Euler Products and Twisted Euler Products

Transcription

1 * Vol. I?? Euler Products and Twisted Euler Products Solomon Friedberg* Abstract We describe in brief some aspects of the Langlands program, focussing on Euler products, and also some new constructions, where Euler products are replaced by twisted Euler products. The twisted Euler products are related to automorphic forms on metaplectic groups Mathematics Subject Classification: 11F66, 11F55, 11F70, 11M41, 22E55. Keywords and Phrases: Euler product, L-function, Langlands Conjectures, Twisted Euler product, Metaplectic group, Functional equation. 1 Introduction The Langlands program is fundamental to modern number theory. The first part of this article describes in brief some aspects of the Langlands program, focussing on Euler products. For the purpose of exposition, most discussion of the Euler factors at the finite number of bad places (including the archimedean ones) is omitted. The remaining parts describe some constructions where Euler products are replaced by twisted Euler products. Though the first examples of twisted Euler products are over 50 years old, this notion is a new one; in fact, we are just now recognizing that a range of earlier constructions may be fit into this general framework and that it is possible to construct other families of twisted Euler products. Naturally one wonders the extent to which the Langlands program, as well as the geometric Langlands program when one works over a function field, can be meshed with this new context. This paper is an expanded version of the author s lectures at the International Instructional Conference on the Langlands and Geometric Langlands Program in Guangzhou, China, June 18-21, The author heartily thanks the conference organizers for the opportunity to attend. The Weyl group multiple Dirichlet series described in Sections 6 9 below are joint work with Brubaker, Bump, Chinta, and Hoffstein, and the author is delighted to have the opportunity to warmly thank *Department of Mathematics, Boston College, Chestnut Hill, Massachusetts, , USA. friedber@bc.edu.

2 190 Solomon Friedberg these co-authors for many years of helpful conversations and shared insights. This work was supported by NSF grants DMS and DMS and by NSA grant H Euler Products 2.1 A classical paradigm We begin with a classical paradigm that is common in number theory. A local problem or construction gives rise to local L-functions L p (s), defined for (almost all) primes p, of the form L p (s) = h(p c )p cs c=0 and these series converge for Re(s) sufficiently large. The local objects may be put together into an Euler product L(s) = p L p (s). This product converges for Re(s) sufficiently large. It is an important (and oftentimes deep) fact that the global L-function L (s) = L (s)l(s), where L (s) is a Gamma factor arising from the archimedean places, has analytic continuation and a functional equation under the transformation s k s for suitable k. This fact is typically connected to the theory of automorphic forms on some reductive group. The functional equation above is of the form L (s) = ǫ(s) L (k s) where L is a similarly-constructed function and ǫ(s) is a factor of the form ab s for some constants a C, b R. Here are some examples of the paradigm. 2.2 L-functions of elliptic curves Let E be an elliptic curve defined over Q. For p a prime of good reduction, let a p = p + 1 E(F p ), and let the local L-functions L p (s, E) be given by L p (s, E) = 1 1 a p p s + p 1 2s. Then, thanks to the work of Wiles et al., the L-function L(s, E) (where the local factors L p (s, E) may be defined at the remaining places p) 1, originally defined for 1 If p divides the conductor of E then the local factors are of the form L p(s, E) = (1 a pp s ) 1 with a p = ±1 or 0 depending on the reduction of E at p.

3 Euler Products and Twisted Euler Products 191 Re(s) sufficiently large, has analytic continuation to all complex s and functional equation under s 2 s. This is explained as Wiles and Taylor [Wi, WT] in the semistable case, and Breuil, Conrad, Diamond, and Taylor [BCDT] in general, showed that the L-function L(s, E) is the L-function of a holomorphic modular form of weight 2, as conjectured by Shimura and Taniyama. 2.3 Artin L-functions Let K be a global field and K denote an algebraic closure. If σ : Gal( K/K) GL(V ) is a finite-dimensional continuous Galois representation, then the local Artin L-function is defined for all unramified (finite) places v by L v (s, σ) = det(i V σ(frob v )Nv s ) 1, where I V is the identity map on V, Frob v is an element in the Frobenius conjugacy class in Gal( K/K) and Nv is the absolute norm of v. There is a similar definition at the ramified places. Then the Artin conjecture states that the resulting L- function L(s, σ) is entire. It is a conjecture of Langlands [La1] that L(s, σ) should satisfy a functional equation, and in fact be the standard L-function of a cuspidal automorphic representation, so that Artin s conjecture follows from the properties of automorphic L-functions. So this is (conjecturally) another instance of the paradigm. We refer to Gelbart s article in this volume [Ge] for more details, connections to reciprocity laws, etc. 2.4 Langlands L-functions Let G be a connected reductive algebraic group defined over a number or function field K. Suppose for simplicity that G is quasisplit over K. Let π = π v be a factorizable irreducible unitary automorphic representation of the adelic points of G, G(A K ). Then at almost all places v the representation π v is unramified i.e. has a vector fixed under a maximal compact subgroup of G(K v ), and the Satake isomorphism attaches to π v a semisimple conjugacy class t π,v in L G, the L-group of G. Recall that the connected component L G 0 of L G is a complex Lie group. Given a finite dimensional complex vector space V and a homomorphism ρ: L G GL(V ) such that ρ restricted to L G 0 is a complex analytic representation, Langlands defined the local L-function at almost all finite places v by L v (s, π v, ρ) = det ( I V ρ(t π,v )Nv s) 1. It is a conjecture of Langlands [La1] that the resulting global L-function L(s, π, ρ) has continuation to C and functional equation; more precisely, local L-factors may be defined at the remaining places such that the global L-function has functional equation under s 1 s (and π π). Moreover, this L-function is conjectured (an instance of the Langlands functoriality conjecture) to be the standard Godement-Jacquet L-function of an automorphic representation Π on GL(dim V ) regarded as an algebraic group over K. So these conjectures fit the above paradigm.

4 192 Solomon Friedberg 2.5 Rankin-Selberg L-functions The Langlands conjectures assert roughly that all operations in multilinear algebra have an analogue in the theory of automorphic representations which is governed by their L-functions. The above example was that of a single homomorphism. There is also a conjecture for the tensor product. For example, let π i be factorizable automorphic representations of GL(n i ) for 1 i r. For each place v such that all π i,v are unramified (this is all but a finite number of places), let t i,v be the associated semisimple conjugacy classes in GL(n i, C). Then one defines the local Rankin-Selberg L-function by L v (s, π 1,v π r,v ) = det(i n1n 2...n r t 1,v t 2,v t r,v Nv s ) 1. Once again this is expected to satisfy the paradigm above, and the resulting L- function is expected to be the standard L-function of an automorphic representation of GL(n 1... n r ). 2.6 Implications The continuation and functional equation asserted in the classical paradigm has significant consequences in many of the above examples (even if one does not know that it can be explained automorphically). For example, if π is on GL(d), then establishing the properties of the L-functions attached to symmetric powers L(s, π, Sym r ) for all r 1 would imply the Ramanujan conjecture for π. In addition, fine enough knowledge of the continuation and functional equation of Rankin-Selberg L-functions is enough in many cases to imply that the continuation is explained automorphically. This is illustrated by the Converse Theorem of Cogdell and Piatetski-Shapiro [CPS1, CPS2], one version of which is given as follows. Let Π be an irreducible admissible representation of GL(n, A K ), n 3, that is unitary and generic but not necessarily automorphic. It has a decomposition Π = Π v, where Π v is an irreducible admissible generic representation of GL(n, K v ), and to each Π v one can associate its local L-function L v (s, Π v ), which for almost all v is the local Langlands L-function attached to the standard representation of L G 0 = GL(n, C) as above. As in the paradigm, form the Euler product L(s, Π) = v L v(s, Π v ). For example, if π is a cuspidal automorphic representation on GL(d, A K ) and an integer r > 1 is given, then one can define Π r,v for almost all v by the equation L v (s, Π r,v ) = L v (s, π v, Sym r ). (One can in fact define suitable Π r,v at the remaining places by making use of the local Langlands conjecture for GL n, established by Harris and Taylor [HT].) It would be highly desirable to know the properties of Π r = Π r,v and in particular to establish that, as the Langlands conjectures predict, Π r is automorphic. To address this, returning to the case of an arbitrary irreducible admissible unitary generic representation Π = Π v of GL(n, A K ), suppose that L(s, Π)

5 Euler Products and Twisted Euler Products 193 converges in some half plane Re(s) 0, and that the central character ω Π of Π is automorphic. Also, for π = π v a cuspidal automorphic representation of GL(m, A K ) with m < n, define the Rankin-Selberg Euler product L(s, Π π ) = v L v(s, Π v π v). Using the cuspidality of π, one may show that this product converges absolutely for Re(s) 0. Theorem 2.1. (Cogdell-Piatetski-Shapiro) Let Π be an irreducible admissible unitary generic representation of GL(n, A K ). Suppose that for each cuspidal automorphic representation π of GL(m, A K ) with m < n 1, the Rankin-Selberg Euler product L(s, Π π ) has analytic continuation to an entire function of s which is bounded in vertical strips of finite width, the same is true for the contragredient Rankin-Selberg Euler product, and L(s, Π π ) satisfies a standard functional equation. Then Π is a cuspidal automorphic representation. There are variations on this Theorem where one twists only by automorphic representations π that are unramified at a finite set of places, or even by automorphic representations π which are unramified outside a finite set of places (containing the archimedean ones). See [CPS1,CPS2] for details. 2.7 Approaches to proving the continuation and functional equation There are two main approaches to directly proving the continuation and functional equation of L-functions attached to one or more automorphic representations as described above 2. The first direct approach, the Rankin-Selberg method, going back historically to Riemann s second proof of the functional equation of the zeta function, obtains the desired L-function as an integral whose integrand involves one or more automorphic forms. The properties of the L-function are then obtained from the integral. For example, the Hecke integral expresses the standard L-function of a GL(2) automorphic form in terms of its Mellin transform, and the classical Rankin-Selberg integral for GL(2) GL(2) expresses this L-function with degree 4 Euler factors as an integral of the product of two automorphic forms against a GL(2) Eisenstein series (with the functional equation resulting from the functional equation of the Eisenstein series). See Bump [Bu] for an extensive survey. The second, the Langlands-Shahidi method, based on the continuation of the Eisenstein series obtained by Langlands using spectral methods [La3], obtains the desired L- function when it appears in the constant term of an Eisenstein series. In order to control the pieces of this constant term (which is expressed in terms of a product of L-functions), one restricts to generic automorphic representations and makes use of the Whittaker coefficients of the Eisenstein series. As an advantage, one obtains a natural way of defining the L-factors at the remaining ( bad ) places. 2 One can also attempt to establish the functoriality conjecture first, for example by the trace formula, and deduce the continuation and functional equation as consequences. A striking example of the use of the trace formula to establish a case of functoriality is the work on base change of Arthur and Clozel [AC]. For an introduction to the trace formula, see Lapid [Lap].

6 194 Solomon Friedberg This approach was suggested by Langlands [La2], and developed extensively by Shahidi, Kim and others. For very useful accounts of these methods and their consequences, see Gelbart and Miller [GM] and Shahidi [Sh1], [Sh2]. It is important to emphasize that these methods do not account for all automorphic L-functions. Indeed, the Langlands-Shahidi method only allows one to study L-functions on a specific list [La2]. The Rankin-Selberg method allows one to continue several L-functions that are not of Langlands-Shahidi type, but the list of L-functions that have been treated by it is still quite limited. For example, as far as I know, for split groups G and representations (ρ, V ρ ) of L G there is no example of a Rankin-Selberg integral for an L-function L(s, π, ρ), π on G(A), such that the L G-invariants in C[V ρ ] is not a polynomial ring. 2.8 Multi-variable version of the classical paradigm There is also a generalization of the classical paradigm to multi-variable objects. The requirement that L (s) satisfies a functional equation under s k s is replaced by the requirement that L (s 1,, s r ) satisfies a group of functional equations. An example is provided by the Whittaker coefficients of the Eisenstein series attached to a non-maximal parabolic subgroup. These gives rise to a function of several complex variables that can be expressed as a multiple Dirichlet series h(c1,...,c r )c s1 1...cr sr where the coefficients are multiplicative: if (c 1... c r, c 1... c r ) = 1, then h(c 1 c 1,..., c rc r ) = h(c 1,..., c r )h(c 1,...,c r ). Such a function satisfies a group of transformations obtained from a Weyl group. 3 Twisted Euler Products 3.1 The twisted paradigm We now introduce a new paradigm, the Twisted Paradigm. A local problem or construction gives rise to local objects D p (s 1,, s r ) which are power series in p s1,..., p sr, convergent for Re(s i ) sufficiently large: D p (s 1,, s r ) = k 1,...,k r=0 H(p k1,...,p kr )p k1s1 krsr.

7 Euler Products and Twisted Euler Products 195 The local objects may be put together into a global object by twisted multiplicativity. This says that if then (C 1... C r, C 1...C r ) = 1, H(C 1 C 1,..., C r C r) = µ C,C H(C 1,...,C r )H(C 1,..., C r), where µ C,C is a certain root of unity that depends on C = (C 1,..., C r ) and C = (C 1,..., C r ). Then one can form the global object (multiple Dirichlet series) D(s 1,..., s r ) = H(C 1,...,C r )C s Cr sr C 1,...,C r 0 convergent for Re(s i ) sufficiently large. This series is not an Euler product. It is once again an important (and oftentimes deep) fact that the global multiple Dirichlet series D (s 1,..., s r ) = D (s 1,..., s r )D(s 1,..., s r ), where D (s 1,..., s r ) is a factor arising from the archimedean places, has analytic continuation and a functional equation under a group of transformations. This fact is typically connected to the theory of automorphic forms on a metaplectic cover on some reductive group. As in Section 2.8 above, the group of transformations of D(s 1,..., s r ) is related to a Weyl group. To illustrate this, we give one example of such a group. Consider the action on C r generated by transformations σ i defined by σ i (s 1,..., s r ) = (s 1,...,s r) with 1 s i if j = i s j = s i + s j 1/2 if j = i + 1 or j = i 1 otherwise. s j Notice that if r = 1, then this is just the action s 1 1 s 1 that is familiar from the theory of automorphic forms. Then the group of transformations generated by σ i is isomorphic to S r+1, the Weyl group of A r. It arises in one set of multiple Dirichlet series later in these notes. 3.2 The metaplectic group We have mentioned the metaplectic group, so let us say a few words about it. The n-fold metaplectic group is a central simple extension 1 µ n G G(A F ) 1

8 196 Solomon Friedberg where G is a simply connected algebraic group over F and F µ n. (It is not essential that G be simply connected.) The group is constructed by means of a 2- cocyle defined by Matsumoto [Ma] after Weil (n = 2), Moore, and Steinberg. The Matsumoto cocycle, locally a product of Hilbert symbols, is described in terms of generators and relations, and this makes computations difficult. Also, if n > 1, then G is not the adelic points of an algebraic group. 4 Examples of the Twisted Paradigm, I: Sums of Quadratic Twists 4.1 Construction of a twisted multiple Dirichlet series via metaplectic Eisenstein series We give a first example of such a series, indicate how it first arose in the literature, and describe an application. Let Ẽ(z, s) be the half-integral weight Eisenstein series Here Ẽ(z, s) = ζ(2s) j(γ, z)im(γ z) s/2+1/4. (( a b j c d ) ), z ( c = ǫ 1 d (cz + d) d) 1/2 is the multiplier for the Jacobi theta function, which involves a quadratic Kronecker symbol ( c d), and the sum is over a suitable quotient of a congruence subgroup of SL(2, Z). Though the Fourier coefficients of the classical, integral weight, Eisenstein series are Ramanujan sums, the Fourier coefficients in this metaplectic context are more complicated. Let m = m 0 m 2 1 with m 0 square-free, and let χ m0 (a) = ( m 0 ) a. Then Maass essentially showed that with m-th coefficient of Ẽ(z, s) = L(s, χ m 0 )a(s, m) times factors at 2, a(s, m) = d2 1 2s χ m0 (d 3 )µ(d 3 )d s 3. d 1d 2d 3=m 1 Here µ is the Möbius function. (We wrote essentially above since Maass worked with holomorphic Eisenstein series of higher half-integral weight; the calculation in the present case is virtually the same.) Siegel [Si] took the Mellin transform in a new variable w of Ẽ(z, s). This gives, roughly, Z(s, w) = L(s, χ m0 )a(s, m)m w, m a weighted sum of quadratic Dirichlet L-functions. Note that because this is obtained as the Mellin transform of an automorphic form, the series Z(s, w) has

9 Euler Products and Twisted Euler Products 197 continuation in w as well as in s. Goldfeld and Hoffstein [GH] used this construction to get information about the growth of L(s, χ m0 ) as one varies m 0. By sieving and analyzing the asymptotics of the archimedean contributions to this Mellin transform, they showed: Theorem 4.1. (Goldfeld-Hoffstein) 1<±m<X m squarefree c(ρ)x + O(X 1 2 +ǫ ) Re(ρ) 1 L(ρ, χ m ) = c(ρ)x + c ± (ρ)x 3 2 ρ + O(X θ+ǫ 1 ) 2 Re(ρ) < 1, ρ 1 2 c 1 X log X + c 1 X + O(X 19/32+ǫ ) ρ = where c(ρ), c ± (ρ), c 1, c 1 are explicitly given constants. 2 2 This theorem is the first example of an application of what we now view as a multiple Dirichlet series satisfying the twisted paradigm. 4.2 Connection to the twisted paradigm To explain the connection to the twisted paradigm, let us write the series studied by Siegel and Goldfeld-Hoffstein Then a calculation shows that: If (C 1 C 2, C 1 C 2 ) = 1, then Z(s, w) = L(s, χ m0 )a(s, m)m w = H(C 1, C 2 )C s 1 C w 2. H(C 1 C 1, C 2C 2 ) = χ C 1 (C 2 )χ C 2 (C 1 )H(C 1, C 2 )H(C 1, C 2 ). On prime powers, p b/2 b even, b a H(p a, p b ) = p a/2 a even, b a 0 otherwise. In some ways this is remarkable! An infinite sum of quadratic Dirichlet L- functions is obtained by creating a mutliple Dirichlet series from these two simple ingredients using the twisted paradigm. We also remark that the local factor is given by a,b H(p a, p b )p as bw = 1 + p s + p w p 2s w p s 2w p 2s 2w (1 p 2s )(1 p 2w )(1 p 1 2s 2w. (4.1) ) We shall return to this remark later.

10 198 Solomon Friedberg 4.3 Other sums of twisted L-functions The above example is a multiple Dirichlet series which may be described as a sum of quadratic twists of a GL(1) L-function (indeed, working over a number field one may sum the quadratic twists of a given Hecke character by using a suitable Eisenstein series). A great deal of work that has been done to generalize this example to higher rank and to twists of higher fixed order. We mention a few of the papers, and briefly describe how such series arise. In every case, the resulting multiple Dirichlet series satisfies the twisted paradigm. Goldfeld, Hoffstein and Patterson [GHP] used half-integral weight Eisenstein series over an imaginary quadratic field together with an Asai integral to get similar results for L-functions attached to CM elliptic curves. Though this is a GL(2) example, it is essentially obtained from GL(1) via automorphic induction. Sums of general quadratically-twisted GL(2) L-functions were first obtained and studied by Bump, Friedberg and Hoffstein [BuFH1]. We obtained the multiple Dirichlet series from an Eisenstein series on the double cover of GSp(4) by using an integral transform due to Novodvorsky. Doing so, we were able to prove that infinitely many quadratic twists of a given GL(2) L-function vanish at the center of the critical strip to order 0 resp. 1 (depending on the sign of the functional equation), a result with applications to elliptic curves 3 4. Friedberg and Hoffstein [FH] gave a different construction of the multiple Dirichlet series based on a Rankin-Selberg convolution with a GL(2) half-integral weight Eisenstein series, and we used this to generalize this nonvanishing theorem to number fields. Bump, Friedberg and Hoffstein [BuFH2, BuFH3] realized that it was not necessary to find a Rankin-Selberg integral in order to obtain such results; in fact, one can establish them directly from the functional equations for the L- functions that one is summing, by carefully summing over one index of summation, using standard convexity estimates, and invoking Bochner s theorem concerning continuation to the convex hull of a tube domain. This is often technically simpler than using a Rankin-Selberg approach. Fisher and Friedberg [FF1, FF2] used this in the case of quadratic twists of GL(1) and GL(2) over a general function field to show that the resulting multiple Dirichlet series is a rational function with prescribed denominator in these cases; we also linked the weighted series of GL(2) twists to a weighted series of GL(1) twists. Sums of GL(3) quadratic twists were taken up by Bump, Friedberg and Hoffstein [BuFH3], again using Bochner s tube theorem. A residue of this series 3 The proof of nonvanishing is this: fix π on GL(2). Then a twisted Euler product that is an infinite sum of quadratic twists, of the form P d L(s, π, χ d)a(s, π, d)d w, is shown to have a pole in w at s = 1/2 unless π is self-contragredient and the signs of the twists are all 1. If there is a pole in w at s = 1/2, then infinitely many summands must be nonzero at s = 1/2, and L(1/2, π, χ d ) 0 for infinitely many d. If π is self-contragredient and the signs are all 1, then the same is shown to be true for the partial derivative of this series with respect to s, and it follows that infinitely many L(s, π, χ d ) vanish to order exactly 1 at s = 1/2. 4 For π on PGL 2, a different proof of the nonvanishing of infinitely many L(1/2, π, χ d ), based on the Shimura correspondence, had previously been given by Waldspurger.

11 Euler Products and Twisted Euler Products 199 is shown to be the symmetric square of the GL(3) representation, so that in fact the twisted paradigm establishes the continuation of a Langlands L-function as well. In fact, there are several Rankin-Selberg integrals that would also lead to this series. Note that for rank greater than 3, the sum of quadratic twists is not expected to continue to the entire product of complex planes; see Bump, Friedberg and Hoffstein [BuFH2], Section 4, for a discussion. Sums of n-fold twists of GL(1), for fixed n > 2, were studied by Friedberg, Hoffstein and Lieman [FHL]. Once again there is a direct construction but a Rankin-Selberg construction is also available. And Brubaker, Friedberg and Hoffstein [BFH] studied sums of cubic twists of GL(2) and used this to show that the central values of cubic twists of a given GL(2) automorphic L-function again are nonvanishing infinitely often. A residue of the multiple Dirichlet series series gives the symmetric cube, so once again one obtains a new way of studying this Langlands L-function. For additional references and details concerning the above works, see the paper of Chinta, Friedberg and Hoffstein [CFH]. 5 Examples of the Twisted Paradigm, II: Sums of Gauss Sums 5.1 Eisenstein series on the n-fold cover of GL(2) and the Kubota Dirichlet series Let n > 1 be a fixed integer and F be a number field containing n n-th roots of unity. (For convenience, we assume that 1 is an n-th power, though this could be avoided.) Let ( ) d c be the n-th power residue symbol. Write O for the integers n of F. Kubota observed that the map ( ) { ( d ) a b c if c 0 κ = n c d 1 if c = 0 is a homomorphism from a congruence subgroup of SL 2 (O) to the group of n-th roots of unity. Using this, one can form a standard GL(2) Eisenstein series with this multiplier, that is, a series of the form E(g, s) = γ κ(γ) Im(γ g) s. In a more modern language, such a series may be regarded as an automorphic form on the n-fold cover of GL(2). What are the Fourier coefficients of such an Eisenstein series? We pass to a slightly different set-up, working with a finite set S of places of F, containing the archimedean ones and all those ramified over Q (including those dividing n), that is sufficiently large that the ring of S-integers O S is a principal ideal domain.

12 200 Solomon Friedberg If m, c O S are nonzero define the n-th order Gauss sum g(m, c) = ( ) ( ) d md, c c d mod c (d,c)=1 where ψ is an additive character of conductor O S. (The power residue symbol also depends on S in this set-up.) Then Kubota showed ([Ku], Section 5) that the m-th Fourier coefficient of E(g, s) is a Dirichlet series that is roughly of the form D(s, m) = 0 c O S/O S ψ n g(m, c)nc 2s (this is not exactly right, as the summand as written is not invariant under units; see Kubota [Ku] or Brubaker and Bump [BB] for exact formulas). 5.2 Properties of the series D(s, m) We make the following observations. By the Chinese remainder theorem, the Gauss sum g satisfies a twisted multiplicativity. If (c 1, c 2 ) = 1, then ( ) ( ) c1 c2 g(m, c 1 c 2 ) = g(m, c 1 )g(m, c 2 ). c 2 n c 1 n By a generalization of the classical theory of Eisenstein series, these series have analytic continuation and functional equation under s 1 s. More precisely, when completed by archimedean factors, indicated by asterisk, one has D (s, m) = Nm 1 2s D (1 s, m). Thus this series fits the Twisted Paradigm. What is the p-part of this series? Suppose for convenience that m = 1. Notice that, by elementary properties of Gauss sums, g(1, p j ) = 0 if j 2. Thus the p-part is simply H(p j )p js = 1 + g(1, p)p s. j=0 As we shall see, this is an instance of a far broader contruction, that of Weyl group multiple Dirichlet series. 6 Weyl Group Multiple Dirichlet Series 6.1 Construction of the Weyl group multiple Dirichlet series (n large) Weyl group multiple Dirichlet series are certain multiple Dirichlet series that satisfy the twisted paradigm, formed from infinite sums of n-th order Gauss sums.

13 Euler Products and Twisted Euler Products 201 They have been defined and studied by Brubaker, Bump, Chinta, Friedberg and Hoffstein [BBCFH], [BBF1] for n sufficiently large. Before giving an exact description, we describe some basic features of the construction. These series have coefficients that are products of n-th order Gauss sums. The recipe for constructing these coefficients reflects the combinatorics of a given reduced root system. The series converge absolutely for Re(s i ) sufficiently large. We show that they have analytic continuation to C r and functional equation. The series are expected to be connected to the theory of metaplectic Eisenstein series. To give more details, we need some notation. Let Φ be a reduced root system in R r, Φ + (resp. Φ ) be the positive (resp. negative) roots of Φ, Σ = {α 1,...,α r } be the set of simple roots, W be the Weyl group of Φ, ρ be the Weyl vector α Φ + α, and let <, > be a W-invariant inner product <, > on Rr. For ρ = 1 2 convenience, suppose that all short roots have norm 1 (all roots in the simply-laced case). Let n be sufficiently large (depending on Φ) ( the stable case ); we do not give the exact constraint on n here. In the above-cited papers, we define and study a series of the form r where Z Ψ (s) = c i H(c 1,, c r )Ψ(c 1,, c r ) s = (s 1,...,s r ) i=1 Nc 2si i, and the H and Ψ are certain functions. The arithmetic information is encoded in H while the Ψ runs over a finite dimensional vector space of functions on ( v S F v )r ; see [BBF1] for details. We focus on the function H below. The function H satisfies (and is determined by) the following properties. (1) The function H is twisted multiplicative: if (C 1... C r, C 1... C r) = 1, then H(C 1 C 1,, C r C r) = r ( ) αi 2 ( Ci C i i=1 C i n C i ) αi 2 n ( i<j H(C 1,, C r )H(C 1,, C r ) ) 2 αi,α j C i C j n ( C i C j ) 2 αi,α j Due to this multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. (2) The number of nonzero coefficients in the p-part is equal to the order of the Weyl group W. More precisely, H(p k1,, p kr ) 0 if and only if ki α i = ρ w(ρ) for some w W. (3) In this case H(p k1,, p kr ) is a product of l(w) Gauss sums, where l is the length function on W. n

14 202 Solomon Friedberg To keep the notation simple, we give the exact formula in the simply laced case. If ρ w(ρ) = i k iα i, then H(p k1,..., p kr ) = α Φ + w(α) Φ g(p d(α) 1, p d(α) ) where d(α) = 2 < ρ, α >. In the non-simply laced case, the coefficients involve Gauss sums formed from different powers of the residue symbol. Note that the series H(p k 1,..., p kr )p k1s1 krsr is a finite polynomial in the p si. In fact, for Φ = A 2, this polynomial is supported at the same powers of p that appeared in the numerator of (4.1) above. This will be explained below. 6.2 Continuation and functional equation Remarkably, after multiplying by a suitable factor, each Weyl group multiple Dirichlet series has continuation to C r and satisfies a functional equation, so that it does fit the twisted paradigm. The factor involves both zeta functions and Gamma functions. We omit the description, but will denote the resulting normalized multiple Dirichlet series by Z Ψ. Theorem 6.1. (Brubaker, Bump, Friedberg [BBF1])The function Z Ψ (s) has meromorphic continuation to all s C r and satisfies a group of functional equations isomorphic to the W, the Weyl group of Φ. The precise functional equations, given in [BBF1], involve an action of W on the space of functions Ψ. Let us sketch the proof of this result. Philosophically, it follows the familiar approach of reduction to the rank one case. Thus extensive use is made of the functional equation for the Kubota Dirichlet series, as refined by Brubaker and Bump [BB]; this is essentially the case Φ = A 1. To make use of this information, we utilize the combinatorics of the root system Φ to write the multiple Dirichlet series as an iterated sum where the inner sum, obtained by isolating one of the indices of summation, is in fact a Kubota Dirichlet series. That is, we write (in r different ways) ZΨ as an infinite sum of such Kubota Dirichlet series. Each such expression allows us, using the convexity estimates for the Kubota Dirichlet series that follow from their continuation and functional equation, to continue ZΨ to a larger tube domain in C r and to establish a functional equation there. The last step is to apply Bochner s tube theorem from several complex variables (this

15 Euler Products and Twisted Euler Products 203 states that a holomorphic function on a tube domain is holomorphic on the convex hull of that tube domain) to continue ZΨ (multiplied by some linear factors that remove the poles) to the convex hull of the larger tube domain. We show that this convex hull is C r, and this completes the proof. 6.3 Relation to Automorphic Forms As indicated in the Twisted Paradigm, the continuation discussed above should be explained by a connection to automorphic forms, but this time on a metaplectic cover of an algebraic group. Let G be a simply connected algebraic group over F whose root system is the dual of the root system Φ; in other words, Φ is the root system of the L-group L G. Then Brubaker, Bump and Friedberg conjecture [BBF1] Conjecture 6.1. (The Eisenstein Conjecture)The Weyl group multiple Dirichlet series Z Ψ (s 1,, s r ) is a Whittaker coefficient of an Eisenstein series on an n-fold metaplectic cover G of G(A F ). This has been proved in full for A 1, A 2 only, but additional unpublished calculations support it in other cases too. 5 We note that this Conjecture together with the continuation and functional equations of the metaplectic Eisenstein series would imply the continuation and functional equations for the Weyl group multiple Dirichlet series. However, the precise functional equations established in [BBF1] and mentioned above would only follow from a precise description of the scattering matrix for the metaplectic Eisenstein series. Even for the metaplectic covers of GL(2), this scattering matrix has only recently been given in a form suitable for this application [BB]. Actually, an Eisenstein series has many (nondegenerate) Whittaker coefficients, corresponding to the different additive characters. Can we describe them all? We shall return to this question later. 7 The Case of n Small (Relative to Φ) When n is not large, one may still construct Weyl group multiple Dirichlet series (WMDS) that satisfy the twisted paradigm and whose prime-power coefficients are sums of n-th order Gauss sums. However, the details are considerably more complicated, and in some cases they are still being worked out. Let us introduce this topic by describing two approaches. First, for Φ = A r, Brubaker, Bump, Friedberg and Hoffstein [BBFH1] have given a description of Weyl group multiple Dirichlet series coefficients for all n, not just n sufficiently large. The description is made by attaching Gauss sums to certain combinatorial objects called Gelfand-Tsetlin patterns. (The Weyl group multiple Dirichlet series are more general than those above in that they should 5 Recently, Brubaker, Bump and the author have found a proof for A r for all r [BBF5]. It is possible that our approach will extend to other groups.

16 204 Solomon Friedberg describe all coefficients of the underlying metaplectic Eisenstein series and not only the first.) We will describe this in more detail soon. Second, Chinta and Gunnells [CG1, CG2] have given a definition of the WMDS that works for any root system Φ. In contrast to our work, their construction is not coefficient-by-coefficient. Rather they construct the entire p-part as an average over the finite group W. The relation between these two approaches is a matter of on-going investigation; see Chinta, Friedberg and Gunnells [CFG]. The work in [BBFH1] is based on detailed computations in the case Φ = A 2. We turn to a description of this, which will then help motivate the general description of the series given in [BBFH1]. 8 A 2 Metaplectic Computations 8.1 Overview In [BBFH1], we compute the (m 1, m 2 )-th Whittaker coefficient of the minimal parabolic metaplectic Eisenstein series on GL(3). We find that (m 1, m 2 )-th coefficient= H Wh (C 1, C 2 ; m 1, m 2 )C s1 1 C s2 2. C 1,C 2 Though this coefficient is given by a complicated exponential sum, in fact it has remarkable properties. We show: The coefficients H Wh satisfy twisted multiplicativity. Hence the general coefficients may be determined by the coefficients H Wh (p k1, p k2 ; p l1, p l2 ) for prime p. The (1, 1)-coefficient H Wh (p k1, p k2 ; 1, 1) is nonzero if and only if (k 1, k 2 ) is one of the six points (0, 0), (1, 0), (0, 1), (2, 1), (1, 2), (2, 2). This is the stable WMDS case discussed above; 6 = W(A 2 ). Notice that these are exactly the six points where the numerator of the Siegel-Goldfeld- Hoffstein series L(s, χ m0 )a(s, m)m w is located compare Eqn. (4.1) above. This corresponds to the (1,1)-Whittaker coefficient of the GL(3) metaplectic Eisenstein series. Thus this identification holds because one may obtain the same series, satisfying the twisted paradigm, by taking the Mellin transform of a GL(2) metaplectic Eisenstein series or by taking the Whittaker coefficient of a GL(3) metaplectic Eisenstein series. This remarkable observation was made many years ago by Bump and Hoffstein. For general coefficient, the vertices of this hexagon expand. More precisely, if n is sufficiently large, then H Wh (p k1, p k2 ; p l1, p l2 ) is nonzero if and only if (k 1, k 2 ) is one of the six points (0, 0), (L 1, 0), (0, L 2 ), (L 1 + L 2, L 2 ), (L 1, L 1 + L 2 ), (L 1 + L 2, L 1 + L 2 ),

17 Euler Products and Twisted Euler Products 205 where L i = l i + 1. For smaller n, other coefficients appear. These coefficients are contained in the convex hull of the stable support, i.e. in the convex hull of this hexagon. When (k 1, k 2 ) is in this convex hull, the coefficient H Wh (p k1, p k2 ; p l1, p l2 ) is a sum of products of Gauss sums. 8.2 A rough description of the interior A 2 coefficients We now give a description of the coefficients corresponding to indices in the interior of the convex hull. (These coefficients vanish if n is sufficiently large, and this is why the stable case is easier.) As above, rather than giving complicated but difficult formulas, we begin with a less detailed but more conceptual and surprising description. The coefficients may be described in terms of combinatorial objects called Gelfand-Tsetlin patterns! The description is as follows. (1) To each (k 1, k 2 ) in the hexagon, we assign a set of combinatorial objects called strict Gelfand-Tsetlin patterns. Gelfand-Tsetlin patterns arise in constructing a specific basis for the representation of GL(3, C) of given highest weight, and the strict patterns are a subset. (2) To each Gelfand-Tsetlin pattern T we assign a product of Gauss sums G(T). 8.3 Main Theorem for A 2 In [BBFH1], we show Theorem 8.1. The coefficient H Wh (p k1, p k2 ; p l1, p l2 ) is given by H Wh (p k1, p k2 ; p l1, p l2 ) = T G(T), where the sum is over all strict Gelfand-Tsetlin patterns T attached to (k 1, k 2 ). 8.4 A 2 coefficients: detailed description Here is a complete description of the A 2 coefficients at a given prime p. The notation continues as in the previous subsection. This material is taken directly from [BBFH1]. A Gelfand-Tsetlin pattern is a triangular-shaped array of integers a 00 a 01 a 02 a 0r a 11 a 12 a 1r with r rows that interleave; that is, a i 1,j 1 a i,j a i 1,j. The pattern is called strict if each row is strictly decreasing. a rr

18 206 Solomon Friedberg For the rank 2 case, we are concerned with strict Gelfand-Tsetlin patterns of the form l 1 + l l T = a b. c This pattern is attached to (k 1, k 2 ) provided k 1 = a + b l 2 1, k 2 = c. To a pattern T we assign the product of Gauss sums G(T) = g(p a b 1, p c b ) g(p l2, p b )g(p l1+b, p a+b l2 1 ) unless a = l 2 + 1; in that case we let l 1 + l l G l b = Np b g(p a b 1, p c b ) g(p l2, p b ). c For example, consider the coefficient H Wh (p 4, p 4 ; p, p 3 ). The following two patterns contribute to this case: T 1 = 6 2, T 2 = 4 The formula defining G(T) gives G(T 1 ) = g 32 g 32 g 34, G(T 2 ) = g 11 g 33 g 44, where g ij = g(p i, p j ). And indeed a computation shows that H Wh (p 4, p 4 ; p, p 3 ) = g 32 g 32 g 34 + g 11 g 33 g The A 2 computation We close Section 8 with a description of the computation. We include this so that the reader may see exactly how the metaplectic group enters, and also get a sense for the complexity of these objects. Let n = 3, F = Q(ρ) with ρ = e 2πi/3, λ = 1 ρ, and f = (λ 2 ). Let Γ r (f) denote the principal congruence subgroup of SL r (O). Kubota showed that the map κ 2 : Γ 2 (f) C given by ( a κ 2 c ) { ( c ) b d if c 0 = d 1 if c = 0

19 Euler Products and Twisted Euler Products 207 where ( c d) denotes the cubic residue symbol, is a homomorphism. One can extend κ 2 to a homomorphism κ 3 of Γ 3 (f). Let f s be a smooth function on SL 3 (C) satisfying f s y 1 y 2 g = y 1 2s2 y 3 2s1 f s (g). y 3 Then the metaplectic Eisenstein series is given for Re(s 1 ), Re(s 2 ) 0 by E f (g,s) = γ Γ 3, (f)\γ 3(f) κ 3 (γ)f s (γg), with Γ 3, (f) is the set of upper triangular unipotents in Γ 3 (f). The (m 1, m 2 )-th Whittaker coefficient of E f (g,s) is given by f 3 \C 3 E f w 1 x 1 x 3 1 x 2 g,s ψ( m 1 x 1 m 2 x 2 )dx 1 dx 2 dx 3 1 where w is the long element of the Weyl group. To compute these coefficients, we must get a formula for κ 3, which is closely related to the Matsumoto cocycle. It suffices to describe κ 3 on Γ 3, (f)\γ 3 (f), and this space may be described by Plücker invariants. Recall that given γ = (a ij ) SL 3 (F), its the Plücker invariants are the six-tuple (A 1, B 1, C 1, A 2, B 2, C 2 ) of bottom 1 1 and 2 2 minors of γ: A 1 = a 31 B 1 = a 32 C 1 = a 33 A 2 = a 21 a 22 a 31 a 32 B 2 = a 21 a 23 a 31 a 33 C 2 = a 22 a 23 a 32 a 33. Theorem 8.2. Suppose that γ Γ 3 (f) has Plücker invariants as above. Then there exists a factorization C 1 = r 1 r 2 C 1 C 2 = r 1 r 2 C 2 B 1 = r 1 B 1 B 2 = r 2 B 2, where r 1 r 2 C 1 C 2 1 modulo f, and gcd(c 1, C 2) = 1. We have gcd(b 1, C 1) = gcd(b 2, C 2) = gcd(a 1, r 1 ) = gcd(a 2, r 2 ) = 1 and ( )( ) ( ) B κ 3 (γ) = 1 B 2 C 1 ( ) ( ) 1 A1 A2. C 1 C 2 C 2 r 1 r 2

20 208 Solomon Friedberg This formula is proved in [BBFH2]; for related formulas see Proskurin [Pr]. Then, we must compute the exponential sums ( m1 B 1 H Wh (C 1, C 2 ; m 1, m 2 ) = κ 3 (γ)ψ + m ) 2B 2. C 1 C 2 A 1,B 1modC 1 A 2,B 2modC 2 gcd(a 1,B 1,C 1)=1 gcd(a 2,B 2,C 2)=1 A 1 B 1 A 2 B 2 0 modf A 1C 2+B 1B 2+C 1A 2 0 modc 1C 2 In [BBFH1], we show that this exponential sum may be expressed as a sum of products of Gauss sums where the sum is parametrized by Gelfand-Tsetlin patterns, as described above! 9 Generalization to A r In their 2007 paper [BBFH1], Brubaker, Bump, Friedberg and Hoffstein formulated a conjectured generalization of this to A r. We defined coefficients H by means of twisted multiplicativity and a formula involving assigning products of n-th order Gauss sums to Gelfand-Tsetlin patterns. (We suppress the formulas here; see [BBFH1].) This allowed us to define new, more general, multiple Dirichlet series Z Ψ (s) for all n including n not stable. We made the following two conjectures. Conjecture 9.1. Z Ψ (s) has meromorphic continuation to all s and satisfies the functional equations above. Conjecture 9.2. Z Ψ is a Whittaker coefficient of an Eisenstein series on the n-fold metaplectic cover of GL r+1. Notice that Conjecture 9.2 asserts that there is a uniform description of the Whittaker coefficients of the GL r+1 minimal parabolic n-fold metaplectic Eisenstein series that is valid simultaneously for all n, including n = 1 (the nonmetaplectic case). At the moment there is no conceptual reason why such a description should exist. Nonetheless this description raises the question as to whether a non-metaplectic cuspidal automorphic representation could ever be fit into a similar family, with a similar uniform description for the Whittaker coefficients in the family. For n = 1, the cover is trivial, and one can obtain Conjectures 9.1 and 9.2 by using the formula of Shintani for the p-adic Whittaker function on GL r+1 (this formula was extended to more general groups by Casselman and Shalika and often carries their names) together with a result of Tokuyama. This was carried out in [BBFH1]. In that paper, these conjectures were also proved for r = 2, and Conjecture 9.1 was proved for n = 2 and r 5 by reconciling it with work of Chinta [Ch]. In [BBF2], Brubaker, Bump and Friedberg proved Conjecture 9.1 for all r in the twisted stable case, n sufficiently large. We also linked the approaches of [BBF] (root systems) and [BBFH1] (Gelfand-Tsetlin diagrams), and showed that they give the same formulas when they both obtain.

21 Euler Products and Twisted Euler Products 209 Finally, in the past year, Brubaker, Bump and Friedberg have been able to prove these conjectures in full. In [BBF3], [BBF4], we give a proof of Conjecture 9.1. The proof is combinatorial; in particular, it makes use of the Schützenberger involution on Young tableaux (this involution was translated into the context of Gelfand-Tsetlin patterns by Kirillov and Berenstein). Moreover, we show that the crucial combinatorial information that is packaged into the coefficients G(T) is related to information obtained from crystal graphs, using a result of Littelmann [Li]. This connection is rather striking. Very recently, we have been able to obtain a direct proof of Conjecture 9.2 [BBF5]. As noted above, the form of Conjecture 9.1 that we prove in [BBF4] is stronger than the result that follows from Conjecture 9.2, since the form of Conjecture 9.1 that we establish gives a precise description of the scattering matrix in full generality. Generalizations to other parabolics and to other groups are under investigation. Just as the study of the Whittaker coefficients of non-metaplectic Eisenstein series leads to a rich class of Euler products (Langlands-Shahidi theory), this work suggests that the study of the Whittaker coefficients of metaplectic Eisenstein series will lead to a rich class of twisted Euler products. 10 Concluding Remarks The constructions above indicate that twisted Euler products, that is, multiple Dirichlet series satisfying the twisted paradigm, arise in many ways. Recall that in the context of reductive groups, both Rankin-Selberg integrals and the Whittaker coefficients of Eisenstein series give rise to Euler products with continuation and functional equation. As we have now described, in the context of metaplectic groups the same methods, judiciously applied, give rise to twisted Euler products with similar properties. It is reasonable to expect that other features of the nonmetaplectic case generalize; for example, analogously to the existence of Eulerian Rankin-Selberg integrals that are not of Langlands-Shahidi type, there should be twisted Euler products with functional equation that arise via metaplectic Rankin- Selberg type integrals but which are not themselves the Whittaker coefficients of metaplectic Eisenstein series. But at the moment no such examples are known. Nor we do not have a complete description of all series that should arise, nor an analogue of functoriality. This seems a rich area for further investigations. In particular, one wonders if there are arithmetic or geometric situations that give rise to twisted Euler products with continuation and functional equations. References [AC] J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989.

22 210 Solomon Friedberg [BCDT] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercices, JAMS 14 (2001), [BB] B. Brubaker and D. Bump, On Kubota s Dirichlet series, J. Reine Angew. Math. 598 (2006), [BBCFH] B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series I, in: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory (D. Bump, S. Friedberg, D. Goldfeld, and J. Hoffstein, eds.), Proc. Sympos. Pure Math., 75, Amer. Math. Soc., Providence, R.I., 2006, pp [BBF1] B. Brubaker, D. Bump, and S. Friedberg., Weyl group multiple Dirichlet series II. The stable case, Invent. Math. 165 (2006), [BBF2], Twisted Weyl group multiple Dirichlet series: the stable case, in: Eisenstein Series and Applications (Gan, Kudla and Tchinkel, eds.), Progress in Mathematics Vol. 258, Birkhäuser, Boston, 2008, pp [BBF3], Gauss sum combinatorics and metaplectic Eisenstein series, to appear in the Volume in honor of the 60th birthday of Steve Gelbart, Isr. Math. Conf. Proceedings. [BBF4], Weyl group multiple Dirichlet series and Gelfand-Tsetlin patterns, preprint (Jan. 2008). [BBF5], Manuscript in preparation. [BBFH1] B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable A r, Ann. Math. 166 (2007), [BBFH2], Metaplectic Eisenstein series on GL(3), unpublished manuscript, available on the author s website. [BFH] B. Brubaker, S. Friedberg and J. Hoffstein, Cubic twists of GL(2) automorphic L-functions, Invent. Math. 160 (2005), [Bu] D. Bump, The Rankin-Selberg method: an introduction and survey, in: Automorphic Representations, L-functions and Applications: Progress and Prospects (Cogdell, Jiang, Kudla, Soudry, and Stanton, eds.), Ohio State University Mathematical Research Institute Publications, vol. 11, de Gruyter, 2005, pp [BuFH1] D. Bump, S. Friedberg and J. Hoffstein, Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives, Ann. Math. 131 (1990), [BuFH2], On some applications of automorphic forms to number theory, Bull. A.M.S. 33 (1996), [BuFH3], Sums of twisted GL(3) automorphic L-functions, in: Contributions to Automorphic Forms, Geometry and Arithmetic (H. Hida, D. Ramakrishnan and F. Shahidi, eds.), Johns Hopkins Univ. Press, 2004, pp [Ch] G. Chinta, Mean values of biquadratic zeta functions, Invent. Math. 160 (2005),