A peculiar modular form of weight one

A peculiar modular form of weight one by Stephen S. Kudla1 Michael Rapoport arXiv:math/9808143v1 [math.NT] 31 Aug 1998 and Tonghai Yang2 Introduction. In this paper we construct a modular form φ of weight one, attached to an imagi√ nary quadratic field, which has several curious properties. Let k = Q( −q) where q > 3 is a prime congruent to 3 modulo 4 . The first curious property concerns the Fourier expansion of φ , which is nonholomorphic and is not a cusp form. For a positive integer n , the n th Fourier coefficient an (φ) of φ vanishes unless either n or n/p , for some prime p , inert in k , is a norm of an integral ideal of k . If n or n/p is a norm, then an (φ) has a simple formula. The nonholomorphic part of φ occurs in the negative Fourier coefficients, which are nonvanishing only for integers −n where n ≥ 1 is a norm and involve the exponential integral. The second curious property concerns the Mellin transform of φ . It has a simple expression in terms of the Dedekind zeta function of k and the difference of the logarithmic derivatives of the Riemann zeta function and the Dirichlet L-series associated to k . The third curious property concerns the Fourier coefficients an (φ) , for positive n , which are connected with the theory of complex multiplication. More precisely, they are the degrees of certain 0 -cycles on the moduli scheme of elliptic curves with complex multiplication by Ok . We now describe these results in more detail. In [13], a certain family of (incoherent) Siegel Eisenstein series was introduced which have an odd functional equation and hence have a natural zero at their center of symmetry ( s = 0 ). It was suggested that the derivatives of such series at s = 0 should have some connection with arithmetical algebraic geometry, and some evidence was provided in the case of genus 2 and weight 3/2 . In that case, certain of the Fourier coefficients of the central derivative were shown to involve (parts of) the height pairing of Heegner points on Shimura curves. Higher dimensional cases are studied in [15] and [17]. In the present paper, we consider the simplest possible example of an incoherent 1 Partially 2 Partially supported by NSF Grant DMS-9622987 supported by NSF Grant DMS-9700777 Typeset by AMS-TEX 1 2 Eisenstein series and its central derivative. More precisely, let q > 3 be a prime con√ gruent to 3 modulo 4 . Associated to the imaginary quadratic field k = Q( −q) ,
there is a nonholomorphic Eisenstein series of weight 1 and character χq (·) = −q · for the congruence subgroup Γ0 (q) of Γ = SL2 (Z) . In classical language, this series has the following form. For τ = u + iv in the upper half plane and s ∈ C with Re(s) > 1 , let E(τ, s) = v s/2 (0.1) X (cτ + d)−1 |cτ + d|−s Φq (γ), γ∈Γ∞ \Γ where, for γ = (0.2)  a b c d  ∈ Γ, Φq (γ) =  χq (a) −iq −1/2 χq (c) if c ≡ 0 mod (q), if c is prime to q. After analytic continuation to the whole s plane, the normalized Eisenstein series (0.3) E ∗ (τ, s) = q (s+1)/2 Λ(s + 1, χq ) E(τ, s) satisfies the functional equation E ∗ (τ, −s) = −E ∗ (τ, s) . Here (0.4) Λ(s, χq ) = π −(s+1)/2 Γ s + 1
L(s, χq ). 2 Our ‘peculiar’ modular form of weight 1 is then     ∂ ∂ ∗ (0.5) φ(τ ) := −hk · E(τ, s) E (τ, s) =− ∂s ∂s s=0 , s=0 so that, up to sign, φ(τ ) is the leading term of the Laurent expansion of E ∗ (τ, s) at the central point s = 0 . Here hk is the class number of k . Theorem 1. The nonholomorphic modular form φ has a Fourier expansion: φ(τ ) = a0 (φ, v) + X an (φ, v) e(nτ ) + n<0 X an (φ) e(nτ ), n>0 where the coefficients are as follows. Let ρ(n) = |{a ⊂ Ok | N (a) = n}|, so that ζk (s) = ∞ X n=1 ρ(n) n−s , 3 is the Dedekind zeta function of k . The constant term of φ is given by   Λ′ (1, χq ) . a0 (φ, v) = −hk · log(q) + log(v) + 2 Λ(1, χq ) For n > 0 , X

an (φ) = 2 log(q) ordq (n) + 1 ρ(n) + 2 log(p) ordp (n) + 1 ρ(n/p), p6=q where the sum runs over primes p which are inert in k . For n < 0 , an (φ, v) = 2β1 (4π|n|v) ρ(−n). Here β1 is essentially the exponential integral: Z ∞ u−1 e−ut du. (0.6) β1 (t) = − Ei (−t) = 1 Since one of the aims of this paper is to illustrate the constructions of [13] in the simplest possible case, we work adelically, rather than classically, and obtain both the Fourier expansion of E(τ, s) and its derivative by computing local Whittaker functions and their derivatives. This is done in sections 1–3. Following the tradition of Hecke, we next compute the Mellin transform of φ , or, more precisely, of φ with its constant term omitted:  Z ∞ φ(iv) − a0 (φ, v) v s−1 dv. (0.7) Λ(s, φ) := 0 This can be done termwise. Let Λ(s) = π −s/2 Γ( 2s ) ζ(s) , and Λk (s) = Λ(s)Λ(s, χq ) . Theorem 2. Λ′ (s, χ) Λ′ (s) Λ(s, φ) = Λk (s) log(q) + − Λ(s, χ) Λ(s)   What seems striking (to us) here is the occurrence of the difference of the logarithmic derivatives of Λ(s, χ) and Λ(s) . The computation is done in section 4. Next, we show that the Fourier coefficients of φ are connected with certain 0-cycles on an arithmetic curve. This arithmetic curve is defined as the moduli scheme M of elliptic curves with complex multiplication ι by the ring of integers Ok in k . For each positive integer n , there is a 0-cycle Z(n) on M , which is the locus on which the elliptic curves have an additional endomorphism y with y 2 = −n , Galois commuting with the action of Ok , i.e, with y · ι(a) = ι(ā) · y , for a ∈ Ok . The following result is proved in section 5 as an application of the classical theory of complex multiplication and its further development due to Gross and Zagier, [10], [11]. 4 Theorem 3. (i) For n ∈ Z>0 , suppose that neither n nor n/p , for any prime p which is inert in k is a norm of an integral ideal of Ok , i.e., suppose that ρ(n) = 0 and ρ(n/p) = 0 for all inert primes p . Then the cycle Z(n) is empty and an (φ) = 0 . (ii) If ρ(n) 6= 0 ( resp. ρ(n/p) 6= 0 ), then the 0 -cycle Z(n) is supported in the fiber at q ( resp. the fiber at p ), and deg(Z(n)) = an (φ). We obtain the following Fourier expansion: (0.8) φ(τ ) = a0 (φ, v) + X deg(Z(n)) q n + 2 X ρ(n) β1 (4πnv)q −n , n>0 n>0 in which, for a moment, we write, q n = e(nτ ) = e2πinτ . As pointed out above, this expression provides some evidence in favor of the general proposals of [13]. In section 6, we propose a ‘modular’ definition of an Arakelov divisor Z(t, v) for t < 0 for which deg(Z(t, v)) = at (φ, v) as well. It would be nice to have a geometric interpretation of the constant term a0 (φ, v) . Also in section 6, we suggest the modifications which are necessary to extend Theorems 1 and 3 to the most general incoherent Eisenstein series on SL(2) . As far as we know, the modular form φ does not appear in the classical literature. It resembles, however, the modular form of weight 3/2 considered by Zagier [24], whose Fourier expansion is (0.9) φZagier (τ ) = − X X 2 1 + deg(Z(n))q n + v −1/2 β3/2 (4πm2 v)q −m , 12 n>0 m∈Z where (0.10) 1 β3/2 (t) = 16π Z ∞ u−3/2 e−ut du. 1 Here we have momentarily changed notation and have denoted by Z(n) certain 0-cycles on a modular curve (over C ), cf., for example, [1]. We conclude with the observation of Zagier that the coefficients of φ actually occur in [11]. Let −d be a fundamental discriminant with (d, q) = 1 and with d > 4 . 5 Let (0.11) J(−q, −d) = Y [τ1 ], [τ2 ] disc(τ1 ) = −q disc(τ2 ) = −d j(τ1 ) − j(τ2 )
be the integer of (1.2) of [11]. Then Theorem 1.3 and (1.4) of that paper can be written as (0.12) 2 log(J(−d, −q)) = X am (φ), n n < dq 2 n ≡ dq(4) 2 where, in the sum, m = (dq − n2 )/4 . This paper has its origin in a question raised by Dick Gross at the Durham conference in the summer of 1996 when we presented results on the higher dimensional cases. We thank him for his insistence that we consider this ‘simplest’ case. Notation. We will use the following notation for elements of SL(2) :         1 b 1 cos(θ) sin(θ) a , n(b) = ,w= , kθ = , m(a) = 1 −1 − sin(θ) cos(θ) a−1 for θ ∈ R . Let ψ : QA → C1 be the standard nontrivial additive character, as defined in Tate’s thesis. §1. Incoherent Eisenstein series for SL(2) . The simplest examples of incoherent Eisenstein series, as considered in [13], occurs for the group SL(2) . In this section, we sketch their construction. √ Let χ be a nontrivial quadratic character of Q× \Q× A , and let k = Q( D) be the associated quadratic extension. We assume that D < 0 is the discriminant of the extension k/Q , and write Ok for the ring of integers of k . Let V be a two dimensional vector space over Q with quadratic form x 7→ Q(x) . Let χV (x) = (x, − det V ) be the quadratic character associated to V , and assume that χV = χ . Then V ≃ k with quadratic form Q(x) = κ · N (x) , κ ∈ Q× , given by a multiple of the norm form. Up to isomorphism over Q , the space V is determined by its collection of localizations {Vp } , for p ≤ ∞ . These localizations 6 Vp are determined by their Hasse invariants. They have the following form, up to isomorphism: (i) If p = ∞ , V∞ has signature (2, 0) or (0, 2) , and these spaces have Hasse invariants ǫ∞ (V∞ ) = 1 and ǫ∞ (V∞ ) = −1 , respectively. (ii) If p is inert or ramified in k/Q , then Vp = kp with Q(x) = N (x) or Q(x) = κp · N (x) , where χp (κp ) = −1 , i.e., with κp ∈ / N k× p . These spaces have Hasse invariants ǫp (Vp ) = 1 and ǫp (Vp ) = −1 , respectively. (iii) If p splits in k , Vp = V1,1 , the split binary quadratic space. In this case, the Hasse invariant is ǫp (Vp ) = 1 . If V = k with Q(x) = κ · N (x) , then ǫp (Vp ) = χp (κ) = (κ, D)p . The Hasse Q invariants satisfy the product formula p≤∞ ǫp (Vp ) = 1 . An incoherent collection C = {Cp } is a collection of such local quadratic spaces with the following property. For some (and hence for any) global binary quadratic space V with χV = χ , there is a finite set S of places, with (i) |S| is odd, (ii) for any finite place p ∈ S , p does not split in k , and (iii) for all p ≤ ∞ , (1.1) Cp =  (Vp , κp Q) (Vp , Q) if p ∈ S, otherwise. Q Here κp is as in (ii) above and κ∞ = −1 . By the definition p≤∞ ǫp (Cp ) = −1 , and there is no global binary quadratic space with these localizations. Let G = SL(2) over Q , and let B = T N be the upper triangular Borel subgroup. Associated to quadratic character χ is the global induced representation G(A) I(s, χ) = IndB(A) (χ| |s ), of G(A) . Here Φ(s) ∈ I(s, χ) satisfies (1.2) Φ(n(b)m(a)g, s) = χ(a)|a|s+1 Φ(g, s), for b ∈ A and a ∈ A× . At s = 0 , we have a decomposition into irreducible representations of G(A) : (1.3) I(0, χ) =   ⊕V Π(V ) ⊕   ⊕C Π(C) . Here V runs over binary quadratic spaces with χV = χ , and C runs over incoherent collections with χC = χ . Let S(V (A)) be the Schwartz space of V (A) . Recall that the irreducible subspace Π(V ) is the image of the map λV : S(V (A)) → I(0, χ) defined by λV (ϕ)(g) = (ω(g)ϕ)(0) , where ω = ωψ denotes the action of 7 G(A) on S(V (A)) via the global Weil representation determined by the fixed additive character ψ . Analogously, let S(CA ) = ⊗p≤∞ S(Cp ) be the Schwartz space of C (restricted tensor product of the spaces S(Cp ) ). The group G(A) acts on this space by the restricted product of the local Weil representations, and Π(C) is the image of the equivariant map λC : S(CA ) → I(0, χ) defined by (1.4)
λC (⊗p ϕp )(g) = ⊗p (ωp (g)ϕp )(0) , where ωp = ωψp denotes the action of G(Qp ) on S(Cp ) via the local Weil representation determined by the local component ψp of ψ . For g ∈ G(A) , write g = n(b)m(a)k , where k ∈ K∞ K , K∞ = SO(2) , and K = SL2 (Ẑ) . Here a ∈ A× is not uniquely determined, but the absolute value |a(g)| := |a|A is well defined. For Φ ∈ I(0, χ) , let (1.5) Φ(g, s) = Φ(g, 0)|a(g)|s be its standard extension. For such a standard section Φ(s) , the Eisenstein series (1.6) E(g, s, Φ) = X Φ(γg, s) γ∈B(Q)\G(Q) converges absolutely for Re(s) > 1 and has an entire analytic continuation. If Φ = Φ(0) ∈ Π(C) for some incoherent collection C , the resulting incoherent Eisenstein series E(g, s, Φ) vanishes at s = 0 , [13], Theorem 2.2, and [14], Theorem 3.1. In the sections which follow, we will compute the central derivative E ′ (g, 0, Φ) of such a series in the simplest possible case. §2. The case of prime discriminant. √ Fix a prime q > 3 with q ≡ 3 mod 4 , and let k = Q( −q) . Let R = Ok be the ring of integers of k . Let χ be the character of Q× A associated to k , so that, χ(x) = (x, −q)A , where ( , )A is the global Hilbert symbol. For a prime p , let χp (x) = (x, −q)p , for the local Hilbert symbol ( , )p at p . Let V = k , viewed as a Q -vector space with quadratic form Q(x) = −N (x) , and let C be the incoherent collection defined by Cp = Vp , for all finite primes p , and sig(C∞ ) = (2, 0) . We identify C∞ with k∞ with quadratic form N (x) . We will consider the incoherent Eisenstein series associated to the standard fac- torizable section Φ(s) ∈ I(s, χ) with Φ(0) = λ(ϕ) for ϕ = ⊗p ϕp ∈ S(CA ) and 8 λ = λC : S(CA ) → I(0, χ), as above. We choose ϕ = ⊗p≤∞ ϕp ∈ S(CA ) as follows. For p < ∞ , let ϕp be the characteristic function of Rp = R ⊗Z Zp , and let ϕ∞ (x) = exp(−πN (x)) . In the range of absolute convergence, Re(s) > 1 , the incoherent Eisenstein series determined by Φ(s) has a Fourier expansion X (2.1) E(g, s, Φ) = Et (g, s, Φ), t∈Q with (2.2) Et (g, s, Φ) = Z E(n(b)g, s, Φ) ψ(−tb) db. Q\A For t 6= 0 , (2.3) Et (g, s, Φ) = Y Wt,p (gp , s, Φp ), p≤∞ where (2.4) Wt,p (gp , s, Φp ) = Z Φp (wn(b)gp , s) ψp (−tb) db Qp is the local Whittaker integral. Here db is the self dual measure with respect to ψp . Similarly, (2.5) E0 (g, s, Φ) = Φ(g, s) + M (s)Φ(g), where the global intertwining operator has a factorization M (s)Φ = ⊗p Mp (s)Φp , and (2.6) Mp (s)Φ(g) = W0,p (g, s, Φ). To obtain an explicit formula for our particular choice of ϕ ∈ S(CA ) it is necessary, first, to determine the section Φ(s) = ⊗p Φp (s) defined by ϕ and then to compute the integrals Wt,p (gp , s, Φp ) . For p 6= q , these are completely standard matters, and we will spare the reader the details of the computations and just summarize the results. Also, in the end, we will assume that, for τ = u + iv in the upper half plane, g = gτ = n(u)m(v 1/2 ) ∈ G(R) , so that we may as well assume that gp = 1 for all finite places. Since our section Φ(s) has been fixed, we will omit it from the notation and simply write: (2.7) ∗ Wt,p (s) = Lp (s + 1, χ) Wt,p (e, s, Φp ), for p a finite prime, and (2.8) ∗ Wt,∞ (τ, s) = L∞ (s + 1, χ) Wt,∞ (gτ , s, Φ∞ ). 9 Lemma 2.1. (i) For p 6= q , ∞ , the function ϕp is invariant under Kp = SL2 (Zp ) , and so Φp (s) is the unique Kp -invariant vector in Ip (s, χ) with Φp (e, s) = 1 . (ii) For p = ∞ , and for kθ ∈ SO(2) = K∞ , ω(kθ )ϕ∞ = eiθ · ϕ∞ . Thus, Φ∞ (s) is the unique weight 1 eigenvector for K∞ in I∞ (s, χ∞ ) with Φ∞ (e, s) = 1 . For p = q , let Jq ⊂ Kq = SL2 (Zq ) be the Iwahori subgroup. The ramified character χq , χq (t) = (t, −q)q defines a character of Jq by   a b χq ( ) = χq (a). qc d (2.9) Recall that Kq = Jq ∪ (Jq ∩ N )wJq . Thus, the subspace of Iq (s, χ) consisting of χq -eigenvectors of Jq is two dimensional, and is spanned by the ‘cell’ functions Φ0q (s) and Φ1q (s) , determined by (2.10) Φiq (wj , s) = δij , where w0 = 1 and w1 = w . By a direct calculation, given in section 3, we find:   a b Lemma 2.2. For k = ∈ Jq , ω(k)ϕq = χq (a)ϕq , so that Φq (s) lies qc d in the 2 dimensional subspace of Iq (s, χ) consisting of χq eigenvectors of Jq . √ Explicitly Φq (s) = Φ0q (s) + cq · Φ1q (s) , where cq = −1 q −1/2 . It follows that our section Φ(s) is a χ -eigenvector under the action of the group Q J = Jq p6=q Kp , and hence that the Eisenstein series E(g, s, Φ) is right χ - equivariant under J . Lemma 2.3. If g ∈ G(R) , then Et (g, s, Φ) = 0 for t ∈ / Z. Proof. For t ∈ Q − Z , take b ∈ Ẑ so that ψ(tb) 6= 1 . Then, since n(b) ∈ J with χ(n(b)) = 1 , we have (2.11) Et (g, s, Φ) = Et (gn(b), s, Φ) = Et (n(b)g, s, Φ) = ψ(tb) · Et (g, s, Φ), and hence Et (g, s, Φ) = 0 .  From now on we will assume that t is integral. ∗ ∗ Next, we compute the normalized local Whittaker functions Wt,p (s) , Wt,∞ (τ, s) , and their derivatives. 10 Lemma 2.4. For a finite prime p 6= q , let X = p−s . Then ordp (t) ∗ Wt,p (s) = X (χp (p)X)r . r=0 Here χp (p) = (p, −q)p is 1 if p is split in k and −1 if p is inert. If t = 0 , so that ordp (t) = ∞ , ∗ W0,p (s) = (1 − χ(p)X)−1 , and hence Mp∗ (s)Φ0p = Lp (s, χ) Φ0p (−s). Here Mp∗ (s) = Lp (s + 1, χ) Mp (s). (2.12) Let ρp (t) = ρ(pordp (t) ), (2.13) so that, for t > 0 , (2.14) ρ(t) = Y ρp (t). p Note that ρ(t) = 0 for t < 0 . The following facts, which follow immediately from Lemma 2.4, will be useful below. Lemma 2.5. At s = 0 , ∗ Wt,p (0) = ρp (t). If χp (p) = −1 and ordp (t) is odd, i.e., if ρp (t) = 0 , then 1 ∗,′ Wt,p (0) = log(p) (ordp (t) + 1) ρp (t/p). 2 Note that, ρp (t/p) = 1 here. Next consider the archimedean factor. In section 3, we indicate the proof of the following result. 11 Proposition 2.6. For τ = u + iv in the upper half plane, and gτ = n(u)m(v 1/2 ) , as above, ∗ Wt,∞ (τ, s) =v (1−s)/2 2iπ s/2 e2πtv e(tu) Γ(s/2) Z e−2πu us/2 (u − 2tv)s/2−1 du. u>0 u>2tv This formula can be found in Siegel [21], eq.(14), p. 88. For t 6= 0 , it extends to an entire function of s . ∗ (i) If t > 0 , then Wt,∞ (τ, 0) = 2i · v 1/2 e(tτ ) . (ii) If t = 0 , then
s+1 ∗ ∗ ), W0,∞ (τ, s) = M∞ (s)Φ∞ (gτ ) = i · v (1−s)/2 · π −(s+1)/2 Γ( 2 ∗ and W0,∞ (τ, 0) = iv 1/2 . ∗ (τ, 0) = 0, and (iii) If t < 0 , then Wt,∞ ∗,′ Wt,∞ (gτ , 0, Φ1∞ ) = i v 1/2 e(tτ ) · β1 (4π|t|v), where β1 is given by (0.6). Finally, in the case p = q , and noting that Lq (s + 1, χ) = 1 , a direct calculation given in section 3 below yields: Proposition 2.7. For p = q , let cq = √ −1q −1/2 , as in Lemma 2.2 above. Then ∗ Wt,q (s) = (1 − χq (t)q −s(ordq (t)+1) ) · cq . ∗ ∗ If χq (t) = −1 , then Wt,q (0) = 2cq · ρq (t) . If χq (t) = 1 , then Wt,q (0) = 0 , and ∗,′ (0) = cq log(q) (ordq (t) + 1) ρq (t). Wt,q Note that the factor ρq (t) = 1 . ∗ ∗ Allowing ordq (t) to go to infinity in Wt,q (s) , gives W0,q (s) = cq and hence (2.15) Mq∗ (s)Φq = cq · Φq (−s). We can now assemble these facts. For t 6= 0 , and for Re(s) > 1 , we have (2.16) ∗ ∗ Et∗ (gτ , s, Φ) = q (s+1)/2 · Wt,∞ (τ, s) · Wt,q (s) · Y p6=q ∗ Wt,p (s), 12 Since the factors in the product for p with ordp (t) = 0 are equal to 1 , the expressions given in Lemmas 2.4, 2.6 and 2.7 provide the entire analytic continuation of the right side of (2.16). At s = 0 the factors on the right side vanish as follows: ∗ (i) Wt,∞ (τ, 0) = 0 , if t < 0 , i.e., if χ∞ (t) = −1 , ∗ (ii) Wt,q (0) = 0 if χq (t) = 1 , and (iii) Wt,p (0) = 0 if χp (t) = −1 for p 6= q , ∞ , . Note that case (iii) occurs only when p is inert in k and ordp (t) is odd. Since (2.17) χ∞ (t)χq (t) Y χp (t) = χ(t) = 1, p6=q an odd number of factors in (2.16) are forced to vanish at s = 0 , via (i)–(iii). Precisely one factor will vanish when one of the quantities ρ(−t) , ρ(t) or ρ(t/p) is nonzero. Specifically, this factor will be ∗ (i) Wt,∞ (τ, 0) when ρ(−t) 6= 0 , i.e., when χ∞ (t) = −1 , χq (t) = −1 and χp (t) = 1 for p 6= q , ∞ ; ∗ (ii) Wt,q (0) when ρ(t) 6= 0 , i.e., when χ∞ (t) = 1 , χq (t) = 1 and χp (t) = 1 for p 6= q , ∞ ; and (iii) Wt,p (0) when ρ(t/p) 6= 0 , i.e., when χ∞ (t) = 1 , χq (t) = −1 , χp (t) = −1 and χℓ (t) = 1 for ℓ 6= p, q , ∞ . These are the only cases which contribute to the derivative at s = 0 . The corresponding nonzero Fourier coefficients of E ∗,′ (gτ , 0, Φ) can then be obtained by combining the values of normalized local Whittaker functions and their derivatives described in the Lemmas above. Proposition 2.8. (i) If ρ(−t) 6= 0 , then Et∗,′ (gτ , 0, Φ) = −2v 1/2 e(tτ ) · β1 (4π|t|v) · ρ(−t). (ii) If ρ(t) 6= 0 , then
Et∗,′ (gτ , 0, Φ) = −2v 1/2 e(tτ ) · log(q) ordq (t) + 1 · ρ(t). (iii) If ρ(t/p) 6= 0 , then
Et∗,′ (gτ , 0, Φ) = −2v 1/2 e(tτ ) · log(p) ordp (t) + 1 · ρ(t/p). Since the classical Eisenstein series of the introduction is (2.18) E ∗ (τ, s) = v −1/2 E ∗ (gτ , s, Φ), 13 we obtain the expressions of Theorem 1 for the nonconstant Fourier coefficients of φ. Finally, we compute the constant term and its derivative: E0∗ (gτ , s, Φ) = q (s+1)/2 Λ(s + 1, χq ) · Φ(gτ , s) + q (s+1)/2 M ∗ (s)Φ(gτ ) (2.19) = q (s+1)/2 Λ(s + 1, χq ) · v (s+1)/2 + q (s+1)/2 · i · cq · Λ(s, χq ) · v (1−s)/2 = q (s+1)/2 Λ(s + 1, χq ) · v (s+1)/2 − q (1−s)/2 Λ(1 − s, χq ) · v (1−s)/2 . Thus (2.20) E0∗,′ (gτ , 0, Φ)   ∂ (s+1)/2 (s+1)/2 q Λ(s + 1, χq ) · v =2 ∂s s=0   ′ Λ (1, χq ) 1/2 =v · hk · log(q) + log(v) + 2 Λ(1, χq ) This gives the constant term of φ , and finishes the proof of Theorem 1. §3. Some computations. In this section, we provide the details for the computations of the values and derivatives of the Whittaker functions used in section 2, in the case p = q or ∞ . We begin with the case p = q . Proof of Lemma 2.2. We compute the action of SL2 (Zq ) on S(V (Qq )) for the Weil representation. Write Jq = (Jq ∩ Nq− )(Jq ∩ Mq )(Jq ∩ Nq ) . For b ∈ Zq , (3.1) ω(n(b))ϕq (x) = ψq (−bN (x))ϕq (x) = ϕq (x), since x ∈ Rq implies that N (x) ∈ Zq , and ψq is unramified. Similarly, for a ∈ Z× q , (3.2) ω(m(a))ϕq (x) = χq (a)|a|ϕq (xa) = χq (a)ϕq (x). Finally, note that n− (qc) = ω(w (3.3) −1  1 qc 1  )ϕq (x) = −i = −i = wn(−qc)w−1 . Then Z Z ϕq (y) ψq (−tr(xy σ )) dy Vq ψq (−tr(xy σ )) dy Rq = −i meas(Rq )ϕ̂q (x), 14 where ϕ̂q is the characteristic function of the dual lattice R̂q = (3.4) √1 Rq −q . Then ω(n(−qc))ω(w−1 )ϕq (x) = −i meas(Rq ) ψq (qcN (x)) ϕ̂q (x), and so ω(n− (qc))ϕq (x) = meas(Rq ) (3.5) = meas(Rq ) Z ψq (qcN (y)) ψq (tr(xy σ )) ϕ̂q (y) dy Vq Z ψq (qcN (y)) ψq (tr(xy σ )) dy R̂q = meas(Rq ) Z ψq (tr(xy σ )) dy R̂q = meas(Rq ) meas(R̂q ) ϕq (x). Here meas(Rq ) meas(R̂q ) = 1 , since the measure dy is self dual with respect to ψq . The second statement is clear, since (3.6) λq (ϕq ) = λq (ϕq )(w0 )Φ0q (0) + λq (ϕq )(w1 )Φ1q (0), so that cq = λq (ϕq )(w) (3.7) = ω(w)ϕq (0) Z =i ϕq (y) dy Vq = i meas(Rq ) = i q −1/2 .  ∗ Proof of Proposition 2.7. Recall that Wt,q (s) = Wt,q (e, s, Φq ) , where, for Re(s) > 1 , the Whittaker integral Wt,q (g, s, Φq ) is as in (2.4). We will actually determine this function completely. It is easy to check that (3.8) Wt,q (n(b)m(a)g, s, Φ) = χq (a)|a|−s+1 ψq (tb) Wa2 t,q (g, s, Φ), so that, by Lemma 2.2 and the decomposition (3.9) G(Qq ) = Nq Mq w0 Jq ∪ Nq Mq w1 Jq , it will suffice to compute the functions (3.10) Wij (s, t) := Wt,q (wi , s, Φjq ) = Z Qq Φjq (wn(b)wi , s) ψq (−tb) db, 15 for i, j ∈ {0, 1} and the cell functions Φjq as in (2.10). Let (3.11) Wij (s, t) (0) Z = Zq Φjq (wn(b)wi , s) ψq (−tb) db, and (3.12) Wij (s, t) (1) = Z Qq −Zq Φjq (wn(b)wi , s) ψq (−tb) db. For b ∈ Zq , wn(b)wi ∈ Kq , so the integrand in Wij (s, t)(0) is independent of s . Then W0j (s, t) (0) (3.13) = = Z Zq Φjq (wn(b), s) ψq (−tb) db Φjq (w, s) Z ψq (−tb) db Zq = δ1j char(Zq )(t), where char(Zq ) is the characteristic function of Zq . Next, (3.14) W1j (s, t) (0) = Z Zq Φjq (−wn(b)w−1 , s) ψq (−tb) db. Since wn(qx)w−1 ∈ Jq , the function b 7→ Φjq (−wn(b)w−1 , s) depends only on b modulo qZq , and thus the integral vanishes if t ∈ / q −1 Zq . If t ∈ q −1 Zq , then (3.15) W1j (s, t)(0) = q −1 χq (−1) X b∈Z/qZ Φjq (n− (−b)) ψq (−tb). Note that n− (−b) ∈ Jq if and only if b ∈ qZq . If b ∈ Z× q , then write (3.16) n− (−b) = n(x)wk = n(x)w  A qC B D  . This yields (3.17) (0, 1)n− (−b) = (−b, 1) = (−1, 0)k = (−A, −B), and thus A = b and B = −1 . Hence (3.18) Φjq (n− (−b)) = χq (b) · δ1j , 16 and W1j (s, t) (0) =q −1  χq (−1) δ0j + δ1j X χq (b) ψq (−tb) b∈(Z/qZ)×   −1 q χq (−1)δ0j if ordq (t) ≥ 0,      = q −1 χq (−1) δ0j + δ1j χq (t) g(χq ) if ordq (t) = −1.    0 otherwise. (3.19) Here (3.20) X g(χq ) = χq (b) ψ(−b/q) b∈(Z/qZ)× is the Gauss sum for χq . We note that χq (q) = (q, −q)q = 1 and χq (b) = for b ∈ (Z/qZ)× , so that χq (−1) = −1 . Next consider Wij (s, t)(1) . Here, for ordq (b) < 0 ,   −1    −b 1 1 −b−1 (3.21) wn(b) = = 1 −1 −b −b  1 b−1 1    b q . Thus Φjq (wn(b)wi , s) = χq (−b)|b|−s−1 δij , (3.22) since n− (b−1 )wi = wi wi−1 n− (b−1 )wi and wi−1 n− (b−1 )wi ∈ (Jq ∩ N− ) ∪ (Jq ∩ N ) (3.23) for either i . Therefore, recalling that χq (q) = 1 , Wij (s, t) (1) = χq (−1)δij ∞ X r=1 (3.24) =  q −rs Z Z× q χq (b) ψq (−tb/q r ) db χq (−1) δij q −1−s(ordq (t)+1) χq (t) g(χq ) 0 Here we use the fact that  Z 0 r (3.25) χq (b) ψq (−tb/q ) db = q −1 χq (t) g(χq ) Z× q if ordq (t) ≥ 0, otherwise. if r 6= ordq (t) + 1, if r = ordq (t) + 1. These results can be summarized as follows: If ordq (t) ≥ 0 , then, writing r = ordq (t) + 1 ,    −sr  q χq (t) g(χq ) qχq (−1) −1 (3.26) Wij (s, t) = q χq (−1) . 1 q −sr χq (t) g(χq ) 17 If ordq (t) = −1 , then     0 0 −1 Wij (s, t) = q χq (−1) . 1 χq (t)g(χq ) (3.27) Otherwise (Wij (s, t)) = 0 . √ Finally, recall that Φq (s) = Φ0q (s) + cq Φ1q (s) , with cq = √ g(χq ) = −1 q 1/2 . Thus, if ordq (t) ≥ 0 , (3.28)  Wt,q (w0 , s, Φq ) Wt,q (w1 , s, Φq )  =     1 Wij (s, t) · cq = (1 − χq (t)q −s(ordq (t)+1) )  −1 q −1/2 , and that cq −q −1  . If ordq (t) = −1 , then (3.29)  Wt,q (w0 , s, Φq ) Wt,q (w1 , s, Φq )  = (1 − χq (t))  0 −q −1  . This completes the proof of Proposition 2.7, and a little more.  We next turn to the case p = ∞ . Of course, these calculations are rather well known, but we include them for convenience. Proof of Proposition 2.6. Recall that, by (2.8), (3.30) 1 ∗ ∗ (τ, s) = v 2 (1−s) e(tu) Wtv,∞ Wt,∞ (e, s) 1 = v 2 (1−s) e(tu) L∞ (s + 1, χ) W (s, tv), where (3.31) L∞ (s + 1, χ) = π −s/2−1 Γ as in (0.4), and where we set (3.32)
s +1 , 2 W (s, t) := Wt,∞ (e, s, Φ∞ ). A simple computation of the Iwasawa decomposition of wn(b) yields (3.33) Φ1∞ (wn(b), s) = i · (1 + ib) . (1 + b2 )s/2+1 18 Thus (3.34) W (s, t) = i · ∞ Z (1 + ib)−s/2 (1 − ib)−s/2−1 e(−tb) db −∞ s/2+1 π =i Γ(s/2 + 1) Z ∞ (1 + ib) −s/2 Z ∞ e−π(1−ib)u us/2 du e(−tb) db. 0 −∞ Recall that Z (3.35) ∞ e(−xv) e−xz xs−1 dx = (z + 2πiv)−s Γ(s), 0 and hence, Z (3.36) ∞ e(vx) (z + 2πiv) −s dv = −∞  1 Γ(s) 0 · e−xz xs−1 if x > 0, if x ≤ 0. Thus (3.37) ∞ 1 (1 + ib)−s/2 e(b( u − t)) db 2 −∞ ( 1 (2π)s/2 · e−2π( 2 u−t) ( 12 u − t)s/2−1 Γ(s/2) = 0 Z if 21 u − t > 0, if 21 u − t ≤ 0. This gives (3.38) π s/2+1 (2π)s/2 W (s, t) = i Γ(s/2 + 1) Γ(s/2) 1 1 e−πu us/2 e−2π( 2 u−t) ( u − t)s/2−1 du 2 Z u>0 u>2t s+1 2πt π e = 2i · Γ(s/2)Γ(s/2 + 1) Z e−2πu us/2 (u − 2t)s/2−1 du. u>0 u>2t This yields the first statement of Proposition 2.6. We are interested in the behavior at s = 0 . If t > 0 , the substitution of u + 2t for u yields (3.39) Z ∞ π s+1 e−2πt e−2πu (u + 2t)s/2 us/2−1 du W (s, t) = 2i · Γ(s/2)Γ(s/2 + 1) 0   π s+1 e−2πt −s/2 s/2 = 2i · Γ(s/2)(2π) (2t) + O(s) . Γ(s/2)Γ(s/2 + 1) 19 Thus, if t > 0 , W (0, t) = 2πi · e−2πt . If t = 0 , we get π s+1 W (s, 0) = 2i · Γ(s/2)Γ(s/2 + 1) (3.40) = 2πi · Z ∞ e−2πu us−1 du 0 2−s Γ(s) Γ(s/2)Γ(s/2 + 1) = iπ 1/2 · Γ( s+1 ) 2 , ) Γ( s+2 2 via duplication. Finally, when t < 0 , (3.41) π s+1 e2πt W (s, t) = 2i · Γ(s/2)Γ(s/2 + 1) Z ∞ e−2πu us/2 (u + 2|t|)s/2−1 du. 0 The integral here converges for Re(s) > −2 , so that the Γ(s/2) in the denominator yields W (0, t) = 0 when t < 0 . Note that Z ∞ e−2πu (u + 2|t|)−1 du 0 (3.42) = Z ∞ e−4π|t|u (2|t|u + 2|t|)−1 2|t| du 0 ∞ e−4π|t|u du u+1 0 Z ∞ −4π|t|(u+1) e 4π|t| du =e · u+1 0 = Z = −e4π|t| · Ei(−4π|t|). Therefore (3.43)   ∂ W (s, t) ∂s s=0 = −2πi · 1 2πt 4π|t| ·e ·e · Ei(−4π|t|) 2 = −iπ · e2π|t| · Ei(−4π|t|), when t < 0 . This completes the proof of Proposition 2.6.  §4. The Mellin transform. In this section, we compute the Mellin transform Λ(s, φ) . 20 Lemma 4.1. The Mellin transform (0.7) can be computed termwise. Proof. It suffice to check the termwise absolute convergence of the expression (4.1) − Z 0 ∞ ∞X t=1 ρ(t) · Ei(−4πtv) e2πtv v s−1 dv. It will turn out to be more convenient to consider the expression (4.2) − ∞ X ρ(t) Z ∞ Ei(−4πtv) e2πtv v σ−1 dv, 0 t=1 with σ > 1 . First, by a change of variable in each integral, we obtain − (4.3) ∞ X ρ(t)(2πt) −σ ∞ Z Ei(−2v) ev v σ−1 dv 0 t=1 X  Z ∞ −σ · =− ρ(t)(2πt) v Ei(−2v) e v σ−1 0 t=1 = −(2π) ∞ −σ ζk (σ) · Z ∞ dv  Ei(−2v) ev v σ−1 dv. 0 Thus, it suffices to prove that that the integral is convergent when σ > 1 . Note that the expression (4.4) −Ei(−2v) = Z ∞ 1 e−2vu du u is positive. If v > δ > 0 , and if 0 < ǫ < 1 , we can write: v −Ei(−2v) e < (4.5) Z ∞ 1 e−(1−ǫ)vu du e−ǫv u ∞ e−(1−ǫ)δu du e−ǫv u 1 −ǫv = Cδ,ǫ · e . < Z On the other hand, if 0 < v < δ , we have v δ −Ei(−2v) e < e (4.6) < eδ Z ∞ Z1 ∞ e−2vu du u e−2vu du 1 δ = e (2v)−1 . 21 Returning to the integral, (4.7) − Z ∞ v Ei(−2v) e v σ−1 dv < 0 Z δ δ e (2v) −1 v σ−1 dv + 0 Z ∞ δ Cδ,ǫ · e−ǫv v σ−1 dv. The second integral on the right side converges for all σ , while the first is convergent for Re(s) > 1 , as required.  Computing (0.7) termwise, the contribution of the holomorphic terms is simply (4.8) 2(2π) −s Γ(s) ∞  X log(q)(ordq (t) + 1) ρ(t) + t=1 X  log(p)(ordp (t) + 1)ρ(t/p) t−s , p6=q, ∞ where the sum on p is over primes inert in k . If p is such a prime, then ρ(pr ) = 1 if r is even and 0 if r is odd, and we have ∞ X (ordp (t) + 1) ρ(t/p) t −s = ∞ ∞ X X (r + 1) ρ(pr−1 t) p−rs t−s t=1 r=0 (t,p)=1 t=1 (4.9) =  X ∞ ρ(t) t −s = (1 − p = (4.10) t=1 r=0 2p−s · ζ (s). 1 − p−2s k (ordq (t) + 1) ρ(t) t−s = (1 − q −s )−1 · ζk (s). Thus, the holomorphic part of the Fourier expansion contributes (4.11)  X 2p−s 1 Λk (s) log(q) + log(p) 1 − q −s 1 − p−2s p6=q   ′ L (s, χ) ζ ′ (s) = Λk (s) log(q) + , − L(s, χ) ζ(s) as one easily checks.  −rs )p  X  ∞ −s(2r+1) ) · ζk (s) · (2r + 2) p Similarly, ∞ X r−1 (r + 1) ρ(p r=0 t=1 (t,p)=1 −2s  X ∞ 22 The termwise transform of the nonholomorphic part of the Fourier expansion is 2 ∞ X ρ(t) t=1 (4.12) = 2(2π) ∞ Z β1 (4πtv) e2πtv v s−1 dv 0 −s ∞ X ρ(t) t −s = 2(2π) v β1 (2v) e v s−1 dv 0 t=1 −s ∞ Z   ′ Γ( s+1 Γ( 2s )′ 2 ) ζk (s) Γ(s) . − Γ( 2s ) ) Γ( s+1 2  Thus, the contribution of these terms is   s+1 ′ Γ( 2 ) Γ( 2s )′ (4.13) Λk (s) . − s ) Γ( Γ( s+1 ) 2 2 Combining this with the contribution of the holomorphic terms, we obtain the expression of Theorem 2 for Λ(s, φ) . In (4.12) we have used the following identity, noting that Γ( 2s )′ = 21 Γ′ ( 2s ) . Lemma 4.2. Z ∞ v β1 (2v) e v 0 s−1   ′ s+1 Γ ( 2 ) Γ′ ( 2s ) 1 dv = Γ(s) . − 2 Γ( 2s ) Γ( s+1 2 ) Proof. Recall that, [18], (4.14) Γ′ (s) = −γ + ψ(s) := Γ(s) Z 0 1 1 − xs−1 dx. 1−x Therefore (4.15) ψ(s + 1/2) − ψ(s) = Z = Z = Z 1 xs−1 − xs−1/2 dx 1−x 0 1 xs−1 0 1 0 =2 Z 0 1 − x1/2 dx 1−x xs−1 dx 1 + x1/2 1 u2s−1 du. 1+u Thus we have the useful formula: Z 1 s−1 w s+1 s (4.16) 2 dw = ψ( ) − ψ( ). 2 2 0 1+w 23 But now Z ∞ β1 (2v) ev v s−1 dv 0 = (4.17) Z ∞ 0 = Γ(s) ∞ Z e−2uv du ev v s−1 dv u 1 Z ∞ 1 = Γ(s) Z = Γ(s) Z (2u − 1)−s u−1 du ∞ r −s (1 + r)−1 dr 1 1 0 This gives the claimed expression. ws−1 dw. 1+w  §5. Algebraic-geometric aspects. In the beginning of this section we will consider a slightly more general situation than before. We let k be an imaginary quadratic field with ring of integers Ok . We denote by a 7→ a the non trivial automorphism of k . We will consider the following moduli problem M over (Sch/Ok ) . To S ∈ (Sch/Ok ) we associate the category M(S) of pairs (E, ι) where • E is an elliptic curve over S (i.e. an abelian scheme of relative dimension one over S ) • ι : Ok → EndS (E) is a homomorphism such that the induced homomorphism (5.1) Lie(ι) : Ok → EndOS (Lie E) = OS coincides with the structure homomorphism. The morphisms in this category are the isomorphisms. A pair (E, ι) will be called an elliptic curve with CM by Ok . We denote by M the corresponding set valued functor of isomorphism classes of objects of M . Proposition 5.1. a) The moduli problem M is representable by an algebraic stack (in the sense of Deligne-Mumford) which is finite and étale over Spec Ok . b) The functor M has a coarse moduli scheme which is also finite and étale over Spec Ok . Proof. a) Let M̃ be the algebraic moduli stack of elliptic curves over Spec Ok . The forgetful morphism i : M → M̃ is relatively representable by an unramified 24 morphism (rigidity theorem). Since the generic elliptic curve in any characteristic has only trivial (i.e., Z ) endomorphisms, M has finite fibres over Spec Ok . Now the finiteness of M over Spec Ok follows from the valuative criterion of properness, since an elliptic curve with complex multiplication has potentially good reduction. The étaleness of M over Spec Ok will follow by the Serre-Tate theorem from a study of the p -divisible group of an elliptic curve with CM by Ok . b) That M has a coarse moduli scheme follows from a) (or use level- n -structures). Let ξ be a geometric point of M and ξ the corresponding geometric point of M . Let ÔM,ξ and ÔM,ξ be the completions of the local rings. If ξ corresponds to (E, ι) , we have (5.2) Aut(ξ) = Aut(E, ι) = Ok× , comp. cases 1 and 2 below. This group acts on ÔM,ξ and (5.3) ÔM,ξ̄ = (ÔM,ξ )Aut(ξ) . However, since M is étale over Spec Ok , we have that ÔM,ξ is equal to the strict completion of Ok in the prime ideal corresponding to the image of ξ in Spec Ok . Hence the action of Aut(ξ) on ÔM,ξ is trivial. Therefore the canonical morphism (5.4) M→M is étale and the last assertion follows.  Corollary 5.2. Let ξ be a geometric point of M and let ξ¯ be the corresponding point of M . Then ÔM,ξ̄ = ÔM,ξ . Let p be a prime number and ℘ a prime ideal of Ok above p . Let κ be an algebraically closed extension of the residue field κ(℘) of ℘ . Let (E, ι) ∈ M(κ) , and let X be the p -divisible group of E . Then the p -adic completion Op = Ok ⊗ Zp acts on X . We distinguish two cases. Case 1: p splits in Ok . Then p = ℘ · ℘ with ℘ 6= ℘ and (5.5) Op = O℘ ⊕ O℘ = Zp ⊕ Zp . By the condition on Lie(ι) the corresponding decomposition of X is of the form (5.6) X = Ĝm × Qp /Zp . 25 In this case it is obvious that (X, ι) deforms uniquely, i.e. the base of the universal deformation is Spf W (κ) , where W (κ) denotes the ring of Witt vectors of κ . In this case the elliptic curve E is ordinary and (5.7) ∼ ι : Ok −→ End(E) . Case 2: p not split in Ok . In this case Op = O℘ and (X, ι) is a formal O℘ - module of height 2 over κ , in the sense of Drinfeld [7], comp. also [10]. Let WO (κ) be the ring of relative Witt vectors of κ , [2]. Then again (X, ι) deforms uniquely, i.e. the base of the universal deformation of (X, ι) is Spf WO (κ) ([19], comp. [10], 2.1). In this case the elliptic curve E is supersingular. Let Bp be the definite quaternion algebra over Q which ramifies at p . Then End(E) is isomorphic to a maximal order in Bp . The main theorem of complex multiplication may be summarized as follows (comp. [20],[3]) ¯ be the algebraic closure of k . The Galois group Gal(k ¯ /k) Proposition 5.3. Let k ¯ ) through its maximal abelian quotient, which we identify via class field acts on M (k theory with (k ⊗ Af )× /k× . There is a bijection ¯ ) = k× \(k ⊗ Af )× /(O ⊗ Ẑ)× , M (k k ¯ /k)ab represented by an idèle x ∈ (k ⊗ Af )× acts such that an element of Gal(k by translation by x on the right side. Corollary 5.4. Let H be the Hilbert class field of k and let OH be its ring of integers. Then M ≃ Spec(OH ). Corollary 5.5. Fix p and a prime ℘ above p . Then M(κ(℘)) forms a single isogeny class. There is a bijection M (κ(℘)) = k× \ (k ⊗ Af )× /(Ok ⊗ Ẑ)× The action by the Frobenius automorphism over κ(℘) on the left corresponds to the translation by the idèle with component a uniformizer at ℘ and 1 at all other finite places on the right. Remark 5.6. The bijection in Corollary 5.5 is given explicitly as follows. Choose a base point (Eo , ιo ) ∈ M(κ(℘)) , and an identification End0 (Eo ) = Bp . Let 26 O(Eo ) ⊂ Bp be the maximal order corresponding to End(Eo ) . To g ∈ (k ⊗ Af )× there is associated the point (E, ι) ∈ M(κ(℘)) whose Tate module T̂ (E) = Q ℓ Tℓ (E) (with p -component the covariant Dieudonné module) is the lattice (5.8) g · T̂ (Eo ) ⊂ T̂ (Eo ) ⊗ Q . In particular (5.9) O(E) = Bp ∩ g(O(Eo) ⊗ Ẑ)g −1 , is a maximal order in Bp . We now turn to the definition of special cycles on the moduli space/stack M . Definition 5.7. Let (E, ι) ∈ M(S) . A special endomorphism of (E, ι) is an element y ∈ End(E) with (5.10) y ◦ ι(a) = ι(a) ◦ y, for all a ∈ Ok . Let V (E, ι) be the set of special endomorphisms of (E, ι) . Then V (E, ι) is an Ok -module. Let us first consider the case where S = Spec κ with κ an algebraically closed field. If char κ = 0 , then End(E) = Ok and hence V (E, ι) = (0) . The same is true if char κ = p and p splits in Ok , cf. (5.7). Let now char κ = p where p does not split. Then End(E) can be identified with a maximal order O(E) in the quaternion algebra Bp introduced earlier and the Ok -module V (E, ι) is of rank one. Let a 7→ a′ be the main involution of Bp . Then V (E, ι) is stable under this involution. Let y ∈ V (E, ι) . Since (5.11) y + y ′ ∈ Z ∩ V (E, ι) = (0) , we obtain y 2 = −y · y ′ ∈ Z . We define a Z -valued quadratic form Q on the Z module V (E, ι) of rank 2 by (5.12) Q(y) = −y 2 = y · y ′ . The definition of this quadratic form extends to an elliptic curve with CM by Ok over any connected scheme S . 27 Definition 5.8. For a positive integer t , let Z(t) be the moduli problem over M which to S ∈ (Sch/Ok ) associates the category of triples (E, ι, y) where (E, ι) ∈ M(S) and where y ∈ V (E, ι) satisfies Q(j) = −y 2 = t. (5.13) We denote by (5.14) pr : Z(t) −→ M the forgetful morphism. It is obvious that pr is relatively representable, hence Z(t) is representable by an algebraic stack. By the rigidity theorem the morphism pr is finite and unramified. We also denote by pr the induced morphism between coarse moduli schemes (5.15) pr : Z(t) −→ M . ˜ = {±1} . Indeed, if We note that for a geometric point ξ˜ ∈ Z(t) we have Aut(ξ) ξ˜ corresponds to (E, ι, y) then an automorphism is given by a unit u ∈ Ok× with uy = yu , i.e. with u = u . Proposition 5.9. The fibre of Z(t) in characteristic 0 or in characteristic p where p splits in k is empty. In particular, Z(t) is an artinian scheme. For p not split in k , let ℘ be the prime ideal of Ok dividing p . The fibre of pr over a geometric point (E, ι) ∈ M (κ(℘)) is empty if t is not represented by the quadratic form on V (E, ι) . Let p be inert or ramified in k . We choose a base point (Eo , ιo ) ∈ M(κ(℘)) and denote by V (p) the quadratic space V (Eo , ιo ) ⊗Z Q over Q . Then (5.16) End0 (Eo , ιo ) = Bp = k ⊕ V (p) , where the decomposition is orthogonal with respect to the quadratic form given by the reduced norm on Bp . Choosing a set of coset representative gi , i = 1, . . . , hk for the cosets on the right side of Corollary 5.5 and applying the construction of Remark 5.6, we obtain Ok -lattices (5.17) in V (p) . V (Ei , ιi ) = V (p) ∩ O(Ei ) 28 Corollary 5.10. Let p be inert or ramified in k and ℘ ⊂ Ok over p . Then X |Z(t)(κ(℘))| = |{x ∈ V (Ei , ιi ) | Q(x) = t}|. i We now quote the result of Gross [10], Prop. 3.3 and 4.3. Let p be an inert or ramified prime and κ an algebraically closed extension of κ(℘) . If ξ ∈ M(κ) , there is an isomorphism (5.18) ÔM,ξ = WO (κ) , cf. case 2 above. Here again WO (κ) = WO℘ (κ) is the ring of relative Witt vectors. We let π denote a uniformizer of O℘ and hence of WO (κ) . Let ξ˜ ∈ Z(t)(κ) with ˜ = ξ , and ξ˜ ∈ Z(t)(κ) its image in Z(t) . Here pr is as in (5.14). pr(ξ) Theorem 5.11. (Gross): We have ÔZ(t),ξ̃ = ÔZ(t),ξ̃ = WO (κ)/(π ν ) where ν = νp (t) = ordp (t) + dp − 1 + 1, fp dp = ordp (disck/Q ), and fp = [κ(℘) : Fp ]. In particular, if p is unramified in k , we have fp = 2 , and dp = 0 and ordp (t) ≡ 1(mod 2) , and ordp (t) + 1 . 2 If p is ramified in k , then fp = 1 and dp ≥ 1 and, if p 6= 2 , νp (t) = νp (t) = ordp (t) + 1, √ If p = 2 is ramified in k , write k = Q( −q) with q square free. Then    ordp (t) + 1 if q ≡ 3 mod 4, νp (t) = ordp (t) + 2 if q ≡ 1 mod 4,   ordp (t) + 3 if q ≡ 2 mod 4. Note that the length of ÔZ(t),ξ̃ only depends on ordp (t) . Combining Corollary 5.10 and Theorem 5.11, we therefore obtain an expression for the degree of Z(t) . Here we define the degree as (5.19) deg(Z(t)) = X ξ∈Z(t) [22], [9].
log #(OZ(t),ξ ) , 29 Theorem 5.12. deg(Z(t)) = X p fp log(p) νp (t) · X i |{x ∈ V (Ei , ιi ) | Q(x) = t}|. Here p ranges over the primes which are not split in k . For a fixed p which is not split in k , we now describe the lattices V (Ei , ιi ) of (5.17) more explicitly. The basic technique is from [11], [6]. Let ∆ be the discriminant of the field k . Fix an auxillary prime p0 ∤ 2p∆ as follows. If p is inert in k , we require that (5.20)  (∆, −pp0 )v = −1 1 if v = p or ∞, and otherwise. If p is ramified in k , we require that (5.21) (∆, −p0 )v =  −1 1 if v = p or ∞, and otherwise. Let (5.22) κp =  pp0 p0 if p is inert in k, and if p is ramified in k. Then Bp is the cyclic algebra (∆, −κp ) and thus can be written in the form (5.23) Bp = k ⊕ k · y for an element y with tr(y) = 0 , y 2 = −κp , and a · y = y · ā for all a ∈ k . Let R = Ok ⊕ Ok · y , so that R is an order in Bp with R ∩ k = Ok . Suppose that O is a maximal order of Bp containing R . If x1 ∈ R and x2 ∈ O , then tr(x1 x2 ) ∈ Z . Using the coordinates from (5.23), this gives (5.24) tr((α1 + β2 y)(α2 + β2 y)) = trk (α1 α2 − κp β1 β̄2 ) ∈ Z. Therefore (5.25) where D −1 = (5.26) −1 R = Ok ⊕ Ok · y ⊂ O ⊂ D −1 ⊕ κ−1 · y, p D √ −1 ∆ Ok is the inverse different of k . Setting V (O) = V (p) ∩ O, 30 we conclude that (5.27) −1 Ok · y ⊂ V (O) ⊂ κ−1 · y. p D The conditions imposed on p0 imply that p0 splits. Let p0 Ok = ℘0 ℘¯0 and let pOk = ℘ep . Then, it is easily checked that (5.28) R′ := Ok ⊕ ℘−1 0 y and R′′ := Ok ⊕ ℘¯−1 0 y are orders in Bp containing R . Note that R′ and R′′ cannot be contained in the −1 same order O . Indeed, if R′ + R′′ ⊂ O , then (℘−1 ¯−1 0 +℘ 0 ) · y = p0 Ok · y ⊂ O . −1 2 −2 But Q(p−1 / Z. 0 y) = −(p0 y) = p0 κp ∈ √ −1 −1 ∆ βy ∈ Suppose that O is a maximal order containing Ok ⊕ ℘−1 0 y . If x = κp V (O) , with β ∈ Ok , then (5.29) This forces (5.30) −1 Q(x) = −x2 = κ−1 N (β) ∈ Z. p ∆ √ β ∈ ℘−1 ∆ Ok , κ p 0 or √ β ∈ ℘¯0−1 κp ∆ Ok , x ∈ ℘−1 0 y or x ∈ ℘¯−1 0 y. so that (5.31) Therefore, (5.32) V (O) = ℘−1 0 · y. Fix a maximal order O containing R′ , and note that O ∩ k = Ok , i.e., that Ok is optimally embedded in O , in Eichler’s terminology, [8]. By the Chevalley-Hasse- Noether Theorem (comp. [8], Satz 7), it follows that if O′ is any maximal order in Bp in which Ok is optimally embedded, then there exists a finite idèle g ∈ k× Af , such that (5.33) O′ = g(O ⊗Z Ẑ)g −1 ∩ Bp . If then
a = gOk ⊗Z Ẑ ∩ k, (5.35) V (O′ ) = aā−1 ℘−1 0 y. (5.34) Conversely, any ideal a arises in this way. Now suppose that O′ = End(E) for a supersingular elliptic curve (E, ι) ∈ M(κp (℘)) over F̄p with complex multiplication by Ok . Note that Ok is optimally embedded in O′ and that (5.36) V (E, ι) = O′ ∩ ky = V (O′ ). 31 Proposition 5.13. Fix an auxillary prime p0 as above. Then there is an element y0 in V with Q(y0 ) = κp , and the lattices V (Ei , ιi ) in Theorem 5.12 have the form −1 V (Ei , ιi ) ≃ ai ā−1 i ℘0 · y0 , as the ai run over representatives for the ideal classes of k . In particular, the isomorphism class of V (Ei , ιi ) runs over the genus [[℘0 ]] of the ideal ℘0 . Remark. The genus [[℘0 ]] is independent of the choice of ℘0 and is characterized by the values of the genus characters. Recall that a basis for the characters of the group of genera are given as follows, [12]. Let qi , 1 ≤ i ≤ t be the primes dividing ∆ . Let (5.37) χi (a) = (∆, N (a))qi , where ( , )qi is the Hilbert symbol for Qqi . (∆, N (x))qi = 1 , and that (5.38) Note that, if x ∈ k× , then (∆, N (a)N (b)2)qi = (∆, N (a))qi . Thus χi depends only on the ideal class of b modulo squares, i.e., defines a character of the group of genera Cl(k)/2Cl(k) . Any t − 1 of the χi ’s give a basis for the characters of this group. The conditions (5.20) and (5.21) on the auxillary prime p0 imply that  (∆, −p)qi if p is inert,   (5.39) χi (℘0 ) = (∆, p0 )qi = (∆, −1)qi if p is ramified and qi 6= p, and   −(∆, −1)qi if p is ramified and qi = p. The genus of ℘0 is determined by these conditions. Corollary 5.14. (i) The rational quadratic space V (p) is given by (V (p) , Q) ≃ (k, κp N ), where κp is given by (5.22). (ii) Let Ci be the ideal class 2 Ci = [a−1 i a¯i ℘0 ] = [ai ] [℘0 ]. Then {x ∈ V (Ei , ιi ) | Q(x) = t} = |Ok× | · {c ⊂ Ok | [c] = Ci , and N (c) = p0 t/κp } . 32 Note that (5.40) p0 /κp =  p if p is inert, and 1 if p is ramified. −1 Proof. We just note that, if x = β · y ∈ V (Ei , ιi ) , with β ∈ ai ā−1 i ℘0 , then t = Q(x) = κp N (β) . The ideal c = βa−1 i a¯i ℘0 is integral, lies in the ideal class Ci , and has (5.41) N (c) = N (β)p0 = tp0 /κp .  Note that, by genus theory, as i runs from 1 to hk , [ai ] runs over the ideal class
group and Ci runs over 2Cl(k) [℘0 ] = [[℘0 ]] , the genus of ℘0 , with each class in [[℘0 ]] occurring 2t−1 times. Here again t is the number of rational primes which ramify in k . Finally, we specialize to the case of prime discriminant considered in earlier sections √ of this paper, i.e., let k = Q( −q) with q > 3 a prime congruent to 3 modulo 4 . In this case, the class number of k is odd, there is only one genus, and the Ci ’s run over all ideal classes. Combining the results above, we obtain: Theorem 5.15. deg(Z(t)) = 2 log(q) · (ordq (t) + 1) · ρ(t) + 2 X p log(p) · (ordp (t) + 1) · ρ(t/p). Here p runs over the primes which are inert in k . This is precisely the expression given in Theorem 3 of the introduction. §6. Variants. In this section, we will sketch a few variations and extensions of the results of the previous sections. However, none of them touch on the most tantalizing problem of establishing a direct connection between the Fourier coefficients and the degrees of special cycles. The central derivatives of all of the incoherent Eisenstein series constructed in section 1 should have an arithmetic interpretation analogous to Theorem 3. A full description of such a result would require several extensions of what we have done above, which is specialized in several respects. 33 First of all, we have restricted to the case of a prime discriminant ∆ = −q . This eliminates complications involving genus theory. Of course, the moduli problem of section 5 has been set up to allow elliptic curves with complex multiplication √ by Ok where k = Q( ∆) for an arbitrary fundamental discriminant ∆ < 0 . On the analytic side, the machinery of section 1 and 2 provides corresponding incoherent Eisenstein series, whose Fourier expansions can be computed by the methods of section 2. It should be then a routine matter to work out the analogues of Theorems 1, 2 and 3 in this case, except that the group of genera will now play a nontrivial role. Second, in section 2, we have considered only those Eisenstein series built from sections which are defined by the characteristic functions of the completions of Ok . More general sections could be considered at the cost of (i) more elaborate calculations of Whittaker functions and their derivatives used in computing the Fourier expansion of the central derivative, and (ii) incorporation of a level structure in the moduli problem. Third, in section 2, we have restricted to the simplest type of incoherent collection C , i.e., a collection C which is obtained by considering the global quadratic space (V, Q) = (k, −N ) and switching the signature from (0, 2) to (2, 0) . Thus, at each finite place, Cp = (kp , −N ) . The most general collection C is obtained by switching the signature of the space (V, Q) = (k, κN ) where κ ∈ Q× . In this situation, a slightly more elaborate moduli problem is necessary. We give a brief sketch of this. Let B be a indefinite quaternion algebra over Q , with a fixed embedding i : k ֒→ B . Fix a maximal order OB , in B , a positive involution ∗ of B stabilizing OB , and assume that i(Ok ) ⊂ OB . We consider polarized abelian surfaces A with an action ι : OB ⊗ Ok → End(A) compatible with the positive involution b ⊗ a 7→ (b ⊗ a)∗ = b∗ ⊗ ā . For such an (A, ι) , let (6.1) V (A, ι) = {y ∈ End(A) | yι(b ⊗ a) = ι(b ⊗ ā)y } be the space of special endomorphisms. Observe that (6.2) M2 (k) ≃ B ⊗ k ֒→ End0 (A), so that A is isogenous to a product A20 for an elliptic curve A0 with complex multiplication. Thus (6.3) End0 (A) ≃ M2 (End0 (A0 )). 34 If A0 is not a supersingular elliptic curve in characteristic p , for p not split in k , then End0 (A) ≃ M2 (k) and V 0 (A, ι) = V (A, ι) ⊗Z Q = 0 . If A0 is supersingular in characteristic p , then (6.4) End0 (A) ≃ M2 (Bp ), where Bp = End0 (A0 ) is, as before, the definite quaternion algebra ramified at infinity and p . The commutator of ι(B ⊗ 1) in End0 (A) is then isomorphic to B ′ , the definite quaternion algebra over Q with (6.5) invℓ (B ′ ) = ( invℓ (B) if ℓ 6= p, and −invℓ (B ′ ) if ℓ = p. Note that ι yields an embedding of k into B ′ , and we have a decomposition, analogous to (5.16), (6.6) B ′ = k ⊕ V 0 (A, ι), where V 0 (A, ι) = k · y for an nonzero element y ∈ B ′ with ya = āy for all a ∈ k . The quadratic form on V 0 (A, ι) is then (6.7) Q(βy) = −(βy)2 = −N (β)y 2 , so that the binary quadratic space (V 0 (A, ι), Q) is isomorphic to (k, κN ) where κ = −y 2 . Note that B ′ is then isomorphic to the cyclic algebra (∆, −κ) . An analysis of the corresponding integral moduli problem like that of section 5 should be possible and give the desired generalization of the results above. The case B = M2 (Q) will reduce precisely to the situation of section 5. √ Returning to Theorem 3 in the case k = Q( −q) , from the point of view of Arakelov theory, it is possible to given an interpretation of the negative Fourier coefficients of φ as degrees of cycles. First recall, [22], that an Arakelov divisor for M = Spec(OH ) is an expression of the form h (6.8) D= k X λ rλ λ + X np p, p where p runs over the nonzero prime ideals of OH , λ runs over the complex places of H , the np ’s are integers (almost all zero), and the rλ ’s are real numbers. Let 35 Divc (M ) be the group of Arakelov divisors, and let Picc (M ) be its quotient by the group of principal Arakelov divisors, [22]. We can view the cycles Z(t) defined in section 2 as elements of Divc (M ) as follows. √ Fix the embedding of k into C , where −q has positive imaginary part, and let √ ω = (1 + −q)/2 so that Ok = Z + Zω . Let (6.9) j0 = j(ω) = j(Ok ) ∈ OH be the corresponding singular value of the j -function. Let p be a rational prime which does not split in k and let pOk = ℘ep . Then ℘ is principal and hence splits completely in H . For any prime p of OH above ℘ , the image of j0 under the reduction map OH → OH /p = Fp3−ep is the j - invariant of an elliptic curve Ep over F̄p , unique up to isomorphism, with complex multiplication by Ok . For each such p , this gives a bijective map from the primes over ℘ to the fiber M (F̄p ) . Then, for t > 0 , (6.10) Z(t) = XX p np (t)p, p|℘ where (6.11) np (t) = νp (t) · {x ∈ V (Ep , ι) | Q(x) = t} . Here V (Ep , ι) is the lattice of special endomorphisms of Ep , as in Definition 5.7, and νp (t) is the integer given in Theorem 5.11. Let Cp = [V (Ep , ι)]−1 , (6.11) where [V (Ep , ι)] ∈ Pic(Ok ) ≃ Cl(k) is the class of the rank one Ok -module [V (Ep , ι)] . Using Corollary 5.14, (6.10) becomes (6.12) np = νp (t) · |{c ⊂ Ok | [c] = Cp and N (c) = t/κp }|, where (6.13) κp = ( p if p is inert in k, and 1 if p = q. We view Z(t) as an element of Divc (M ) with zero archimedean component. To define an element Z(t) ∈ Divc (M ) when t < 0 , we use a construction based on an idea explained to us by Dick Gross. For a point (E, ι) of M(C) , let E top 36 be the underlying real torus. By analogy with Definition 5.7, we define the space of special endomorphisms (6.14) V (E, ι) = {y ∈ End(E top ) | yι(a) = ι(ā)y}. If we write E = C/L for a lattice L with a chosen basis, then End(E top ) ≃ M2 (Z), (6.15) and so, by analogy with the case of a supersingular elliptic curve in characteristic p , End(E top ) is isomorphic to a maximal order in the quaternion algebra M2 (Q) . It is not difficult to check that V (E, ι) is a rank one Ok -module with a Z -valued quadratic form given by Q(y) = −y 2 . For each embedding λ of H = k(j) into C , let Eλ be the elliptic curve over C with j -invariant λ(j0 ) . This curve has complex multiplication by Ok and space of special endomorphism, in the sense just defined, V (Eλ , ι) . Let Cλ = [V (Eλ , ι)]−1 ∈ Pic(Ok ) . Then, for t ∈ Z<0 and for τ = u + iv in the upper half plane, we associate the Arakelov divisor X (6.16) Z(t) = Z(t, v) := rλ λ, λ where (6.17) rλ (t) = rλ (t, v) = 2β1 (4π|t|v) · |{y ∈ V (Eλ , ι) | Q(y) = t} = 2β1 (4π|t|v) · |{c ⊂ Ok | [c] = Cλ and N (c) = −t}|. Here β1 is the exponential integral (0.6). Thus we have defined for all t 6= 0 an Arakelov divisor with ( at (φ) if t > 0, and (6.18) deg(Z(t)) = at (φ, v) if t < 0, and hence (6.19) φ(τ ) = a0 (φ, v) + X deg(Z(t)) q t , t6=0 extending (0.8). It remains to find a natural definition of an Arakelov divisor Z(0) = Z(0, v) such that (6.20)  Λ′ (1, χ) . deg(Z(0)) = a0 (φ, v) = −hk log(q) + log(v) + 2 Λ(1, χ)  Supposing this done, one would be tempted to consider the generating series X (6.21) [Z(t)] q t t∈Z where [Z(t)] is the image of Z(t) in Picc (M ) ⊗Z C . We do not know if such a series has a meaning (e.g., converges). 37 References. [1] R. E. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, preprint (1997). [2] J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorèmes de Cerednik et de Drinfeld, in: Courbes modulaires et courbes de Shimura, Astérisque 196–197, 1991, pp. 45–158. [3] P. Deligne, Travaux de Shimura. Sem. Bourbaki 389, Springer Lecture Notes in Math. 244, 1971. [4] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Pub. Math. IHES 36 (1969), 75–109. [5] P. Deligne and M. Rapoport, Les schémas de modules des courbes elliptiques, Modular Functions of One Variable II, Springer Lecture Notes in Math. 349, 1973, pp. 143–316. [6] D. Dorman, Special values of the elliptic modular function and factorization formulae, J. Reine Angew. Math. 383 (1988), 207–220. [7] V. G. Drinfeld, Coverings of p-adic symmetric regions, Funct. Anal. Appl. 10 (1977), 29–40. [8] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math. 195 (1955), 127–151. [9] H. Gillet, Introduction to higher dimensional Arakelov theory, Contemporary Math.: Current trends in arithmetic algebraic geometry, vol. 67, AMS, Providence, 1987. [10] B. Gross, On canonical and quasi-canonical liftings, Inventiones math. 84 (1986), 321–326. [11] B. Gross and D. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. [12] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Akad. Verlagsges., Leipzig, 1923. [13] S. Kudla, Central derivatives of Eisenstein series and height pairings, Annals of Math. 146 (1997), 545–646. [14] S. Kudla and S. Rallis, A regularized Siegel-Weil formula, the first term identity, Annals of Math. 140 (1994), 1–80. 38 [15] S. Kudla and M. Rapoport, Cycles on Siegel 3-folds and derivatives of Eisenstein series, preprint (1997). [16] S. Kudla and M. Rapoport, Height pairings on Shimura curves and p -adic uniformization, in preparation. [17] S. Kudla and M. Rapoport, Arithmetic Hirzebruch Zagier cycles, in preparation. [18] N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972. [19] J. Lubin and J. Tate, Formal complex multiplication in local fields, Annals of Math. 81 (1965), 380–387. [20] J.-P. Serre, Complex multiplication, in: Algebraic Number Theory, J.W.S. Cassels and A. Fröhlich, eds., Academic Press, London, 1967, pp. 292–296. [21] C. L. Siegel, Indefinite quadratische Formen und Modulfunktionen, Ges. Abh., III, Springer Verlag, Berlin, 1966, pp. 85–91. [22] L. Szpiro, Degrés, intersections, hauteurs, in: Séminare sur les pinceaux arithmétiques: La conjecture de Mordell, Astérisque 127 (1985), 11–28. [23] Tonghai Yang, An explicit formula for local densities of quadratic forms, to appear in J. Number Theory. [24] D. Zagier, Nombres de classes et formes modulaires de poids 3/2, C.R. Acad. Sc. Paris 281 (1975), 883–886. S. Kudla Department of Mathematics University of Maryland College Park MD 20742 USA M. Rapoport Mathematisches Institut der Universität zu Köln Weyertal 86-90 50931 Köln Germany T. Yang Department of Mathematics SUNY at Stony Brook Stony Brook, NY 11794 USA