NON-VANISHING TWISTS OF GL(2) AUTOMORPHIC L-FUNCTIONS

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1 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN Abstract. Let π be a cuspidal automorphic representation of GL2, A K. Suppose there exists a single non-anishing n th order twist of the L-series associated to π at the center of the critical strip. We use the method of multiple Dirichlet series to establish that there exist infinitely many such non-anishing n th order twists of the L-series of the representation at the center.. Introduction and Statement of the Main Result Let E be an elliptic cure defined oer a number field K. The behaior of the rank of the L-rational points EL as L aries oer some family of algebraic extensions of K is a problem of fundamental interest. The conjecture of Birch and Swinnerton-Dyer proides a means to inestigate this problem ia the theory of automorphic L-functions. Assume that the L-function of E coincides with the L-function Ls, π of a cuspidal automorphic representation of GL2 of the adele ring A K. Let L/K be a finite cyclic extension and χ a Galois character of this extension. Then the conjecture of Birch and Swinnerton-Dyer equates the rank of the χ-isotypic component EL χ of EL with the order of anishing of the twisted L-function Ls, π χ at the central point s = 2. In particular, the χ-component EL χ is finite according to the conjecture if and only if the central alue L 2, π χ is non-zero. Thus it becomes of arithmetic interest to establish non-anishing results for central alues of twists of automorphic L-functions by characters of finite order. For quadratic twists this problem has receied much attention in recent years. In this paper we address this question for twists of higher order. Our main result is contained in the following theorem. Theorem.. Fix a prime integer n > 2, a number field K containing the n th roots of unity, and a sufficiently large finite set of primes S of K. Let π be a self-contragredient cuspidal automorphic representation Date: October 2, Mathematics Subject Classification. Primary F.

2 2 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN of GL2, A K which has triial central character and is unramified outside S. Suppose there exists an idèle class character χ 0 of K of order n unramified outside S such that L 2, π χ 0 0. Then there exist infinitely many idèle class characters χ of K of order n unramified outside S such that L 2, π χ 0. We refer the reader to Section 2 for the definition of the finite set S. Fearnley and Kisilesky hae proen a related result for the L-function Ls, E of an elliptic cure defined oer Q. In [5] they show that if the algebraic part L alg 2, E of the central L-alue is nonzero mod n, then there exist infinitely many Dirichlet characters χ of order n such that L 2, E, χ 0. We note that if L 2, E 0 then the hypothesis L alg 2, E 0 mod n is satisfied for all sufficiently large primes n. We find it interesting and frustrating! that, although the methods of [5] based on the arithmetic of modular symbols are completely different from the methods of this paper, both our result and theirs require some nonanishing assumption. An unconditional result in the cubic case n = 3 has recently been established in []. We comment more on this below. The quadratic case n = 2 is particularly accessible because, by the results of Waldspurger [7], Kohnen and Zagier [3], and others, the existence of a quadratic character χ such that L 2, π χ 0 implies the existence of a metaplectic cuspidal automorphic representation π on the double coer of GL2, A K corresponding to π. The correspondent π is related to π in the following way. If Lw, π π denotes the Rankin-Selberg conolution of π with itself then. Lw, π π = d 0 L 2, π χ2 d 0 P d π Nd w, up to corrections at a finite number of places. Here d = d 0 d 2 2, with d 0 square free, the χ 2 d 0 are quadratic characters with conductor d 0, and the P d π are certain non-zero correction factors which are triial when d 2 =. We defer a precise definition of such objects until Section 2. These correction factors are small in the sense that, for any fixed d 0,.2 the sum d 2 0 P d0d 2π 2 conerges absolutely for any w with Rew > 2. Nd 2w 2

3 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 3 This connection between π and π causes the existence of one non-anishing quadratic twist to imply the existence of infinitely many d 0 such that L 2, π χ2 d 0 0. This is because if π 0 then Lw, π π has a pole at w =. Howeer the right hand side of. will conerge at w = if there are only finitely many non-anishing quadratic twists. For n > 2 there are no known results relating n th order twists of the L-series of π to Fourier coefficients of other automorphic objects. In fact een a conjectural generalization of the results of Waldspurger to the case n > 2 remains mysterious. Howeer, in this paper we describe how a generalization of. can still be found by associating π to a certain metaplectic form. This generalization is at least sufficient to answer the question of whether one non-anishing twist of a gien order implies the existence of infinitely many non-anishing twists of that order. It may ultimately shed some light on the question of the correct generalization of Waldspurger s results, but at the moment this aspect remains opaque. We will describe in detail a Dirichlet series that has the rough form.3 Z n s, w = d 0 Ls, π χ n d 0 ɛd 0 P d s, π Nd w, where d = d 0 d n n with d 0 n th power-free see Section 2. Here ɛd 0 denotes an n th order Gauss sum corresponding to the character χ n d 0 d n =. These are also small, in the sense that for Res 2,.4 the sum d n 0 and P d s, π again denotes certain correction factors which are triial when P d0d n s, π n Nd nw n conerges absolutely for any w with Rew > n + 9. The fraction 9 comes from the bound of Kim and Shahidi [2]. The series Z n s, w is natural for the following reasons. First, when n = 2 and π exists, Z n 2, w agrees at almost all places with the Rankin-Selberg conolution Lw, π π. Second, after an interchange in the order of summation, it has an automorphic interpretation as a Rankin-Selberg conolution of π with an Eisenstein series on the n-fold coer of GL2. In the case n = 2, this automorphic interpretation of Z n s, w was exploited by Friedberg and Hoffstein [8]. In the case n = 3 the automorphic interpretation was used by She [6] to establish a non-anishing result for cubic twists of one particular π. In this paper we do not use the automorphic interpretation of Z n s, w. Instead we take the far easier approach of the method of multiple Dirichlet series discussed in brief at the conclusion of this section.

4 4 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN Using this method we establish an analytic continuation and exhibit a finite group of functional equations for Z n s, w in the two ariables s and w. Specializing to s = 2, we obtain a Dirichlet series Zn 2, w with a functional equation in w. The condition.4 implies that if L 2, π χn d 0 0 for only finitely many d 0 then Z n 2, w must conerge for Rew > n + 9. We then show that for n 3 i.e. n + 9 < 2, this is incompatible with the functional equation. It immediately follows that the existence of one non-anishing twist implies the existence of infinitely many. The method can easily be taken a bit further to establish a mean alue result of the form, for Res > 2, n Nd.5 Ls, π χ d 0 ɛd 0 P d 2, π W cs, πx 2 + n, X where W is any suitable smoothing function. The constant cs, π is a ery interesting function; it is a simple multiple of Ls + 2n, π θn, the Rankin-Selberg conolution of π with the theta function on the n-fold coer of GL2, ealuated at the point s + 2n. When n = 2 and s = 2, the series L 3 4, π θ2 is essentially the symmetric square L-series of π ealuated at, the edge of the critical strip. Thus in this case the L-series does not anish, and simple conditions on the sign of the functional equation of π determine whether or not c 2, π equals zero. Because of this, unconditional mean alue and non-anishing results can be deried, as was done in [8]. For n > 2, howeer, the L-series L 2 + 2n, π θn does not hae an Euler product and is ealuated at a point inside the critical strip. Thus the question of anishing becomes quite subtle. It is because of this that we cannot yet eliminate the possibility that the entire class of twists L 2, π χn d 0 anishes identically. In [] a different multiple Dirichlet series is constructed, specific to the case n = 3. In this case the cubic Gauss sum is remoed from the numerator and the constant cs, π becomes essentially L3s, π, sym 3. As a consequence unconditional non-anishing and mean alue results can be obtained in the case n = 3. The question of generalizing this method to n 4 remains open and extremely interesting. We close the introduction with a brief oeriew of the method of multiple Dirichlet series. Multiple Dirichlet series are functions of seeral complex ariables of the form am, m 2,..., m r m s m,...,m ms2 r 2. msr r These can be considered, according to the order of summation, as a Dirichlet series in any one of the ariables whose coefficients are again Dirichlet series. For example, in.3 the multiple Dirichlet series Z n s, w is a Dirichlet series in the ariable w with numerator Ls, π χ n d 0 ɛd 0 P d s, π, a family of Dirichlet series in

5 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 5 the ariable s. If a component Dirichlet series possesses a functional equation, then the multiple Dirichlet series inherits a corresponding functional equation. Interchanges in the order of summation may reeal new families of Dirichlet series in the numerator with new functional equations. Interchanging the order of summation in.3 produces a Dirichlet series formed from n th order Gauss sums. Such series arise in the theory of Eisenstein series on the n-fold coer of GL 2 as introduced by Kubota in [4], and extended by Patterson [5] and Kazhdan-Patterson []. Classical conexity estimates on the constituent Dirichlet series gie a region of absolute conergence for the multiple Dirichlet series. Once exact functional equations are obtained, one can apply them to the domain of conergence to obtain a new domain which has a non-empty intersection with the original. This proides the analytic continuation to the union of the original domain and its translates. An analytic continuation to the conex hull of this union follows from a conexity theorem for seeral complex ariables. In the case of Z n s, w, we will show that we obtain a region whose conex hull is all of the complex space C 2. This approach was first detailed by Bump, Friedberg, and Hoffstein in [2] and [3] where instances of multiple Dirichlet series possessing these properties were catalogued. Fisher and Friedberg [6] generalized these arguments to quadratic twists of automorphic forms on GL2 oer arbitrary function fields. This method was also carried out by Friedberg, Hoffstein and Lieman [9] on a multiple Dirichlet series whose coefficients were weighted n th order Dirichlet L-series in order to determine mean-alue estimates for these L-series. Finally, see [] for a different and considerably more complicated construction in the case n = 3 and GL2. The authors would like to thank Sol Friedberg and Adrian Diaconu for many helpful discussions. 2. Preliminaries and Outline of Method Fix n > 2 and let K be a number field containing the n th roots of unity. Let O denote the ring of integers of K. Let π be a cuspidal automorphic representation of GL2, A K. Let S f be a finite set of non-archimedean places such that S f contains all places diiding n, the ring of S f -integers O Sf has class number, and π is unramified outside S f. Let S denote the set of archimedean places and let S = S f S. Let a be the power residue symbol attached to the extension K n a of K. We extend the n th power residue symbol as in Fisher and Friedberg [6]. We reiew the definition.

6 6 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN For each place, let K denote the completion of K at. For nonarchimedean, let P denote the corresponding ideal of O, and let q = NP denote its norm. Let C = S f P n with n sufficiently large so that if a K, and ord a n, then a K n. Let H C be the ray class group modulo C and let R C = H C Z/nZ. Write the finite group R C as a direct product of cyclic groups, choose a generator for each, and let E 0 be a set of ideals of O prime to S which represent these generators. For each E 0 E 0 choose m E0 K such that E 0 O Sf = m E0 O Sf. Let E be a full set of representaties for R C of the form E 0 E 0 E n E 0 0, with n E0 Z. If E = E 0 E 0 E n E 0 0 is such a representatie, then let m E = E 0 E 0 m n E 0 E 0. Note that EO Sf = m E O Sf for all E E. For conenience we suppose that O E and m O =. Let J S denote the group of fractional ideals of O coprime to S f. Let I, J J S be coprime. Write I = meg n with E E, m K, m mod C, and G J S such that G, J =. Then as in [6], the n th power residue symbol mm E J is defined, and if I = m E G n is another such decomposition, then E = E and m m E J = mm E J. In iew of this define the n th power residue symbol I J by I J = mme J. If I is n th -power-free, we denote by χ I the character χ I J = I J. This character depends on the choices aboe, but we suppress this from the notation. Proposition 2. Reciprocity. [6] Let I, J J S be coprime, and αi, J = χ I Jχ J I. Then αi, J depends only on the images of I and J in R C. Let IS denote the integral ideals prime to S f. Let π be as aboe and let L S s, π χ J be the L-function for π twisted by the character χ J, with the places in S remoed. Note that the Euler factor is also at the places diiding J. If ξ is any idèle class character then the twisted L-function Ls, π ξ satisfies a functional equation 2. Ls, π ξ = ɛs, π ξ L s, π ξ, where ɛs, π ξ is the epsilon factor of π ξ. Proposition 2.2. Let E, J OS be n th -power-free with associated characters χ E, χ J of conductors f E, f J respectiely. Suppose that χ J = χ E χ I with I K, I mod C. Then 2.2 ɛs, π χ J = ɛ/2, χ I 2 χ π f J /f E Nf J /Nf E 2/2 s ɛs, π χ E.

7 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 7 Here ɛ/2, χ I is gien by a normalized n th order Gauss sum, as in Tate s thesis. We henceforth assume that π has triial central character and is self-contragredient. Let L S s, π = α q s β q s = / S I IS ai NI s, where α and β are the Satake parameters associated to π at. For J in IS, write J = J 0 J n n, with J 0 the n th power free part of J. For I in IS, let Ĩ represent the part of I coprime to J 0. For ideals I and J, define the function GI, J by where, for α, β 0, GI, J = GP α, P β, ord I=α ord J=β if β = 0 q if α β, β 0n, β > GP α, P β = q β/2 if α = β, β 0n, β > 0 q β/2 q β /2 if α = β, β 0n, β > 0 0 otherwise. To simplify notation, let ζ denote the Dedekind zeta function of K and ζ S the zeta function with the places in S remoed. Define the following pair of multiple Dirichlet series: 2.4 Z s, w; π, ψ, ψ 2 = ζ S nw n/2 + and 2.5 Z 2 s, w; π, ψ, ψ 2 = ζ S nw n/2 + I,J IS I,J IS aiψ Iψ 2 JGI, Jχ J0 Ĩ ɛj 0 NI s NJ w, aiψ Iψ 2 JGI, Jχ J0 ĨɛJ 0 NI s NJ w, where ψ, ψ 2 are two idèle class characters of R C hence of order diiding n and conductor diiding C. The notation ɛj 0 simply abbreiates ɛ 2, χ J 0. Note that Z and Z 2 are essentially dual objects up to conjugation of χ J0 and ɛj 0. By summing first oer J, we hae 2.6 Z s, w; π, ψ, ψ 2 = I IS aiψ IDw, I, ψ 2 NI s,

8 8 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN where 2.7 Dw, I, ψ 2 = ζ S nw n/2 + J IS ψ 2 JGI, Jχ J0 ĨɛJ 0 NJ w is a Dirichlet series obtained from the Fourier coefficient of an Eisenstein series defined on an appropriately restricted congruence subgroup Γ of the n-fold coer of GL2. This motiates the definition 2.3. These metaplectic Eisenstein series were first formulated by Kubota cf. [4] and were studied in further detail by Kazhdan and Patterson in []. In particular, Kazhdan and Patterson exhibited a functional equation as w w and determined the polar structure: Dw, I, ψ 2 has possible simple poles at w = 2 ± n and is holomorphic elsewhere. For a general introduction to the subject, we refer the reader to [0]. Using the aboe theory together with one-ariable conexity results, one sees that for eery ɛ > w 2 + n w 2 n Dw, I, ψ 2 <<ɛ max{, NI Rew/2+ɛ, NI /2 Rew+ɛ }. Hence, we can obtain a region of absolute conergence for our multiple Dirichlet series as a conolution of a GL2 automorphic L-series and the aboe series Dw, I, ψ 2. That is, for i =, 2 we define 2.9 Zi s, w; π, ψ, ψ 2 = s sw + 2 n w + 2 n Z is, w; π, ψ, ψ 2, and it follows from 2.8 that Z i s, w; π, ψ, ψ 2 for i =, 2 conerges absolutely and uniformly on compacta and uniformly bounded away from the boundary for the region 2.0 R = {s, w Res > max{ 0 9, 29 8 Rew 2, 29 8 Rew} }. This is demonstrated carefully in Section 5. Our multiple Dirichlet series hae another fruitful interpretation upon interchanging the order of summation, so that the inner sum is oer ideals I IS. To present this form, first define the correction polynomials Qs, J; π, χ J0 ψ, for ideals J = P ordj, by 2. Qs, J; π, χ J0 ψ = where 2.2 Qs, P nγ, χ; π = ap nγ ord J=nγ q nγ 2 nγs n Qs, P nγ, χ J0 ψ ; π ap nγ k= ord J=nγ+k χp q nγ 2 nγ+s ap nγ χp q nγ 2 nγ s ap nγ+k ψ P k nγ+k nγ+k s 2 q + ap nγ 2, q nγ 2 nγs,

9 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 9 and where we make the conention that ax = 0 for all non-integral x. Then we will show the following result in Section 3. Proposition 2.3. In the region of absolute conergence R gien in 2.0, 2.3 Z s, w, π; ψ, ψ 2 = ζ S nw n/2 + and 2.4 Z 2 s, w, π; ψ, ψ 2 = ζ S nw n/2 + J IS J IS ɛj 0 ψ 2 JL S s, π χ J0 ψ NJ w Qs, J; π, χ J0 ψ, ɛj 0 ψ 2 JL S s, π χ J0 ψ NJ w Qs, J; π, χ J0 ψ. Gien the results of Proposition 2.3, we can use upper bounds on the Fourier coefficients of our L-series and the finite Dirichlet polynomials Q, together with standard one-ariable conexity arguments, to show that the functions Z i s, w, π; ψ, ψ 2 for i =, 2 conerge absolutely and uniformly on compacta in the region R = {s, w Rew > max{, 9 9 Res, 2 2 Res} }. Since R and R hae a non-empty intersection, we see that the functions Z i s, w, π; ψ, ψ 2 for i =, 2 conerge absolutely and uniformly on compacta in their union, gien by 2.5 R = {s, w Rew > max{2 2 Res, 9 9 Res, 29 9 Res 2, 29 8 Res} }. We use the two interpretations of the multiple Dirichlet series to exhibit functional equations as w w and s s. Translating the region R under these equations will lead to an analytic continuation. We begin with the form of the series as a sum of metaplectic Eisenstein series as written in 2.6. By adding in the contributions at the infinite places, we can state a precise formulation of the functional equation inherited by the multiple Dirichlet series from the Eisenstein series. Define 2.6 Γ n w def = [2π /2 n nw n +] r n 2 2 i= Γw 2 + i n D K nw n 2 +, where D K denotes the discriminant of the field K and r 2 is the number of pairs of complex embeddings. This set of gamma factors comes directly from the Fourier analysis and multiplication formula for the gamma function. Then Γ n wdw, I, ψ 2 has a functional equation as w w, which we exploit to obtain the following proposition.

10 0 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN Proposition 2.4. In the region of absolute conergence R gien in 2.5, Γ n wz s + w /2, w; π, ψ, ψ 2 q n/2 nw = Φw, ψ 2, ξγ n wz s, w; π, ψ ψ 2 ξ, ψ 2, S f ξ where each Φw, ψ 2, ξ is a function of w which is bounded in ertical strips of bounded width. The sum is taken oer all characters ξ with conductor diiding C and order diiding n. Proof: This follows as an immediate corollary of the functional equation for Γ n wdw, I, ψ 2 gien as Corollary II.2.4 of []. Note the necessity of twisting by characters ψ and ψ 2 so that as the functional equation takes Eisenstein series to a linear combination of Eisenstein series at each cusp, the form of our basic Dirichlet series remains the same. Our Φw, ξ is then essentially a scattering matrix for this functional equation. Now equipped with one functional equation, we go in search of a second. By interchanging the order of summation, decomposing the sums in 2.4 and 2.5 according to primes diiding J, we will iew our Dirichlet series as weighted sums of L-series in s associated to π. Thus our series inherit functional equations as s s. To make this precise, we must first include the appropriate Gamma factors which complete the L-series. Define Γ K s by s DK 2.7 Γ K s = Γs + iν r2 Γs iν r2, 2π r2 where again D K is the discriminant of K, r 2 denotes the number of pairs of complex embeddings in our totally complex field K, and 4 + ν2 is the eigenalue corresponding to the automorphic representation π. Proposition 2.5. In the region of absolute conergence R, 2.8 S f α ψ P n q n ns = β ψ P n q n ns Γ K sζ S nw + 2ns 3n 2 + Z s, w, π, ψ, ψ 2 ξ ˆR C Bs; ψ, ξγ K sζ S nw n 2 + Z 2 s, w + 2s ; π, ψ, ψ 2 ψ 2 ξ where the product of functions Γ K sbs; ψ, ξ is bounded in ertical strips of bounded width. The proof is completed in Section 4. Because we will apply both functional equations to our multiple Dirichlet series in order to obtain an analytic continuation, we would like to define Γ-factors at the infinite

11 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS places which are inariant under both functional equations as s s and w w. Thus, according to the preious propositions, we define 2.9 Γs, w = Γ K sγ K s + w /2Γ n wγ n w + 2s, and define 2.20 Z i s, w; π, ψ, ψ 2 = Γs, wζ S nw + 2ns 3n 2 + Z is, w; π, ψ, ψ 2 for i =, 2. The pair of functional equations from Propositions 2.4 and 2.5, repeatedly applied to a region of absolute conergence, proide an analytic continuation to all of C 2. This is demonstrated in Section 5. We will show in Section 6 that the resulting expression is uniformly conergent in some right half-plane. The functional equation together with this conergence will allow us to show in Section 7 that if there is a single nonanishing twist at s = 2, there must in fact be infinitely many nonanishing twists at s = Interchanging the Order of Summation A Proof of Proposition 2.3 Proof of Proposition 2.3: We gie the proof for 2.3, noting that the proof of 2.4 follows identically. Fix an ideal J in IS and decompose it according to J = J 0 J n n where J 0 = J J J n n again denotes the n th -power free part of J. Let S be a place such that ord J = nγ, so that we may write J = P nγ J with J, P =. We must analyze the resulting object GI, J. Writing I = P λ I with I, P =, gies I IS aiψ Iψ 2 JGI, Jχ ĨɛJ J0 0 NI s NJ w = ai ψ I ψ 2 JGI, J χ Ĩ J0 ɛj 0 NI s NJ w I ord I =0 ap λ ψ P λ GP λ, P nγ χ J0 P λ NP λs+nγw. We first ealuate the sum oer λ. If γ = 0, the sum becomes λ 0 λ 0 ap λ ψ P λ χ J0 P λ NP λs = L s, π χ J0 ψ, where L s, π χ J0 ψ denotes the Euler factor associated to the place in the L-series.

12 2 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN 3. If γ, then using 2.3 we obtain λ 0 ap λ ψ P λ GP λ, P nγ χ J0 P λ q λs+nγw = + q nγ ap ψ P nγ nγ nγw+nγ s 2 q + nγ nγw 2 + λ nγ χ J0 P nγ q ap λ ψ P λ χ J0 P λ q λs. We wish to sum this geometric series, pulling out a factor of L s, π χ J0 ψ. We must therefore write the Fourier coefficients ap λ in terms of the th Satake parameters. Recall that for S, α β = and α + β = ap. We hae and L s, π χ J0 ψ = ap λ = αλ+ β λ+, α β α χ J0 P ψ P q s β χ J0 P ψ P q s = ap χ J0 P ψ P q s + χ J0 P ψ P q 2s. Substituting these definitions into the latter sum of 3. and ealuating the geometric sums, we hae λ nγ ap λ ψ P λ χ J0 P λ q λs = α β [ α nγ+ q nγs = α β λ nγ α χ J0 P ψ P q s Therefore, we may rewrite the entire equation 3. as λ 0 ap λ ψ P λ GP λ, P nγ χ J0 P λ q λs+nγw = q nγ nγw+nγs 2 + +q [ ap nγ α λ+ χ J0 P λ ψ P λ q λs λ nγ β λ+ χ J0 P λ ψ P λ q λs ] βnγ+ q nγs β χ J0 P ψ P q s = L s, π, χ J0 ψ q nγs ap nγ [ ap nγ ψ P χ J0 P q s ap nγ ap nγ χ J0 P ψ P q s. ψ P χ J0 P q s ] L s, π χ J0 ψ ] = L s, π χ J0 ψ nγ nγw+nγs 2 q + ap nγ ψ P χ J0 P q s [ ap χ J0 P ψ P q s +χ J0 P ψ P q 2s ] + q [ ap nγ ap nγ ψ P χ J0 P q s ]. Expanding in the second bracket, and using the relation ap nγ ap = α β α nγ β nγ α + β = ap nγ + ap nγ 2,

13 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 3 we find that the aboe expression equals L s, π χ J0 ψ q nγ nγw+nγs 2 + [ q ap nγ ap nγ χ J0 P ψ P q s Putting it all together, we hae ap λ ψ P λ GP λ, P nγ χ J0 P λ q λs+nγw = λ 0 = L s, π χ J0 ψ [ q nγw ap nγ ap nγ q nγ 2 nγs ap nγ χ J0 P ψ P q s + ap nγ ap ]. ap nγ χ J0 P ψ P q nγ χ J0 P ψ P q nγ 2 nγ s + ap nγ 2 2 nγ+s q nγ 2 nγs Recall the conention that ax = 0 for all non-integral x. Then repeating the aboe process for all such places with P not diiding J 0, we hae for any fixed ideal J 3.2 I aiψ Iψ 2 JGI, Jχ J0 ĨɛJ 0 NI s NJ w = ord J=nγ L s, π χ J0 ψ q nγw where Qs, P nγ, χ J0 ψ ; π is as defined in 2.2. Qs, P nγ, χ J0 ψ ; π I I J 0 We must now repeat this analysis for the remaining sum oer I such that I J 0 ]. aiψ Iψ 2 JGI, Jχ J0 Ĩ NI s NJ 0 w,. Let be a place such that P J 0. That is, ord J = nγ + k, for some k {, 2,... n } and denote ord I = λ. Then, writing I = P λ I and J = P nλ+k J, we hae GI, J = GP λ I, P nγ+k J = {q nγ+k 2 GI, J, if λ = nγ + k 0, otherwise. Moreoer, in this case P = since P J 0, so χ J0 P = Thus we may write 3.3 I I J 0 aiψ Iψ 2 JGI, JɛJ 0 NI s NJ w = q ap nγ+k ψ P nγ+k nγ+k nγ+k s+nγ+kw 2 I I J 0 Repeating this for the remaining finite list of places such that P J 0, we hae n aiψ Iψ 2 JGI, JɛJ NI s NJ w = ɛj 0 ψ 2 J k= q I I J 0 ord J=nγ+k ai ψ I ψ 2 JGI, J χ J0 Ĩ ɛj 0 NI s NJ w. ap nγ+k ψ P nγ+k nγ+k nγ+k s+nγ+kw 2.

14 4 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN Combining this result with the information from 3.2 and 2.2 and noting that our characters ψ i hae order n, the original series for fixed J takes form 3.5 I IS aiψ Iψ 2 JGI, Jχ J0 ĨɛJ 0 NI s NJ w = ɛj 0 ψ 2 JL S s, π χ J0 ψ ord J=nγ Qs, P nγ, χ J0 ψ ; π q nγw Summing oer each ideal J IS, the result follows. We will also hae need of the following lemma. n k= ord J=nγ+k q ap nγ+k ψ P k nγ+k nγ+k s+nγ+kw 2. Lemma 3.. Let the notation be as aboe. The correction factor Qs, J; π, χ J0 ψ satisfies the following functional equation in s: 3.6 Qs, J; π, χ J0 ψ = NJ 2 J 2 3 J n 2 n J n n 2s ψ 2 J 2 J 2 3 J n 2 n J n n Q s, J; π, χ J0 ψ. Proof: From the definition made in 2.2, one readily sees that at each prime ideal P with P, J 0 =, Qs, P nγ, χ J0 ψ ; π = q nγ 2s Q s, P nγ, χ J0 ψ ; π. Moreoer, for each of the prime ideals P with P J 0 for any choice of k, we hae the identity ap nγ+k ψ P k nγ+k = q nγ+k nγ+k s 2s ψp 2 k ap nγ+k ψ P k nγ+k. 2 nγ+k s 2 q q The lemma therefore follows by combining the aboe identities. and 4. A Functional Equation as s s: A Proof of Proposition 2.5 Recall from Proposition 2.3 that we hae Z s, w, π; ψ, ψ 2 = ζ S nw n/2 + Z 2 s, w, π; ψ, ψ 2 = ζ S nw n/2 + J IS J IS ɛj 0 ψ 2 JL S s, π χ J0 ψ NJ w Qs, J; π, χ J0 ψ, ɛj 0 ψ 2 JL S s, π χ J0 ψ NJ w Qs, J; π, χ J0 ψ, where Qs, J; π, χ J0 ψ is the correction polynomial defined in 2.. To facilitate the statement of the results of this section, we extend the definitions of Z and Z 2 to include arbitrary linear combinations of

15 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 5 characters in place of ψ and ψ 2. In particular for E E, let δ E be the characteristic function of the class E, and consider 4. Z s, w; π, ψ, δ E ψ 2 = ζ S nw n/2 + where for notational conenience, we put J IS J E ɛ/2, J 0 ψ 2 JL S s, π χ J ψ NJ w, 4.2 Ls, π χ J ψ = Ls, π χ J0 ψ Qs, J; π, χ J0 ψ. We determine the functional equation for this completed L-function in the following lemma. Lemma 4.. With the notation as aboe, 2s JCE NJCE L s, π χ J ψ. 4.3 Ls, π χ J ψ = ε 2, χ I 0 2 ɛs, π χ E ψ ψ 2 f E Nf E Proof: From 2., we hae Ls, π χ J0 ψ = εs, π χ J0 ψ L s, π χ J0 ψ. We will ealuate the factor εs, π χ J0 ψ = εs, π ψ χ J0 using Proposition 2.2. Write J 0 = I 0 E, where E represents the class of J 0 in R C, and where I 0 mod C. The conductor of χ J0 is gien by f J0 = J J 2 J n C E, where C E is a constant depending only on the class E. The central character of π is triial, therefore the central character of π ψ is ψ. 2 Thus we hae 2s 4.4 εs, π χ J0 ψ = ε 2, χ I 0 2 ψ 2 J J n C E NJ J n C E εs, π ψ χ f E Nf E E The lemma then follows by combining the aboe with the functional equation for Qs, J; π, χ J0 ψ gien in Lemma 3.. We are now ready to demonstrate the functional equation for Z as s s. The functional equation in Z 2 can be shown completely analogously. Proof of Proposition 2.5: Using Lemma 4., together with the fact that ε 2, χ I 0 2 ε 2, χ J 0 = ε 2, χ J 0 ε 2, χ E 2, we see that 4.5 Z s, w; π, ψ, δ E ψ 2 = AE, s J IS J E ɛ/2, J 0 ψ 2 ψ 2 JL S s, π χ J0 ψ NJ w+2s S L s, π χ J0 ψ, L s, π χ J0 ψ

16 6 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN where 2s NCE AE, s = ψ 2 CE Nf E f E To proceed further, multiply both sides of 4.5 by [ L s, π χ J0 ψ L n ns, π ψ n = + α χ J0 ψ P q s Hence for each J and each S f the term ɛ/2, χ E 2 ɛs, π ψ χ E. Then L S n ns, π ψ n f. + + αn χ J0 ψ P n [ q n s ] + β χ J0 ψ P q s L s, π χ J0 ψ L n ns, π ψ n L s, π χ J0 ψ + + βn χ J0 ψ P n q n s becomes a finite Laurent polynomial in q s whose dependence on J is through terms of the form χ J0 P. Since S f and J is in a fixed class E of R C, we hae χ J0 P = ξ E for some character ξ of R C. Similarly, the quotient 4.6 L s, π χ ψ J0 L s, π χ S J0 ψ where Γ K s is defined as in 2.7, is independent of J. Since = Γ K s, Γ K s Z s, w; π, ψ, ψ 2 = Γs, wζ S nw + 2ns 3n 2 + Z s, w; π, ψ, ψ 2, where we recall that Γs, w is the complete set of Gamma factors defined in 2.9, we conclude that S f α ψ P n q n ns S f β ψ P n q n ns Z s, w, π, ψ, ψ 2 δ E = As; ψ, EZ 2 s, w + 2s ; π, ψ, ψ 2 ψ 2 δ E. Moreoer, the functions As; ψ, E are finite Laurent polynomials in NJ s. Summing oer E, we find that α ψ P n q n ns β ψ P n q n ns Z s, w, π, ψ, ψ 2 = Z s, w, π, ψ, ψ 2 δ E E E is equal to Bs; ψ, ξz2 s, w + 2s ; π, ψ, ψ 2 ψξ 2 where Bs; ψ, ξ is a linear combination of the As; ψ, E. ξ ].

17 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 7 5. Analytic Continuation We wish to analytically continue the functions Z i s, w; π, ψ, ψ 2 for i =, 2 to C 2, for each choice of ψ and ψ 2. This will be achieed using the functional equations for the Zi, along with properties of the series Dw, I, ψ 2 and the Dirichlet series Ls, π χ d ψ. As aboe, we will restrict our attention to Z, as the arguments for Z 2 will be almost identical. First, we consider the expression for Z s, w; π, ψ, ψ 2 gien in 2.6. Using the bound ai ɛ NI /9+ɛ for the Fourier coefficients, as well as the bound gien in 2.8, we see that Z s, w; π, ψ, ψ 2 conerges absolutely and uniformly in the region of C 2 satisfying Rew > and Res > 0 9. Next, we examine the behaior of Z s, w; π, ψ, ψ 2 when Res /9, utilizing the expression for Z s, w; π, ψ, ψ 2 gien in 2.3. It will be conenient to work with this series as Z s, w; π, ψ, ψ 2 = E R C Z s, w; π, ψ, δ E ψ 2, with Z s, w; π, ψ, δ E ψ 2 as gien in 4.. The full Dirichlet series Ls, π χ J ψ satisfies the functional equation gien in 4.3. This functional equation inoles gamma factors, as we hae 5. Ls, π χ J0 ψ = L s, π χ J0 ψ = Γ K sasl S s, π χ J0 ψ, where As = S f L s, π χ J0 ψ. Combining 5. with 4.3, we obtain 5.2 L S s, π χ J ψ = Γ K s Γ K s where we put A s Bs, ENJ 2s L S s, π χ As J ψ, 2s Bs, E = ε 2, χ I 0 2 εs, π χ E ψ ψ 2 JCE NCE. f E Nf E We set s = /9 + σ + it in 5.2 and examine the factors on the right, as t. For the Gamma factors, using a simplified ersion of Stirling s formula gien by we see that Γσ + it t σ /2 exp π t as t, Γ K 0/9 σ it Γ K /9 + σ + it DK 2π t2 ν 2 r2 r2 where ν is the constant gien in 2.7. As t, we therefore hae Γ K 0/9 σ it Γ K /9 + σ + it M t 22/9 4σr2 /9 2σ

18 8 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN for some positie constant M. The factor independent of t. Finally, the factor L S A 9 σ it A 9 +σ+it is a finite polynomial in powers of q σ, hence it is σ it, π χ Jψ is bounded as a function of t, since the full L-series is absolutely conergent when s = 0 9. Thus we see that L S /9 + σ + it, π χ J ψ has polynomial growth as a function of t, for fixed σ < 0. For Res /9, we therefore hae Z s, w; π, ψ, ψ 2 ɛ cσ t 22/9 4σr2 E R C J IS J E where cσ is a constant independent of t, gien by cσ = max {cσ, E}, with E R C cσ, E = A 0 9 σ it A 9 + σ + it B 9 + σ + it M. NJ Rew+2σ /9+ɛ, From this we see that Z s, w; π, ψ, ψ 2 conerges absolutely and uniformly on compacta in the region satisfying Res < /9 and Rew > 2 2 Res. Now by applying a generalized ersion of the Phragmén- Lindelöf Theorem, we see that for s with real part between /9 and 0/9, Z is absolutely and uniformly conergent, proided Rew > 9/9 Res. Therefore, our region of conergence for Z s, w; π, ψ, ψ 2 is the region R = {s, w Rew > max{, 9 9 Res, 2 2 Res} }. To obtain a second region of conergence for Z, we again consider the expression for Z in terms of Dw, I, ψ 2, as gien in 2.6. If Rew < 0, then 2.8 gies hence we see that w 2 + n w 2 n Dw, I, ψ 2 ɛ NI /2 Rew+ɛ, Z s, w; π, ψ, ψ 2 ɛ I IS NI /9+ɛ NI /2 Rew NI s = I IS NI s+rew /8+ɛ. Consequently, the initial region of conergence of Z s, w; π, ψ, ψ 2, i.e. Rew > and Res > 0 9, is extended to include the region of C 2 satisfying Rew < 0 and Res > 29/8 Rew. Then by a second application of the Phragmén-Lindelöf theorem, we see that Z s, w; π, ψ, ψ 2 conerges absolutely and uniformly on compact in the region R = {s, w Res > max{ 0 9, 29 8 Rew 2, 29 8 Rew} }. These regions oerlap, which means that Z s, w; π, ψ, ψ 2 conerges on their union, R = R R. By an almost identical argument, Z2 s, w; π, ψ, ψ 2 also conerges on the region R. Now we may apply the functional equations for the Z i, represented for conenience as α : s, w s, w + 2s and

19 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 9 β : s, w s + w /2, s, to extend this region of conergence. Applying the transformation α to the region of conergence R, we obtain a region which oerlaps R, and when we take the conex hull of their union, we obtain the half-plane {s, w Res > 29 8 Rew}. Finally, applying the transformation β to this half-plane, we obtain another half-plane which oerlaps it. Therefore when we take the conex hull of their union, we obtain all of C 2, as desired. 6. Absolute Conergence of Sums in a Right Half-Plane We first analyze the indiidual expressions Z i 2, w; π, ψ, ψ 2 for i =, 2. Again, we will restrict our attention to Z in what follows, as the conergence of Z 2 will eidently follow in the same fashion. To begin, we separate the sum oer J in terms of two pieces the first corresponding to n th power free J 0, and the second corresponding to J n. From 2.3, we hae Z 2, w; π, ψ, ψ 2 = J 0 IS n th -power free ɛj 0 ψ 2 J 0 L S 2, π, χ J 0 ψ NJ 0 w J n IS Q 2, J; π, χ J 0 ψ NJ n nw. In the expression for Q 2, J; π, χ J 0 ψ obtained from 2., we abbreiate the notation by defining 6. C J0,ψ P = χ J0 P ψ P + χ J0 P ψ P, and writing Q 2, P nγ, χ J0 ψ ; π = ap nγ ap nγ q 2 C J0,ψ P + ap nγ 2 q. For fixed J 0, we show that the sum oer J n is absolutely conergent in a certain right half-plane in w. We may write J n IS Q 2, J; π, χ J 0 ψ NJ n nw = n k= ord J=nγ+k γ 0 ψ P k NP nγw ap nγ+k ord J=nγ Q 2, P nγ, J; π NP nγw γ 0

20 20 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN and then analyze each of the geometric sums indiidually, using the Satake parameters. First, we hae γ 0 ap nγ+k NP nγw ψ P k = ψ P k α β = ψ P k [ α β = ψ P k γ 0 α nγ+k β nγ+k q nγw α k α n q nw α n q nw β k β n q nw ] β n q nw [ ap k + ap n k q nw ]. Notice that we hae two of the factors in the Euler factor corresponding to the non-archimedean places in the symmetric n th power L-function. Namely, for each such place, we hae the j = 0 and j = n factors in the expression 6.2 L nw, π, sym n = Next, we hae γ 0 Q 2, P nγ, χ J0 ψ ; π NP nγw = + γ After factoring out α n q nw γ 0 Q 2, P nγ, χ J0 ψ ; π NP nγw 0 j n ap nγ ap nγ α n j q 2 β j q nw. C J0,ψ P + ap nγ 2 q nγw [ = + α q 2 C J0,ψ α β P + α q α n [ = α n q nw β n q nw + Therefore the sum oer J n IS becomes q nw + β q 2 C J0,ψ P + β q β n q nw q α n q nw β n q nw ]. β n q nw from the entire expression and simplifying, we obtain n 2 n ap ap C J0,ψ q nw P + q nw+/2 n 2 ap q nw+ + ] q 2nw J n IS Q 2, J; π, χ J 0 ψ NJ n nw = α n q nw β n q nw ord J 0 n n k= ord J k n [ + ψ P k [ ap k n 2 n ap ap C J0,ψ q nw P + q nw+/2 + ap n k n 2 ap q nw+ + q nw ] ] q 2nw+.

21 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 2 In order to express this sum in terms of Lnw, π, sym n, we multiply through by the remaining factors in 6.2 and their reciprocals. If we put R n, w; π = α n j βq j nw = { then we hae where R J0 w; π = α n 2 α n 2 n j n q nw q nw k= ord J 0=k α q nw α q nw J n IS β q nw q nw Q 2, J; π, χ J 0 ψ NJ n nw R n, w; πψ P k ord J 0=0 R n, w; π [ + β n 2 q nw β q nw if n is odd, q nw if n is een, β n 2 = L S nw, π, sym n R J0 w; π, [ ap k + ap n k q nw ] n 2 n ap ap C J0,ψ q nw P + q nw+/2 n 2 ap q nw+ + ] q 2nw+. The factor Lnw, π, sym n conerges absolutely for Rew > n + 9. In the factor R J 0 w; π, the product oer places with P J 0 is a finite product, and therefore it does not affect conergence. In the infinite product oer places with P J 0, for a gien place, it is clear that the terms ap n 2 q nw and determine the region of conergence. Using the fact that ap ɛ q /9+ɛ, ap n C J0,ψ P q nw+/2 [2] we see that the first of these two terms is in fact more restrictie. We find that this infinite product, and hence R J0 w; π, conerges absolutely for Rew > 7 9n + 9. Now suppose there are only finitely many twists for which L S 2, π, χ J 0 ψ is nonzero. Then L S nw, π, sym n J 0 IS n th -power free ɛj 0 ψ 2 J 0 L S 2, π, χ J 0 ψ NJ 0 w R J0 w; π, will conerge absolutely for Rew > n + 9. Note that since we are restricting our attention to the case n 3, this would mean that there exists some δ > 0 such that the sum conerges absolutely for Rew > 2 δ.

22 22 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN 7. Nonanishing twists Proof of Theorem. We now use the results of the preious sections to proe Theorem.. We require the following lemma. Lemma 7.. Suppose the Dirichlet series Lw = bd d w is absolutely conergent for Rew > /2 δ, for some positie δ. Suppose further that there exist Dirichlet series M w, M 2 w,..., M r w and functions γ w, γ 2 w,..., γ r w which satisfy the following conditions: Each M j w is absolutely conergent for Rew > /2 δ. 2 Each γ j w is holomorphic for Rew > 0, and for all k > 0, σ > /2 we hae the estimate γ j σ + it << k,σ t k, as t. 3 There is the functional equation Lw = j γ j wm j w. Then Lw is identically zero. To apply the Lemma, we set s = 2 and iew the functions Z is, w; π, ψ, ψ 2 for i =, 2 as Dirichlet series in w. In particular, after repeated applications of the functional equations, the Dirichlet series Lw = Z 2, w; π,, satisfies the functional equation Lw = ξ,ξ 2 b R c γ ξ,ξ 2 wz 2 /2, w; π, ξ, ξ 2 for some collection of functions γ ξ,ξ 2 satisfying condition 2 of Lemma 7.. To see this, apply the w-functional equation, followed by the s-functional equation, followed by the w-functional equation to Z s, w; π,, at s = 2. If there are only finitely many idèle class characters χ J 0 of order n such that L/2, π χ J0 is nonzero, then as established in the preious section, there exists a positie δ such that the Dirichlet series on both sides of the aboe equation are absolutely conergent for Rew > /2 δ. It follows that Lw is identically zero and the proof of Theorem is complete.

23 NON-VANISHING TWISTS OF GL2 AUTOMORPHIC L-FUNCTIONS 23 Proof of Lemma 7.: Suppose Lw is not identically zero. Choose δ with 0 < δ < δ such that L/2+δ = A 0. Then L/2 + δ + it j γ j /2 + δ + it M j /2 δ 0 as t. Choose X so large that L/2 + δ d<x d < X. Then bd d /2+δ L/2 + δ + it 0 L/2 + δ d<x < A/3. Choose t 0 so that d it 0 < /3 for all bd d /2+δ d it0 + bd d /2+δ d X < 2A/3. Hence L/2 + δ + it 0 > A/3. Howeer, we can find arbitrarily large such t 0. This contradicts the fact that L/2 + δ + it 0 as t. References [] B. Brubaker, S. Friedberg, J. Hoffstein, Cubic Twists of GL2 L-series. Preprint. [2] D. Bump, S. Friedberg, J. Hoffstein, Sums of twisted GL3 automorphic L-functions, To appear in a olume dedicated to Joseph Shalika. [3], On some applications of automorphic forms to number theory, Bull. A.M.S , [4] J. W. S. Cassels and A. Frölich eds., Algebraic Number Theory, Academic Press, London, 967. [5] J. Fearnley and H. Kisilesky, Vanishing and non-anishing Dirichlet twists of L-functions of elliptic cures. Preprint. [6] B. Fisher and S. Friedberg, Double Dirichlet series oer function fields, Compositio Math., To appear. [7], Sums of twisted GL2 L-functions oer function fields, Duke Math J., 7, 2003, [8] S. Friedberg and J. Hoffstein, Nonanishing theorems for automorphic L-functions on GL2, Ann. of Math. 2, 42, 995, [9] S. Friedberg, J. Hoffstein, and D. Lieman, Double Dirichlet series and the n-th order twists of Hecke L-series, To appear. [0] J. Hoffstein, Eisenstein series and theta functions on the metaplectic group, Theta functions: from the classical to the modern, CRM Proc. Lect. Notes M. Ram Murty, ed., American Mathematical Society, Proidence, RI, 993, pp [] D. Kazhdan and S. J. Patterson, Metaplectic Forms, Inst. Hautes Ètudes Sci. Publ. Math., , [2] H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications Duke Math. J., 2, 2002, [3] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Inentiones Math , [4] T. Kubota, On automorphic functions and the reciprocity law in a number field, Lectures in Mathematics: Kyoto Uniersity, Kinokuniya Book Store Co., Tokyo, 969. [5] S. J. Patterson, A cubic analogue of the theta series, J. Reine Angew. Math , [6] X. She, On the non-anishing of cubic twists of automorphic L-series, Trans. Amer. Math. Soc., 35, 999, [7] J. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl ,

24 24 BEN BRUBAKER, ALINA BUCUR, GAUTAM CHINTA, SHARON FRECHETTE, AND JEFFREY HOFFSTEIN Department of Mathematics, Stanford Uniersity, Stanford, CA address: Department of Mathematics, Brown Uniersity, Proidence, RI address: Department of Mathematics, Brown Uniersity, Proidence, RI address: Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 060 address: Department of Mathematics, Brown Uniersity, Proidence, RI address: