...,ε n = τ l,0 where τ l,t was defined in 5. The associated L factor is Ls, τ ε1,...,ε n = π s+ l / Γs + l /, l = k ε i iε i. 16 On the other hand, if l = 0, then τ ε1,...,ε n = τ +,0 τ,0, with associated L factor s Ls, τ ε1,...,ε n = π s/ Γ π s+1/ Γ s + 1 By Legendre s formula for the Γ function, this is the same factor as if we put l = 0 in 16. The formula 14 is obtained by taking the product of all these factors over a certain system of representatives of {±1} n / 1. Examples: For n = 1 we obtain Ls, Π k, ρ = π s+k 1/ Γ s + k 1. This is the archimedean Euler factor in the Jacquet Langlands L function for an automorphic form on PGL, A Q of weight k. Up to a factor and a shift by k 1 it is also the classical Γ factor attached to an elliptic modular form of weight k. For n = and k our formula yields Ls, Π k, ρ = 4π s π 1 k Γ s + k 3 Γ s Replacing s with s k + 3 we get 4π k π s ΓsΓs k +. Up to the constant 4π k this is the factor used by Andrianov in [An] to complete the partial L function for classical Siegel modular forms of degree. The absolute values in formula 14 are inconvenient for computations, and we would like to remove them. Let us define Ls, Π k, ρ by the same formula, but without the absolute values, i.e., Ls, Π k, ρ = π s π iε i k ε i/ Γ s + k ε i iε i εi>0 εi=0, iε i>0 π s π iε i/ Γ s + 1 N/. iεi π Γs s 17

9 .1 Basic computation 9 Then Ls, Π k, ρ = Ls, Π k, ρ n εi>0 π k ε i iε i / Γ s + k ε i iε i π k ε i iε i/ Γ s + k ε. i iε i If we put δ m s := π m Γ s + m Γ s m 18 then we can write Ls, Π k, ρ = δs Ls, Π k, ρ with δs = εi>0 k ε i iε i<0 δ iε i k ε i s. 19 Note that for non-negative integers m the function δ m is a polynomial: using repeatedly the functional equation sγs = Γs + 1 we get δ m s = m j=1 s + m j. It is immediate from this description that δ m s = 1 m δ m 1 s. Let us call a polynomial p a reflection polynomial if ps = ±p1 s for some sign ±, and if all the zeros of p are either integers or half-integers. Then all the δ m, and therefore the function δ also, are reflection polynomials..1. Proposition. With notations as in Proposition.1.1 we have Ls, Π k, ρ = δs Ls, Π k, ρ, where Ls, Π k, ρ is defined in 17 and where δs, defined in 19, is a reflection polynomial depending on n and k. If k is large enough, specifically, k then δs = n + 1 for n odd, 1 8 n + 4n for n even, 0

10 10 THE EULER FACTOR FOR THE SPIN REPRESENTATION Proof: Only the last assertion remains to be proved. By definition, δs = 1 if k ε i iε i 0 for each choice of ε i {±1} such that ε i > 0. Let j be the number of indices i such that ε i = 1. Then 0 j n 1 n odd resp. 0 j n 1 n even and ε i = n j. Since n iε i i=1 we have j i + i=1 n i=j+1 i = 1 n + n j j, k ε i iε i kn j 1 n + n + j + j. Thus we get k ε i iε i 0 if k n + n n j j + j n j = 1 n + j j n j. The expression on the right grows with j, so we get an upper bound for it if we put j = n 1 n odd resp. j = n 1 n even. The upper bound is the one given on the right hand side of 0. Remark: Suppose our local factor Ls, Π k, ρ appears as the archimedean factor in some global L function Ls, Π, ρ. Then, since δs is a reflection polynomial, one can replace Ls, Π k, ρ by Ls, Π k, ρ if one is only concerned with questions of meromorphic continuation and functional equation. Of course, to know the precise sign in the functional equation, or to control poles, one has to take the correct Ls, Π k, ρ.. Recursion formulas We are now seeking a relation between the factors Ls, Π k, ρ in degree n and degree n + 1 and shall therefore more precisely write Ls, Π k,n, ρ n for the factor in degree n...1 Proposition. We have the following recursion formulas for the factors Ls, Π k, ρ defined in 17. Let s 0 = k n 1/. i If n is even, then Ls, Π k,n+1, ρ n+1 = Ls + s 0, Π k,n, ρ n Ls s 0, Π k,n, ρ n δ even s, 1 where δ even s is a quotient of reflection polynomials given by δ even s = δ s0 iε i s δ s0 s N/. εi=0, iε i>0 Here N = #{ε 1,..., ε n {±1} n : ε i = iε i = 0} and δ m s is defined in 18. ii If n is odd, then Ls, Π k,n+1, ρ n+1 = Ls + s 0, Π k,n, ρ n Ls s 0, Π k,n, ρ n δ odd s,

11 . Recursion formulas 11 where δ odd s is a quotient of reflection polynomials given by δ odd s = δ iε i n 1s. εi=1, iε i>n+1 Proof: We shall prove the first statement; the manipulations for the second one are similar. The summations in the following formulas are from 1 to n, except in the first line, where they are from 1 to n + 1. Ls, Π k,n+1, ρ n+1 = = = εi+1>0 εi 1>0 εi>0 εi=0 εi>0 ε 1,...,ε n+1 {±1} εi>0 π s π iε i k ε i/ Γ π s π iε i+n+1 k ε i+1/ Γ π s π iε i n+1 k ε i 1/ Γ π s+s0 π iε i k ε i/ Γ π s+s0 π iε i/ Γ s + s 0 π s s0 π iε i k ε i/ Γ s + k ε i iε i s + k ε i + 1 iε i n + 1 s + k ε i 1 iε i + n + 1 s + s 0 + k ε i iε i iεi s s 0 + k ε i iε i For the last line note that ε i > 1 is equivalent to ε i > 0 since n is even. Now we are going to write the product 4 in the form 4 = π s+s0 π iε i/ iεi Γ s + s εi=0, iε i>0 εi=0, iε i>0 εi=0, iε i>0 π s s0 π iε i/ Γ s s 0 + iεi 7 π s+s0 π iε i/ Γ s + s 0 iεi π s s0 π 8 iε i/ Γ s s 0 + iεi N/ π s+s0 Γs + s 0 9

12 1 THE EULER FACTOR FOR THE SPIN REPRESENTATION N/ π s s0 Γs s 0 30 π s+s 0 N/ Γs + s π s s0 Γs s 0 The terms 3, 6 and 9 combine to Ls + s 0, Π k,n, ρ n. The terms 5, 7 and 30 combine to Ls s 0, Π k,n, ρ n. The terms 8 and 31 combine to the factor δ even s. This factor is indeed a quotient of reflection polynomials since s 0 1 Z we cannot conclude that it is itself a reflection polynomial since it may happen that s 0 iε i < 0. At the end of the paper [An] Andrianov defines an archimedean Euler factor for the spin L function as A n,k s := π n 1s γ n,k s 3 actually the exponent in Andrianov s paper is n s, but this is an obvious misprint, where the γ n,k are defined recursively by γ 1,k s = Γs and γ n+1,k s = γ n,k s γ n,k s k + n + 1, n This Euler factor is designed to fit into a global L function that conjecturally has analytic continuation and a functional equation with center point s = nk nn To compare A n,k s with our factor Ls, Π k,n, ρ n we shall make a shift in the argument to make 1/ the center point, and consider à n,k s := A n,k s + nk nn + 1, γ n,k s := γ n,k s + nk 4 nn We see that both Ãn,ks and Ls, Π k,n, ρ n contain a factor π n 1s. Disregarding constant factors, what we really have to compare is γ n,k s and the Γ functions in our formula 14. It follows from 33 that γ n+1,k s = γ n,k s + s 0 γ n,k s s 0, s 0 = k n 1, 35 a recursion formula very similar to 1 and, except that the δ factors are missing. Thus we can say that, except for n = 1 and n =, Andrianov s factor A n,k s is not the standard automorphic archimedean spin Euler factor, but differs from Ls, Π k,n, ρ n only by a quotient of reflection polynomials. As mentioned before, this does not make a big difference for questions of meromorphic continuation and functional equation..3 Connection with symmetric power L functions In this section we shall show that if the degree n is even, then the L factor Ls, Π k, ρ can be expressed as a reflection polynomial times a product of symmetric power L factors for GL. This

13 .3 Connection with symmetric power L functions 13 was in principle already done in [Sch] and used to express the spin L functions of the lifts constructed by Ikeda in [Ik] in terms of symmetric power L functions for GL. Here we will present a more direct computation, but at one point will also use a result of [Sch]. Throughout we will assume that n is even and write n = n. For an integer l let us abbreviate fl := π s+l/ Γ s + l. Our L factor from Proposition.1.1 may then be written as Ls, Π k, ρ = f k ε i iε i. n / 1 We shall work with the modified factor Ls, Π k, ρ = f k ε i iε i εi>0 εi=0, iε i>0 f iεi f0 N/ 36 defined in 17, which differs from Ls, Π k, ρ at most by a reflection polynomial. For ε 1,..., ε n {±1} let j be the number of negative ε i s. Then ε i = n j, and the condition ε i > 0 allows j to run between 0 and n 1. With { αr, j, n := # ε 1,..., ε n {±1} n : ε i = n j, } iε i = r we may then write Ls, Π k, ρ = n 1 r Z f kn j r αr,j,n r>0 fr αr,n,n f0 α0,n,n/. 37 Actually our interest is in Ls, Π k+n, ρ. The numbers αr, j, n can be described as follows. Let αk, j, n = number of possibilities to choose j numbers from the set 38 {1 n, 3 n,..., n 1} such that their sum equals k. This is the dimension of the weight k space of the representation j V n of SL, C, where V n is the n dimensional irreducible representation of this group n = n. It is an easy exercise to show that αr, j, n = α r + n + 1j n, j, n. We therefore have Ls, Π k+n, ρ = n 1 r Z f k 1n j r αr,j,n Note that for a function of integers gj and integers α j we have m gj αj = m m j 0=0 j=j 0 j j 0 mod r>0 fr αr,n,n f0 α0,n,n /. 39 gj αj 0 αj 0, 40

14 14 THE EULER FACTOR FOR THE SPIN REPRESENTATION provided α = α 1 = 0. Applying this to the above equation we get where n 1 ε j 0 = f k 1n j r n αr,j,n 1 = { 0 if n 1 j 0 mod, 1 if n j 0 mod. j 0=0 n 1 ε j 0 j=j 0 step f k 1n j r βr,j 0,n, 41 and where βr, j, n = αr, j, n αr, j, n. The following properties of the numbers α and β are all easy to see..3.1 Lemma. Let r, j, n be integers with 0 j n. i αr, j, n = 0 unless jj n r jn j and r j mod. Similar for βr, j, n. ii αr, j, n = α r, j, n. Similar for βr, j, n. iii jn j r=jj n step αr, j, n n = j, jn j r=jj n step βr, j, n n = j n j. iv n j n mod βr, j, n = αr, n, n. If we consider a fixed j 0 and put m = n j 0, then n 1 ε j 0 j=j 0 step f k 1n j r = = m 1 ε j 0 step m 1 ε j 0 step f k 1n j 0 j r 4 L s r, τ k 1m j,0. 43 If we assume ε j 0 = 0, or equivalently m odd, then this expression is precisely Ls r/, Dk 1, Sym m, see 1. This is not quite the case for ε j 0 = 1, or m even. In this case we have see 13 n 1 ε j 0 j=j 0 step f k 1n j r = L s r, Dk 1, Symm L s r 1, τ j 0,0

15 .3 Connection with symmetric power L functions 15 where j 0 = { + if n j 0 0 mod 4, if n j 0 mod 4. Putting everything together we obtain with n 1 Ls, Π k+n, ρ = δs = n r Z ε j=1 j 0=0 r Z n 1 j 0=0 r Z ε j=1 L s r/, Dk 1, Sym n j 0 βr,j0,n L s r, τ j 0,0 βr,j0,n r>0 = δs L s r/, Dk 1, Sym n j βr,j,n n r Z L s r, τ j,0 βr,j,n Note that by Legendre s formula Γs/Γs + 1/ = 1 s π 1/ Γs we have fr = L s + r, τ +,0 L s + r, τ,0 for each r Z. Applying Lemma.3.1 iv we therefore have δs = = where we put and n r>0 ε j=1 n ε j=1 n r>0 ε j=1 r>0 Ls + r/, τ+,0 Ls + r/, τ,0 βr,j,n Ls r/, τ j,0 Ls + r/, τ j,0 L β0,j,n s, τ α0,n,n / j,0 Ls, τ +,0 Ls, τ,0 δ r, j s βr,j,n δ 0,+ s K, δ r, s = Ls + r/, τ +,0Ls + r/, τ,0 Ls r/, τ,0 Ls + r/, τ,0 K = 1 n ε j=1 fr αr,n,n f0 α0,n,n / 44 fr αr,n,n f0 α0,n,n / n j/ β0, j, n. 46

16 16 THE EULER FACTOR FOR THE SPIN REPRESENTATION First assume that n is odd. Then K = 0 by Lemma.3.1 i, and our formula reads δs = n j=1 step r odd δ r, j s βr,j,n. 47 It is easy to see that for odd r the factors δ r, are just reflection polynomials. Consequently δs is itself a reflection polynomial. Now assume that n is even, so that δs = n r even step δ r, j s βr,j,n δ 0,+ s K. 48 In this case no δ r, is a polynomial. Nevertheless, it is still true that δs is a reflection polynomial. This was proved in [Sch] using some finite-dimensional representation theory, and we are not going to reprove it here in this paper we wrote δ r,0 for δ r,+ and δ r,1 for δ r,. In [Sch] there is also more precise information on the location and order of zeros of the polynomial δs. To summarize:.3. Theorem. The archimedean spin Euler factor in even degree n = n can be expressed as Ls, Π k+n, ρ = δs L s r/, Dk 1, Sym n j βr,j,n n r Z with a reflection polynomial δs. Remark: The formula in this theorem is important for the following reason. Ikeda constructed a lifting map from elliptic cusp forms of weight k to Siegel modular forms of degree n = n and weight k + n, see [Ik]. A group theoretic interpretation of this lifting was given in [Sch]. If π is a holomorphic cuspform of weight k on PGL, and Π is its Ikeda lift to PGSpn, then the spin L function of Π is expressed through symmetric power L functions of π. The Euler factor at a real place is handled by Proposition.3., while at unramified places a similar formula holds, but without the δs. In this way one reduces the analytic properties of the spin L functions of the lifts to those of symmetric power L functions for GL..4 The ε factor We shall fix the additive character ψx = e πix of R and give some information on the spin ε factor εs, Π k, ρ, ψ. The basic computation is as in the proof of Proposition.1.1. With notation as in this proof, the two-dimensional W R representation τ ε1,...,ε n has ε factor εs, τ ε1,...,ε n, ψ = i l +1, n n l = k ε j jε j, j=1 j=1

17 .4 The ε factor 17 regardless of l being zero or not see Consequently εs, Π k, ρ, ψ = n / 1 = i n 1 εj>0 i k ε j jε j +1 i k ε j jε j ε, jε j>0 i jε j. 49 For example, if n = 1 we get εs, Π k, ρ, ψ = i k. This is the number appearing in the functional equation of the L function of a classical elliptic modular form of weight k. If n = and k note our overall assumption k n then we get εs, Π k, ρ, ψ = 1 k. This is the sign in the functional equation of the spin L function associated to a degree Siegel modular form, see [An] Theorem Lemma. For a positive integer n we have 1 n n n n/ ε j = ε 1,...,ε n {±1} j=1 n 1 εj>0 n n 1/ if n is even, if n is odd. If n is even and 4, this number is divisible by 4. Proof: Assume that n is even; the other case is similar. For ε 1,..., ε n {±1} n let l be the number of negative ε j s. Then, for ε j to be positive, l is allowed to run from 0 to n 1. Thus εj>0 n ε j = j=1 n / l=0 n n l. l The first assertion now follows from the elementary formulas n / l=0 n = n 1 1 n l n/ and n / l=0 n l l = n n 1 n n. n/ The last statement is a consequence of the following lemma..4. Lemma. i For any positive integer n, { n mod 4 if n is a power of, n 0 mod 4 otherwise.

18 18 THE EULER FACTOR FOR THE SPIN REPRESENTATION ii For any odd positive integer n 3, { 1 n mod 4 if n 1 is a power of, n 0 mod 4 otherwise. Proof: i This is an easy exercise; consider the adic valuation of n!. ii We have 1 n n = n 1 n of n n 1. Thus we are reduced to i. The case that n is odd In this case the formula 49 simplifies to n n 1. Since n is odd, the adic valuation of the left hand side equals that εs, Π k, ρ, ψ = i M with M = εj>0 k ε j jε j we are assuming that n 3. Let us assume that k is large enough, so that the absolute values can be omitted see Proposition.1.. Then M = k n + 1 εj>0 εj εj>0 j n + 1 The second sum changes its sign if ε j is replaced by ε j := ε n+1 j and therefore vanishes. The first sum was computed in Lemma.4.1, so we get M = k n + 1 n 1 n. 50 n 1/ Note that the representation Π k, considered as a representation of GSpn, R, has trivial central character. The minimal K type is the character g detcz + D k A B for g = K, C D where K is the standard maximal compact subgroup of GSpn its identity component is isomorphic to Un. Therefore, since n is odd, k must necessarily be even. But n 1/ is also even, so we get nn + 1 n 1 M n + 1 n 1 = n 1 n 1/ + n 1 n 1/ n 1 n 1/ n 1/ n 1 n 1/ mod 4, ε j. the last congruence being true for n 5 see Lemma.4.1. following result. In view of Lemma.4. we get the

19 .4 The ε factor Proposition. Assume that n 3 is an odd integer. Then for the spin ε factor in degree n we have { 1 if n = εs, Π k, ρ, ψ = m + 1 for some m, 1 otherwise, for any even weight k 1 4 n + 1. The case that n is even If we assume that n 4 is even and k is large enough see 0, then formula 49 gives εs, Π k, ρ, ψ = i M with M = k ε j jε j + jεj. εj>0 ε, jε j>0 If we replace k by k + 1, then, by Lemma.4.1, M changes by a number that is divisible by 4. It follows that εs, Π k, ρ, ψ = i M is independent of k note this is not true for n =. If we put fl = i l+1, then we can also proceed as in section.3. In analogy with formula 39 we obtain εs, Π k+n, ρ, ψ = n 1 r Z fr αr,n,n r>0 f k 1n j r αr,j,n f0 α0,n,n /. 51 Note that here we can omit the r from fk 1n j r since r Z rαr, j, n = 0 by Lemma.3.1 ii. We then apply 40 and get a formula similar to 41, namely n 1 f k 1n j n αr,j,n 1 = j 0=0 n 1 ε j 0 j=j 0 step f k 1n j βr,j 0,n, 5 If we consider a fixed j 0 and put m = n j 0, then we get the analogue of equation 4: n 1 ε j 0 j=j 0 step f k 1n j = = m 1 ε j 0 step m 1 ε j 0 step f k 1n j 0 j 53 ε s r, τ k 1m j,0, ψ. 54 Note that archimedean ε factors are independent of s, so we are free to put the shift of r/ here. If m is odd, this equals εs r/, Dk 1, Sym m, ψ. If m is even, there is a correcting factor of

20 0 THE EULER FACTOR FOR THE SPIN REPRESENTATION εs, τ j0,0. Thus we get, as in 44 and 45, εs, Π k+n, ρ, ψ = δ n r Z ε s r/, Dk 1, Sym n j, ψ βr,j,n 55 with δ = = = n r Z ε j=1 n r>0 ε j=1 n ε j=1 n r>0 ε j=1 ε s r, τ j,0, ψ βr,j,n r>0 i r+1 ε s r/, τ j,0, ψ ε s + r/, τ j,0, ψ ε s, τ j,0, ψ βr,j,n i α0,n,n /+1 fr αr,n,n f0 α0,n,n / βr,j,n i r+1 εj βr,j,n i K, 56 where εj = { 0 if n j 0 mod 4, 1 if n j mod 4, 57 and K is as in 46. Consider first the case that n is odd and let us compare the formulas 47 and 56. One easily shows that for each of the reflection polynomials δ r, in 47 the functional equation δ r, j s δ r, j 1 s = ir+1 εj holds r is odd, so this is just a sign. It follows that δ = δs δ1 s 58 is the sign appearing in the functional equation of the polynomial δs. Now assume that n is even. Using some standard formulas for the Γ function, one shows that for each of the factors δ r, appearing in formula 48 one has δ r, j s πs σ, δ r, j 1 s = tan σ = i r εj. In other words, if r εj 0 mod 4, then we get a contribution of tanπs/ to δs/δ1 s, and if r εj mod 4 we get a contribution of tanπs/ 1. From the factors δ 0,+ s we get a

21 .4 The ε factor 1 contribution of tanπs/ K. Furthermore, we know the total contribution is 1, i.e. δs = δ1 s, because δs is known to be a polynomial. Comparing this to 56 we see that each factor tanπs/ ±1 corresponds to a factor i ±1. Since we just saw that there are as many positive as negative exponents, it follows that δ = 1. In particular, 58 holds in all cases. We summarize:.4.4 Proposition. Assume k > 1 8 n and let δs be the reflection polynomial of Proposition.3.. Then the archimedean spin epsilon factor in even degree n = n can be expressed as εs, Π k+n, ρ, ψ = δ n r Z where δ = δs/δ1 s. If n is even, then δ = 1. ε s r/, Dk 1, Sym n j, ψ βr,j,n Remark: This proposition makes sense, since, as part of a global L function, the factor δs obviously contributes the number δs/δ1 s to the global ε factor appearing in the functional equation. In fact, we can say more about the symmetric power part of the ε factor..4.5 Proposition. Under the hypotheses of the previous proposition, we have n r Z Proof: Let M = ε s r/, Dk 1, Sym n j, ψ βr,j,n = { 1 if n 1 is a power of, 1 otherwise. εj>0 k ε j jε j + 1, so that i M is the first part of our ε factor, see 49. Similarly as in 50 we compute M = k n + 1 n n + n 1 1 n n/ n/ n + 1n n 1 n mod n/ n/ see Lemma.4.1. On the other hand, during the course of proving 55, we saw that i M = n r Z n ε j=1 ε s r/, Dk 1, Sym n j, ψ βr,j,n r Z ε s r βr,j,n, τ j,0, ψ

22 THE EULER FACTOR FOR THE SPIN REPRESENTATION with = M = n r Z n ε j=1 ε s r/, Dk 1, Sym n j, ψ βr,j,n i M εjβr, j, n = r Z n j n mod n εj j n j and εj as in 57. Here we used Lemma.3.1 iii. It is an exercise to compute this number; one obtains M = 1 n n n 1. It follows that M + M n + 1n n 4 n/ mod 4. This proves the assertion in view of Lemma Critical points We shall now work with the geometric normalization used in Andrianov s paper [An] and put L g s, Π k, ρ := L s nk nn + 1 +, Π k, ρ, 4 cf. 34. This L factor fits into a global L function that conjecturally admits a functional equation with center point s 0 = nk nn Assume f is a Siegel modular form of degree n and weight k > n, and let Mf be the conjectural associated motive such that Ls, Mf = Ls, f, ρ, where on the right we have the spin L function of f. From the functional equation nn + 1 Ls, f, ρ = εs, f, ρl nk + 1 s, f, ρ we conclude that Mf is of weight w = nk nn + 1 note that this is also the value of k ε i iε i for the extreme case ε 1 =... = ε n = 1. The archimedean Euler factor of Ls, f, ρ and of Ls, Mf is precisely our L g s, Π k, ρ. Following Deligne, an integer n Z is called critical for Mf if neither L g s, Π k, ρ nor L g nk nn + 1/ + 1 s, Π k, ρ has a pole at s = n.

23 .5 Critical points Proposition. Let f be a Siegel modular form of degree n and weight k > n, and let Mf be the motive whose L function coincides with the spin L function of f. Let s 0 = nk nn be the center point of the functional equation. Then the critical points of Mf are given as follows. i Assume n is odd. If k n+1 4, then there are k n+1 4 critical points, namely the integer points in the interval [ n 1 k n ,..., n + 1 k n + 13n + 1 ]. 8 If k < n+1 4 and n 1 mod 4, then we have the single critical point s 0. If k < n+1 4 and n 3 mod 4, then there are no critical points. ii Assume n is even. If n 0 mod 4, then there are no critical points. If n mod 4, then s 0 is the only critical point. Proof: i In the odd case we have L g s, Π k, ρ := L s + 1 s 0, Π k, ρ = gs where rε 1,..., ε n = s 0 1 k ε i iε i εi>0 Γs rε 1,..., ε n, and where gs is an entire function without zeros. The factor Γs rε 1,..., ε n contributes poles at s = rε 1,..., ε n N 0, where N 0 = {0, 1,,...}. Thus we need to find the maximal rε 1,..., ε n, or, equivalently, the minimal k ε i iε i. First assume that k n+1 /4, so that k ε i iε i takes only non-negative values see Proposition.1.. Indeed, the minimum value of this expression is then k n + 1 /4, so that the maximum value of rε 1,..., ε n is n 1 k n 1 8. Note this is indeed an integer. Thus all the integers greater than this number and less or equal s 0 are critical. The critical points lie symmetric about s 0, proving the first assertion. Now assume k < n + 1 /4. Let j be the number of negative ε i s. For fixed j let a j be the minimum and b j the maximum of the expression k ε i iε i. This expression takes every second integer value in [a j, b j ]. It is easy to see that a n 1/ < 0 < b n 1/ for k < Furthermore, we have n k ε i iε i { 1 mod, if n 1 mod 4, 0 mod, if n 3 mod 4 6 note that k is necessarily even in the odd case. It follows from 61 and 6 that the minimal value of k ε i iε i is 1 if n 1 mod 4 and 0 if n 3 mod 4. Thus the maximum value of rε 1,..., ε n

24 4 REFERENCES is s 0 1 resp. s 0 1/. The assertion follows, since in the first case we have s 0 Z, while in the second case s Z. ii These are similar considerations, and we omit the details. Remark: The recursive definition 3 and 33 of archimedean Euler factors is not motivic in the following sense. As is easily seen, the A n,k s thus defined contains a factor Γs n 1k + nn + 1/ 1. The point n 1k nn + 1/ + 1 lies on the right of the center s 0 given by 60. But a motivic Euler factor contains only Γ factors of the form Γs p with p < s 0 1 coming from the Hodge types p, q, q, p with p < q and p + q = w = s 0 1 and of the form Γs + ε p/ with p = w and ε {±1} coming from Hodge type p, p, see [De] 5.3. In particular, using Andrianov s factors one would conclude that there are no critical points as soon as n 3. References [An] [AS] Andrianov, A.: Euler products associated with Siegel modular forms of degree two. Russ. Math. Surveys 9, , Asgari, M., Schmidt, R.: Siegel modular forms and representations. Manuscripta Math , [Bo] Borel, A.: Automorphic L-functions. Proc. Sympos. Pure Math , part, 7 61 [De] Deligne, P.: Valeurs de fonction L et périodes d intégrales. Proc. Sympos. Pure Math , part, [Ik] [Kn] [Sch] [Sh] Ikeda, T.: On the lifting of elliptic cusp forms to Siegel modular forms of degree n. Ann. Math , Knapp, A.: Local Langlands correspondence: The archimedean places. Proc. Sympos. Pure Math , part, Schmidt, R.: On the spin L function of Ikeda s lifts. To appear in Comment. Math. Univ. St. Paul. Shahidi, F.: Symmetric power L functions for GL. In: Elliptic Curves and Related Topics, CRM Proc. Lecture Notes 4, AMS, 1994, [Sp] Springer, T.: Reductive groups. Proc. Sympos. Pure Math , part 1, 3 7 [Ta] Tate, J.: Number theoretic background. Proc. Sympos. Pure Math , part, 3 6