ACTA ARITHMETICA LXXII.2 1995) Theta and L-function splittings by Jeffrey Stopple Santa Barbara, Cal.) Introduction. The base change lift of an automorphic form by means of a theta kernel was first done by Kudla in [2, 3] and Zagier in [6]. Kudla s paper omitted the computation of the Fourier series coefficients; he instead referred to the paper of Niwa [4] on the Shimura lift. Knowledge of these Fourier coefficients lets one write the L-series of the lifted form as a product of the original L-series and its quadratic twist. In this paper the factorization of the L-series is shown directly. Niwa s idea of splitting the theta function lets us explicitly compute the Mellin transform Ls, f) of the lifted form f. It is a Rankin Selberg conolution of the original form f with a Maass wae form coming from the quadratic extension. The factorization of the L-series then follows as in the work of Doi and Naganuma [1]. To aoid excessie notation, only the simplest case is considered: the lift to Q q), with q an odd prime q 1 mod 4 such that h + K) = 1. We use χ to denote the Kronecker symbol q ). We take a cusp form fz) = an) exp2πinz) of weight k for SL2, Z), an eigenfunction of all the Hecke operators. Recall that in Section 3 of [2] Kudla defined the theta kernel θz, z 1, z 2 ) = y l L χm) mz 1 z 2 + αz 1 + σ αz 2 + n) k e{xq + iyr)[l]} where z = x + iy is in H and z 1, z 2 ) is in H 2, the lattice ariable l is written as [ α n m α] with α in O, σ α the Galois σ conjugate, and m, n in Z, Q[l] is the indefinite quadratic form 2 detl), u j / j each z j = u j + i j defines an element g j = [ j ] 1/ in SL2, R). j The pair g = g 1, g 2 ) acts on the ector space by g l = g 1 2 lg 1, R[l] is a majorant for Q defined by tr t g l)g l). [11]
12 J. Stopple Then the lifting f is defined by fz 1, z 2 ) = F k dx dy fz)θz, z 1, z 2 )y y 2, where F is a fundamental domain for Γ q)\h. and Splitting the theta function. Let θ 1,j z, ) = y 1 j)/2 2 j α O H j πyα 1/2 + σ α 1/2 )) θ 2,j z, ) = y 1 j)/2 2 j m,n Z exp2πixnα πyα 2 + σ α 2 /)) χm)h j πym 1/2 + n 1/2 )) exp2πixmn πym 2 + n 2 /)). Lemma. Along the purely imaginary axis z 1, z 2 ) = i 1, i 2 ) in H 2, θz, i 1, i 2 ) = 1) k π k k/2 1) )θ ν 1,2ν z, ) 1 θ 2,k 2ν z, 1 2 ). 2ν 2 P r o o f. Along the imaginary axis R[l] = 1 α 2 + 2 σ α 2 + 1 2 m 2 + 2 1 1 2 and the spherical polynomial term is equal to ) 1/2 1) m k 1 2 ) 1/2 n 1 + + iα + i σ α 1 2 ) 1/2 Apply to this the Hermite identity a + ib) k = 2 k k j= 2 n2 ) k H k j a)h j b)i j j 2 1 ) 1/2 ) k. where H j x) = 1) j expx 2 ) dj dx exp x 2 )) is the jth Hermite polynomial. j Include a factor of πy which will be needed later) to show that the spherical polynomial term is k ) 1/2 ) k πy 2 k 1) k πy) )H k/2 k j mπy 1 2 ) 1/2 + n j j= 1 2 ) 1/2 ) 1/2 ) πy1 H j α + σ πy2 α i j. 2 1
Theta and L-function splittings 13 H j x) is an odd or een function according to whether j is odd or een. If j is odd, the α and α terms in the sum defining g j cancel and g j z) is identically zero. Writing j = 2ν finishes the lemma. The point of this is that the Dirichlet series Ls, f) is gien by the Mellin transform Ls, f) = = R + ) 2 /U + fi1, i 2 ) 1 2 ) s 1 d 1 d 2 R + ) 2 /U + F k dx dy fz)θz, i 1, i 2 )y y 2 1 2 ) s k/2 1 d 1 d 2. Here U + is the group of totally positie units, generated by ε. Change the ariables to 1 = 1 / 2 and 2 = 1 2 and by abuse of notation go back to writing 1 and 2 ). Then using the splitting of θ, the Mellin transform becomes Ls, f) = 2 1 1) k π ) k k/2 1) ν 2ν Let g 2ν z) = ε ε ε 1 F ε 1 θ 1,2ν z, ) k dx dy d 1 fz)θ 1,2ν z, 1 )θ 2,k 2ν z, 2 )y y 2 d and E 2ν z, s, ) = Rearranging the integrals shows that Ls, f) is equal to π k/2 k 1) 2 2ν ) 1) k ν F s k/2 2 1 d 2 2. s k/2 d θ 2,2ν z, ). k dx dy fz)g 2ν z)e k 2ν z, s, )y y 2. Two ugly lemmas. Now two lemmas are required. The first is folklore, the second is sketched in [4]. Lemma 1. g 2ν z) is equal to y 1/2 ν 2 2ν ε ε 1 H 2ν πyα 1/2 + σ α 1/2 )) α O exp2πixnα πyα 2 + σ α 2 /)) d and is a Maass wae form of weight 2ν.
14 J. Stopple P r o o f. Computing the integral will show that this is the Fourier expansion of a Maass form in terms of Whittaker functions. Alternatiely, one could use the method of Vignéras [5] to see that the integral is a Maass form, but in the end the Fourier expansion is wanted to apply the Rankin Selberg method.) From [H], Vol. 2, p. 193) H 2ν ) = 1) ν 2ν!/ν! so the α = term contributes 2 2ν y 1/2 ν 1) ν 2ν!/ν!2 logε)). For the terms α in the sum interchange the sum and the integral and change the ariables by w = ε 2n for n Z. This gies y 1/2 ν 2 2ν α O/U + α H 2ν πyαw 1/2 + σ αw 1/2 )) exp πyα 2 w + σ α 2 /w)) dw w exp2πixnα). To compute the integral of the term corresponding to α in the sum change ariables again to let = αw/ Nα ) 1/2 to get 2 2ν y 1/2 ν exp2πixnα) times 2 H 2ν πy Nα ) 1/2 ± 1/)) exp πy Nα ± 1/) 2 ) exp2πynα) d with the ± chosen according to whether Nα is positie or negatie. A final change of ariables with t = log) gies ) ) 1/2 cosh t 2 H 2ν 2πy Nα ) exp 4πy Nα cosh2 t sinh t sinh 2 exp2πyn α) dt. t For integral ν the parabolic cylinder functions are defined by [H], Vol. 2, p. 117) Thus the integral is 2 ν+1 D 2ν z) = 2 ν exp z 2 /4)H 2ν z/ 2). D 2ν 2a cosh t ) ) exp a 2 sinh2 t sinh t cosh 2 dt t with a = 2πy Nα ) 1/2. For Nα > apply [I], Vol. 2, p. 398, 2)) to see that this is the Whittaker function y ν Nα 1/2 W ν, 4πy Nα ) exp2πixnα) when the omitted constants are included. For Nα <, use the imaginary phase shift cosh t = i sinht + iπ/2) = i sinht iπ/2), sinh t = i cosht + iπ/2) = i cosht iπ/2)
to get 2 ν+1 Theta and L-function splittings 15 D 2ν 2a i cosht iπ/2)) expa 2 sinh 2 t ± iπ/2)) dt. The ± will be chosen later.) The identity [H], Vol. 2, p. 117) gies 1) ν ν+1 2ν! 2 2π D 2ν z) = 1) ν 2ν! 2π D 2ν 1 iz) + D 2ν 1 iz)) {D 2ν 1 2a cosht iπ/2)) + D 2ν 1 2a cosht iπ/2))} expa 2 sinh 2 t ± iπ/2)) dt. In the first cylinder function, moing the 1 inside the cosht iπ/2) adds iπ to the argument, giing 1) ν ν+1 2ν! 2 2π {D 2ν 1 2a cosht + iπ/2)) + D 2ν 1 2a cosht iπ/2))} expa 2 sinh 2 t ± iπ/2)) dt. Write this as two integrals, choosing sinh 2 t+iπ/2) in the first and sinh 2 t iπ/2) in the second. Since D 2ν 1 is an entire function one can shift the line of integration by iπ/2 to get 1) ν ν+2 2ν! 2 2π D 2ν 1 2a cosh t) expa 2 sinh 2 t) dt. Apply [I], Vol. 2, p. 398, 21)) to see that this is the Whittaker function ν Γ ν + 1/2)2 1) y ν Nα 1/2 W ν, 4πy Nα ) exp2πixnα) π when the omitted constants are included. Summarizing, this gies g 2ν z) = 2 1 2ν 1) ν 2ν!/ν! logε)y 1/2 ν ) + 1) ν ν Γ ν + 1/2)2 y π Nα 1/2 W ν, 4πy Nα ) exp2πixnα) α O/U + Nα< + y ν α O/U + Nα> Nα 1/2 W ν, 4πy Nα ) exp2πixnα).
16 J. Stopple Lemma 2. E 2ν z, s, ) is equal to y 1/2 ν 2 2ν m,n Z χm)h 2ν πym 1/2 + n 1/2 )) exp 2πixmn πy m 2 + n2 and is a non-holomorphic) Eisenstein series of weight 2ν. is )) s k/2 d P r o o f. The Fourier transform ft) = fs) exp 2πist) ds of H 2ν mπy) 1/2 + πy/) 1/2 s) exp mπy) 1/2 + πy/) 1/2 s) 2 ) 1) ν 2 2ν π ν /y) ν+1/2 t 2ν exp2πimt) exp πt 2 /y) by [I], Vol. 1, p. 39, 9)) and the usual Fourier transform theorems. The Poisson summation formula using {f } s) = f s) and ealuating at mz) then gies H 2ν mπy) 1/2 nπy/) 1/2 ) n exp mπy) 1/2 nπy/) 1/2 ) 2 + 2πimnz) = π) ν 2 2ν /y) ν+1/2 n mz + n) 2ν exp 2πimmz + n) π y ) mz + n)2 = π) ν 2 2ν /y) ν+1/2 n mz + n) 2ν exp π y ) mz + n 2. Thus E 2ν z, s, ) is equal to the Mellin transform y 2ν π) ν χm)mz + n) 2ν exp π y ) mz + n 2 s+ν+1 k)/2 d m,n = 1) ν π s+k 1)/2 Γ s + ν + 1 k)/2) y s ν+1 k)/2 χm)mz + n) 2ν mz + n 2s 2ν+1 k. m,n The group Γ q) has two cusps, and thus two Eisenstein series. Unfortunately, the aboe is the one for the cusp at, and the one for the cusp at would be more conenient. This is a result of not making the optimal
Theta and L-function splittings 17 definition of the theta function aboe. To fix this, let ω q = [ ] 1 q. Since ωq normalizes Γ q), ωq 1 F is another fundamental domain. Thus the integral in 1) can be written fω q z)g 2ν ω q z)e k 2ν ω q z, s, )yω q z) k dx dy y 2 ω 1 q F Here E 2ν z, s, ) is equal to = q s+1/2 F k dx dy fqz)g 2ν z)e k 2ν z, s, )y y 2. 1) ν π s+k 1)/2 Γ s + ν + 1 k)/2) χm)nz + m) 2ν y s ν+1 k)/2 nz + m 2s 2ν+1 k, m,n n mod q i.e., the Eisenstein series at. To do the Rankin trick write E k 2ν z, s, ) as 1) k/2 ν 2π s+k 1)/2 Γ s + 1/2 ν)l2s k + 1, χ) times a sum oer Γ \Γ q) and unfold the integral. This gies Ls, f) = 1) k/2 π s 1/2 q s+1/2 L2s k + 1, χ) ) k 1 Γ s + 1/2 ν) 2ν = 1) k/2 π s 1/2 q s+1/2 L2s k + 1, χ) n=1 an)tnq) nq) 1/2 s+ν+1/2 dx dy fqz)g 2ν z)y y 2 ) k Γ s + 1/2 ν) 2ν s 1/2 dy exp 2πnqy)W ν, 4πnqy)y y. Here tn) is the cardinality of the set {α O/U + Nα = n}, so by the Euler product for the Dedekind zeta function tnq) = tn). The integral representation of the Whittaker functions shows that W ν, = W ν, and 7.621 11)) in [G] gies the Mellin transform as a ratio of Gamma functions k 2ν ) = 2 k 1. Finally, Doi and Γ s) 2 /Γ s + 1/2 ν). One can show Naganuma [1] hae shown that L2s k + 1, χ) an)tn)n s is equal to Ls, f)ls, f χ). This completes the proof of the Theorem. Ls, f) = 1) k/2 2 k q 1/2 2π) 2s Γ s) 2 Ls, f)ls, f χ).
18 J. Stopple References [I] A. Erdélyi et al., Tables of Integral Transforms, based, in part, on notes left by Harry Bateman, McGraw-Hill, 1954. [H], Higher Transcendental Functions, based, in part, on notes left by Harry Bateman, McGraw-Hill, 1953. [G] I. S. Gradshteĭn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed., Academic Press, 198. [1] K. Doi and H. Naganuma, On the functional equation of certain Dirichlet series, Inent. Math. 9 1969), 1 14. [2] S. Kudla, Theta functions and Hilbert modular forms, Nagoya Math. J. 69 1978), 97 16. [3], Relations between automorphic forms produced by theta-functions, in: Modular Functions of One Variable VI, Lecture Notes in Math. 627, Springer, 1977, 277 285. [4] S. Niwa, Modular forms of half integral weight and the integral of certain thetafunctions, Nagoya Math. J. 56 1974), 147 161. [5] M.-F.Vignéras, Séries thêta des formes quadratiques indéfinies, in: Modular Functions of One Variable VI, Lecture Notes in Math. 627, Springer, 1977, 227 239. [6] D. Zagier, Modular forms associated to real quadratic fields, Inent. Math. 3 1975), 1 46. MATHEMATICS DEPARTMENT UNIVERSITY OF CALIFORNIA SANTA BARBARA, CALIFORNIA 9316 U.S.A. E-mail: STOPPLE@MATH.UCSB.EDU Receied on 29.9.1993 2494)