On the archimedean Euler factors for spin L functions
|
|
- Gabriel Greene
- 5 years ago
- Views:
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
ARCHIMEDEAN ASPECTS OF SIEGEL MODULAR FORMS OF DEGREE 2
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number 7, 207 ARCHIMEDEAN ASPECTS OF SIEGEL MODULAR FORMS OF DEGREE 2 RALF SCHMIDT ABSTRACT. We survey the archimedean representations and Langlands parameters
More informationOn Siegel modular forms of degree 2 with square-free level
On Siegel modular forms of degree with square-free level Ralf Schmidt Introduction For representations of GL() over a p adic field F there is a well-known theory of local newforms due to Casselman, see
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More information(E.-W. Zink, with A. Silberger)
1 Langlands classification for L-parameters A talk dedicated to Sergei Vladimirovich Vostokov on the occasion of his 70th birthday St.Petersburg im Mai 2015 (E.-W. Zink, with A. Silberger) In the representation
More informationEndoscopic character relations for the metaplectic group
Endoscopic character relations for the metaplectic group Wen-Wei Li wwli@math.ac.cn Morningside Center of Mathematics January 17, 2012 EANTC The local case: recollections F : local field, char(f ) 2. ψ
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationTRILINEAR FORMS AND TRIPLE PRODUCT EPSILON FACTORS WEE TECK GAN
TRILINAR FORMS AND TRIPL PRODUCT PSILON FACTORS W TCK GAN Abstract. We give a short and simple proof of a theorem of Dipendra Prasad on the existence and non-existence of invariant trilinear forms on a
More informationMATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 2014) LECTURE 1 (FEBRUARY 7, 2014) ERIC URBAN
MATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 014) LECTURE 1 (FEBRUARY 7, 014) ERIC URBAN NOTES TAKEN BY PAK-HIN LEE 1. Introduction The goal of this research seminar is to learn the theory of p-adic
More informationMod p Galois representations attached to modular forms
Mod p Galois representations attached to modular forms Ken Ribet UC Berkeley April 7, 2006 After Serre s article on elliptic curves was written in the early 1970s, his techniques were generalized and extended
More informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More informationSPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS
SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the
More informationTwists and residual modular Galois representations
Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10 th July 2014 Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual
More informationColette Mœglin and Marko Tadić
CONSTRUCTION OF DISCRETE SERIES FOR CLASSICAL p-adic GROUPS Colette Mœglin and Marko Tadić Introduction The goal of this paper is to complete (after [M2] the classification of irreducible square integrable
More informationON THE STANDARD MODULES CONJECTURE. V. Heiermann and G. Muić
ON THE STANDARD MODULES CONJECTURE V. Heiermann and G. Muić Abstract. Let G be a quasi-split p-adic group. Under the assumption that the local coefficients C defined with respect to -generic tempered representations
More information14 From modular forms to automorphic representations
14 From modular forms to automorphic representations We fix an even integer k and N > 0 as before. Let f M k (N) be a modular form. We would like to product a function on GL 2 (A Q ) out of it. Recall
More informationLECTURE 2: LANGLANDS CORRESPONDENCE FOR G. 1. Introduction. If we view the flow of information in the Langlands Correspondence as
LECTURE 2: LANGLANDS CORRESPONDENCE FOR G J.W. COGDELL. Introduction If we view the flow of information in the Langlands Correspondence as Galois Representations automorphic/admissible representations
More informationRIMS. Ibukiyama Zhuravlev. B.Heim
RIMS ( ) 13:30-14:30 ( ) Title: Generalized Maass relations and lifts. Abstract: (1) Duke-Imamoglu-Ikeda Eichler-Zagier- Ibukiyama Zhuravlev L- L- (2) L- L- L B.Heim 14:45-15:45 ( ) Title: Kaneko-Zagier
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik On projective linear groups over finite fields as Galois groups over the rational numbers Gabor Wiese Preprint Nr. 14/2006 On projective linear groups over finite fields
More informationLocal Langlands correspondence and examples of ABPS conjecture
Local Langlands correspondence and examples of ABPS conjecture Ahmed Moussaoui UPMC Paris VI - IMJ 23/08/203 Notation F non-archimedean local field : finite extension of Q p or F p ((t)) O F = {x F, v(x)
More informationThe Grothendieck-Katz Conjecture for certain locally symmetric varieties
The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-
More informationOn the Notion of an Automorphic Representation *
On the Notion of an Automorphic Representation * The irreducible representations of a reductive group over a local field can be obtained from the square-integrable representations of Levi factors of parabolic
More informationHecke Operators for Arithmetic Groups via Cell Complexes. Mark McConnell. Center for Communications Research, Princeton
Hecke Operators for Arithmetic Groups via Cell Complexes 1 Hecke Operators for Arithmetic Groups via Cell Complexes Mark McConnell Center for Communications Research, Princeton Hecke Operators for Arithmetic
More informationQuadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation
(December 19, 010 Quadratic reciprocity (after Weil Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields k (characteristic not the quadratic norm residue symbol
More information10 l-adic representations
0 l-adic representations We fix a prime l. Artin representations are not enough; l-adic representations with infinite images naturally appear in geometry. Definition 0.. Let K be any field. An l-adic Galois
More informationThe Local Langlands Conjectures for n = 1, 2
The Local Langlands Conjectures for n = 1, 2 Chris Nicholls December 12, 2014 1 Introduction These notes are based heavily on Kevin Buzzard s excellent notes on the Langlands Correspondence. The aim is
More informationThe Galois Representation Associated to Modular Forms (Part I)
The Galois Representation Associated to Modular Forms (Part I) Modular Curves, Modular Forms and Hecke Operators Chloe Martindale May 20, 2015 Contents 1 Motivation and Background 1 2 Modular Curves 2
More informationResidual modular Galois representations: images and applications
Residual modular Galois representations: images and applications Samuele Anni University of Warwick London Number Theory Seminar King s College London, 20 th May 2015 Mod l modular forms 1 Mod l modular
More informationOn the equality case of the Ramanujan Conjecture for Hilbert modular forms
On the equality case of the Ramanujan Conjecture for Hilbert modular forms Liubomir Chiriac Abstract The generalized Ramanujan Conjecture for unitary cuspidal automorphic representations π on GL 2 posits
More informationON RESIDUAL COHOMOLOGY CLASSES ATTACHED TO RELATIVE RANK ONE EISENSTEIN SERIES FOR THE SYMPLECTIC GROUP
ON RESIDUAL COHOMOLOGY CLASSES ATTACHED TO RELATIVE RANK ONE EISENSTEIN SERIES FOR THE SYMPLECTIC GROUP NEVEN GRBAC AND JOACHIM SCHWERMER Abstract. The cohomology of an arithmetically defined subgroup
More informationDiscrete Series and Characters of the Component Group
Discrete Series and Characters of the Component Group Jeffrey Adams April 9, 2007 Suppose φ : W R L G is an L-homomorphism. There is a close relationship between the L-packet associated to φ and characters
More informationarxiv: v1 [math.rt] 11 Sep 2009
FACTORING TILTING MODULES FOR ALGEBRAIC GROUPS arxiv:0909.2239v1 [math.rt] 11 Sep 2009 S.R. DOTY Abstract. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of
More informationKleine AG: Travaux de Shimura
Kleine AG: Travaux de Shimura Sommer 2018 Programmvorschlag: Felix Gora, Andreas Mihatsch Synopsis This Kleine AG grew from the wish to understand some aspects of Deligne s axiomatic definition of Shimura
More information0 A. ... A j GL nj (F q ), 1 j r
CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected
More informationWeyl Group Representations and Unitarity of Spherical Representations.
Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν
More informationCOHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II
COHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II LECTURES BY JOACHIM SCHWERMER, NOTES BY TONY FENG Contents 1. Review 1 2. Lifting differential forms from the boundary 2 3. Eisenstein
More information9 Artin representations
9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation
More informationQuadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation
(September 17, 010) Quadratic reciprocity (after Weil) Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields (characteristic not ) the quadratic norm residue
More informationHalf the sum of positive roots, the Coxeter element, and a theorem of Kostant
DOI 10.1515/forum-2014-0052 Forum Math. 2014; aop Research Article Dipendra Prasad Half the sum of positive roots, the Coxeter element, and a theorem of Kostant Abstract: Interchanging the character and
More informationSERRE S CONJECTURE AND BASE CHANGE FOR GL(2)
SERRE S CONJECTURE AND BASE CHANGE OR GL(2) HARUZO HIDA 1. Quaternion class sets A quaternion algebra B over a field is a simple algebra of dimension 4 central over a field. A prototypical example is the
More informationResults from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 1998
98m:11125 11R39 11F27 11F67 11F70 22E50 22E55 Gelbart, Stephen (IL-WEIZ); Rogawski, Jonathan (1-UCLA); Soudry, David (IL-TLAV) Endoscopy, theta-liftings, and period integrals for the unitary group in three
More information1. Statement of the theorem
[Draft] (August 9, 2005) The Siegel-Weil formula in the convergent range Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ We give a very simple, mostly local, argument for the equality
More informationBASIC VECTOR VALUED SIEGEL MODULAR FORMS OF GENUS TWO
Freitag, E. and Salvati Manni, R. Osaka J. Math. 52 (2015), 879 894 BASIC VECTOR VALUED SIEGEL MODULAR FORMS OF GENUS TWO In memoriam Jun-Ichi Igusa (1924 2013) EBERHARD FREITAG and RICCARDO SALVATI MANNI
More informationTriple product p-adic L-functions for balanced weights and arithmetic properties
Triple product p-adic L-functions for balanced weights and arithmetic properties Marco A. Seveso, joint with Massimo Bertolini, Matthew Greenberg and Rodolfo Venerucci 2013 Workshop on Iwasawa theory and
More informationIVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) =
LECTURE 7: CATEGORY O AND REPRESENTATIONS OF ALGEBRAIC GROUPS IVAN LOSEV Introduction We continue our study of the representation theory of a finite dimensional semisimple Lie algebra g by introducing
More informationMarko Tadić. Introduction Let F be a p-adic field. The normalized absolute value on F will be denoted by F. Denote by ν : GL(n, F ) R the character
ON REDUCIBILITY AND UNITARIZABILITY FOR CLASSICAL p-adic GROUPS, SOME GENERAL RESULTS Marko Tadić Abstract. The aim of this paper is to prove two general results on parabolic induction of classical p-adic
More informationON THE RESIDUAL SPECTRUM OF HERMITIAN QUATERNIONIC INNER FORM OF SO 8. Neven Grbac University of Rijeka, Croatia
GLASNIK MATEMATIČKI Vol. 4464)2009), 11 81 ON THE RESIDUAL SPECTRUM OF HERMITIAN QUATERNIONIC INNER FORM OF SO 8 Neven Grbac University of Rijeka, Croatia Abstract. In this paper we decompose the residual
More informationBackground on Chevalley Groups Constructed from a Root System
Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described
More informationCategory O and its basic properties
Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α
More informationLocal root numbers of elliptic curves over dyadic fields
Local root numbers of elliptic curves over dyadic fields Naoki Imai Abstract We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension
More informationProof of a simple case of the Siegel-Weil formula. 1. Weil/oscillator representations
(March 6, 2014) Proof of a simple case of the Siegel-Weil formula Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ First, I confess I never understood Siegel s arguments for his mass
More informationSOME REMARKS ON REPRESENTATIONS OF QUATERNION DIVISION ALGEBRAS
SOME REMARKS ON REPRESENTATIONS OF QUATERNION DIVISION ALGEBRAS DIPENDRA PRASAD Abstract. For the quaternion division algebra D over a non-archimedean local field k, and π an irreducible finite dimensional
More informationL(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that
ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive
More informationOn the cuspidality criterion for the Asai transfer to GL(4)
On the cuspidality criterion for the Asai transfer to GL(4) Dipendra Prasad and Dinakar Ramakrishnan Introduction Let F be a number field and K a quadratic algebra over F, i.e., either F F or a quadratic
More informationIntroduction to L-functions I: Tate s Thesis
Introduction to L-functions I: Tate s Thesis References: - J. Tate, Fourier analysis in number fields and Hecke s zeta functions, in Algebraic Number Theory, edited by Cassels and Frohlich. - S. Kudla,
More informationFrom K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015
From K3 Surfaces to Noncongruence Modular Forms Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 Winnie Li Pennsylvania State University 1 A K3 surface
More informationLifting Galois Representations, and a Conjecture of Fontaine and Mazur
Documenta Math. 419 Lifting Galois Representations, and a Conjecture of Fontaine and Mazur Rutger Noot 1 Received: May 11, 2001 Revised: November 16, 2001 Communicated by Don Blasius Abstract. Mumford
More informationHecke Operators, Zeta Functions and the Satake map
Hecke Operators, Zeta Functions and the Satake map Thomas R. Shemanske December 19, 2003 Abstract Taking advantage of the Satake isomorphism, we define (n + 1) families of Hecke operators t n k (pl ) for
More informationTraces, Cauchy identity, Schur polynomials
June 28, 20 Traces, Cauchy identity, Schur polynomials Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/. Example: GL 2 2. GL n C and Un 3. Decomposing holomorphic polynomials over GL
More informationHighest-weight Theory: Verma Modules
Highest-weight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or,
More informationRigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture
Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-
More informationWhittaker models and Fourier coeffi cients of automorphic forms
Whittaker models and Fourier coeffi cients of automorphic forms Nolan R. Wallach May 2013 N. Wallach () Whittaker models 5/13 1 / 20 G be a real reductive group with compact center and let K be a maximal
More informationOn Modular Forms for the Paramodular Group
On Modular Forms for the Paramodular Group Brooks Roberts and Ralf Schmidt Contents Definitions 3 Linear independence at different levels 6 3 The level raising operators 8 4 Oldforms and newforms 3 5 Saito
More information1 Classifying Unitary Representations: A 1
Lie Theory Through Examples John Baez Lecture 4 1 Classifying Unitary Representations: A 1 Last time we saw how to classify unitary representations of a torus T using its weight lattice L : the dual of
More informationNumber Theory/Representation Theory Notes Robbie Snellman ERD Spring 2011
Number Theory/Representation Theory Notes Robbie Snellman ERD Spring 2011 January 27 Speaker: Moshe Adrian Number Theorist Perspective: Number theorists are interested in studying Γ Q = Gal(Q/Q). One way
More informationTHE RESIDUAL EISENSTEIN COHOMOLOGY OF Sp 4 OVER A TOTALLY REAL NUMBER FIELD
THE RESIDUAL EISENSTEIN COHOMOLOGY OF Sp 4 OVER A TOTALLY REAL NUMBER FIELD NEVEN GRBAC AND HARALD GROBNER Abstract. Let G = Sp 4 /k be the k-split symplectic group of k-rank, where k is a totally real
More informationGalois Representations
9 Galois Representations This book has explained the idea that all elliptic curves over Q arise from modular forms. Chapters 1 and introduced elliptic curves and modular curves as Riemann surfaces, and
More informationIMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS
IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS MAHDI ASGARI AND FREYDOON SHAHIDI Abstract. We give a complete description of the image of the endoscopic functorial transfer of generic automorphic representations
More information(www.math.uni-bonn.de/people/harder/manuscripts/buch/), files chap2 to chap
The basic objects in the cohomology theory of arithmetic groups Günter Harder This is an exposition of the basic notions and concepts which are needed to build up the cohomology theory of arithmetic groups
More informationPrimitive Ideals and Unitarity
Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)
More informationOn the generation of the coefficient field of a newform by a single Hecke eigenvalue
On the generation of the coefficient field of a newform by a single Hecke eigenvalue Koopa Tak-Lun Koo and William Stein and Gabor Wiese November 2, 27 Abstract Let f be a non-cm newform of weight k 2
More informationA partition of the set of enhanced Langlands parameters of a reductive p-adic group
A partition of the set of enhanced Langlands parameters of a reductive p-adic group joint work with Ahmed Moussaoui and Maarten Solleveld Anne-Marie Aubert Institut de Mathématiques de Jussieu - Paris
More informationHecke Theory and Jacquet Langlands
Hecke Theory and Jacquet Langlands S. M.-C. 18 October 2016 Today we re going to be associating L-functions to automorphic things and discussing their L-function-y properties, i.e. analytic continuation
More informationSome remarks on signs in functional equations. Benedict H. Gross. Let k be a number field, and let M be a pure motive of weight n over k.
Some remarks on signs in functional equations Benedict H. Gross To Robert Rankin Let k be a number field, and let M be a pure motive of weight n over k. Assume that there is a non-degenerate pairing M
More informationNON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES
NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES MATTHEW BOYLAN Abstract Let pn be the ordinary partition function We show, for all integers r and s with s 1 and 0 r < s, that #{n : n r mod
More informationBLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n)
BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) VERA SERGANOVA Abstract. We decompose the category of finite-dimensional gl (m n)- modules into the direct sum of blocks, show that
More informationRaising the Levels of Modular Representations Kenneth A. Ribet
1 Raising the Levels of Modular Representations Kenneth A. Ribet 1 Introduction Let l be a prime number, and let F be an algebraic closure of the prime field F l. Suppose that ρ : Gal(Q/Q) GL(2, F) is
More informationREDUCIBLE PRINCIPAL SERIES REPRESENTATIONS, AND LANGLANDS PARAMETERS FOR REAL GROUPS. May 15, 2018 arxiv: v2 [math.
REDUCIBLE PRINCIPAL SERIES REPRESENTATIONS, AND LANGLANDS PARAMETERS FOR REAL GROUPS DIPENDRA PRASAD May 15, 2018 arxiv:1705.01445v2 [math.rt] 14 May 2018 Abstract. The work of Bernstein-Zelevinsky and
More informationOn values of Modular Forms at Algebraic Points
On values of Modular Forms at Algebraic Points Jing Yu National Taiwan University, Taipei, Taiwan August 14, 2010, 18th ICFIDCAA, Macau Hermite-Lindemann-Weierstrass In value distribution theory the exponential
More informationThe local Langlands correspondence for inner forms of SL n. Plymen, Roger. MIMS EPrint:
The local Langlands correspondence for inner forms of SL n Plymen, Roger 2013 MIMS EPrint: 2013.43 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationReducibility of generic unipotent standard modules
Journal of Lie Theory Volume?? (??)???? c?? Heldermann Verlag 1 Version of March 10, 011 Reducibility of generic unipotent standard modules Dan Barbasch and Dan Ciubotaru Abstract. Using Lusztig s geometric
More informationA brief overview of modular and automorphic forms
A brief overview of modular and automorphic forms Kimball Martin Original version: Fall 200 Revised version: June 9, 206 These notes were originally written in Fall 200 to provide a very quick overview
More informationON THE RESIDUAL SPECTRUM OF SPLIT CLASSICAL GROUPS SUPPORTED IN THE SIEGEL MAXIMAL PARABOLIC SUBGROUP
ON THE RESIDUAL SPECTRUM OF SPLIT CLASSICAL GROUPS SUPPORTED IN THE SIEGEL MAXIMAL PARABOLIC SUBGROUP NEVEN GRBAC Abstract. For the split symplectic and special orthogonal groups over a number field, we
More informationA MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET. March 7, 2017
A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET DIPENDRA PRASAD March 7, 2017 Abstract. Following the natural instinct that when a group operates on a number field then every term in the
More informationOn the Langlands Program
On the Langlands Program John Rognes Colloquium talk, May 4th 2018 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for
More informationON THE MAXIMAL PRIMITIVE IDEAL CORRESPONDING TO THE MODEL NILPOTENT ORBIT
ON THE MAXIMAL PRIMITIVE IDEAL CORRESPONDING TO THE MODEL NILPOTENT ORBIT HUNG YEAN LOKE AND GORDAN SAVIN Abstract. Let g = k s be a Cartan decomposition of a simple complex Lie algebra corresponding to
More informationThe Galois representation associated to modular forms pt. 2 Erik Visse
The Galois representation associated to modular forms pt. 2 Erik Visse May 26, 2015 These are the notes from the seminar on local Galois representations held in Leiden in the spring of 2015. The website
More informationClass groups and Galois representations
and Galois representations UC Berkeley ENS February 15, 2008 For the J. Herbrand centennaire, I will revisit a subject that I studied when I first came to Paris as a mathematician, in 1975 1976. At the
More informationGeometric Structure and the Local Langlands Conjecture
Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure
More informationOn the cohomology of congruence subgroups of SL 4 (Z)
On the cohomology of congruence subgroups of SL 4 (Z) Paul E. Gunnells UMass Amherst 19 January 2009 Paul E. Gunnells (UMass Amherst) Cohomology of subgroups of SL 4 (Z) 19 January 2009 1 / 32 References
More informationMath 249B. Geometric Bruhat decomposition
Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique
More informationA GEOMETRIC JACQUET FUNCTOR
A GEOMETRIC JACQUET FUNCTOR M. EMERTON, D. NADLER, AND K. VILONEN 1. Introduction In the paper [BB1], Beilinson and Bernstein used the method of localisation to give a new proof and generalisation of Casselman
More informationTWO SIMPLE OBSERVATIONS ON REPRESENTATIONS OF METAPLECTIC GROUPS. In memory of Sibe Mardešić
TWO IMPLE OBERVATION ON REPREENTATION OF METAPLECTIC GROUP MARKO TADIĆ arxiv:1709.00634v1 [math.rt] 2 ep 2017 Abstract. M. Hanzer and I. Matić have proved in [8] that the genuine unitary principal series
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationA connection between number theory and linear algebra
A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.
More informationON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS
ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,
More informationFundamental Lemma and Hitchin Fibration
Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental
More informationU = 1 b. We fix the identification G a (F ) U sending b to ( 1 b
LECTURE 11: ADMISSIBLE REPRESENTATIONS AND SUPERCUSPIDALS I LECTURE BY CHENG-CHIANG TSAI STANFORD NUMBER THEORY LEARNING SEMINAR JANUARY 10, 2017 NOTES BY DAN DORE AND CHENG-CHIANG TSAI Let L is a global
More informationGalois groups with restricted ramification
Galois groups with restricted ramification Romyar Sharifi Harvard University 1 Unique factorization: Let K be a number field, a finite extension of the rational numbers Q. The ring of integers O K of K
More informationEXTENSIONS AND THE EXCEPTIONAL ZERO OF THE ADJOINT SQUARE L-FUNCTIONS
EXTENSIONS AND THE EXCEPTIONAL ZERO OF THE ADJOINT SQUARE L-FUNCTIONS HARUZO HIDA Take a totally real field F with integer ring O as a base field. We fix an identification ι : Q p = C Q. Fix a prime p>2,
More information