ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION

S. BÖCHERER AND R. SCHULZE-PILLOT

Abstract. We compute the central critical value of the triple product L-function associated to three cusp forms f_{1}, f_{2}, f_{3} with trivial character for groups Γ_{0}(N_{i}) with square free levels N_{i} not all of which are 1 and weights k_{i} satisfying k_{1} ≥ k_{2} ≥ k_{3} and k_{1} < k_{2} + k_{3}. This generalizes work of Gross and Kudla and gives an alternative classical proof of their results in the case N_{1} = N_{2} = N_{3} with k_{1} = k_{2} = k_{3} = 2.

Introduction

Starting from the work of Garrett and of Piatetskii-Shapiro and Rallis on integral representations of the triple product L-function associated to three elliptic cusp forms the critical values of these L-functions have been studied in recent years from different points of view. From the classical point of view there are the works of Garrett [9], Satoh [22], Orloff [21], from an adelic point of view the problem has been treated by Garrett and Harris [10], Harris and Kudla [12] and Gross and Kudla [11]. Of course the central critical value is of particular interest. Harris and Kudla used the Siegel-Weil theorem to show that the central critical value is a square up to certain factors (Petersson norms and factors arising at the bad and the archimedean primes); the delicate question of the computation of the factors for the bad primes was left open. In the special situation that all three cusp forms are newforms of weight 2 and for the group Γ_{0}(N) with square free level N > 1 Gross and Kudla gave for the first time a completely explicit treatment of this L-function including Euler factors for the bad places; they proved the functional equation and showed that the central critical value is a square up to elementary factors (that are explicitly given).

We reconsider the central critical value from a classical point of view, dealing with the situation of three cusp forms f_{1}, f_{2}, f_{3} of weights k_{i} (i = 1, …, 3) that are newforms for groups Γ_{0}(N_{i}) with N = lcm(N_{i}) a squarefree integer ≠ 1. The weights k_{i} are subject to the restriction k_{1} < k_{2} + k_{3} where k_{1} ≥ max(k_{2}, k_{3}); the distinction whether this inequality holds or not played an important role in [11] and [12] too. We start from the simplest possible Eisenstein series E of weight 2 for Γ_{0}^{(3)}(N) on the Siegel space H_{3} (“summation over C ≡ 0 mod N”). After applying a suitable differential operator (depending on the weights k_{i}) to E we proceed in a way similar to Garrett’s original approach: We restrict the differentiated Eisenstein series in a first step to H_{1} × H_{2} and integrate against f_{1}, the resulting function on H_{2} is then restricted to the diagonal and integrated against f_{2}, f_{3}. The necessary modifications to Garrett’s coset decompositions (that were for level 1) are not

Both authors were supported by MSRI, Berkeley (NSF-grant DMS-9022140). R. Schulze-Pillot also supported by the Deutsche Forschungsgemeinschaft and by the Max-Planck-Institut für Mathematik, Bonn.

2 S. BÖCHERER AND R. SCHULZE-PILLOT

difficult (for the first step they have already been carried out in [2]). The actual computation of the integral is elementary and needs only standard results from the theory of newforms. It yields a Dirichlet series (2.41) whose Euler product decomposition is then computed in Section 3. The cases that p divides one, two or all three of the levels N_{i} or is coprime to N must all be treated separately, which makes the discussion somewhat lengthy. However, the actual computation in each of these cases is again fairly straightforward. In Section 4 we show that the Euler factors defined in Section 3 are the “right ones” by proving the functional equation. In order to exhibit the central critical value as a square (up to elementary factors) we follow a similar strategy as [12]: the Eisenstein series E at s = 0 is expressed as a linear combination of genus theta series of quaternary positive definite integral quadratic forms. At most one of these genera (depending on the levels N_{i} and the eigenvalues of the f_{i} under the Atkin-Lehner involutions) contributes to the integral. Eichler’s correspondence between cusp forms for Γ_{0}(N) and automorphic forms on definite quaternion algebras allows then to express this contribution as an (explicitly computable) square of an element of the coefficient field of the f_{i}; this element arises as value of a trilinear form on a space of automorphic forms on the quaternion algebra and may be interpreted as value of a height pairing similar to [11].

It may be of interest to compare the advantages of the different methods applied to this problem. Although the adelic method makes it easier to obtain general results, the explicit computations needed here appear to become somewhat simpler in the classical context. In particular, by making use of the theory of newforms and of orthogonality relations for the theta series involved from [2] we can use the same Eisenstein series E independent of the f_{i}. This is of advantage since the pullback formalism is especially simple for this type of Eisenstein series and leads to the remarkably simple computations in Sectiions 2 and 3.

Most of this article was written while both authors were guests of the MSRI in Berkeley during its special year on automorphic forms. We wish to thank the MSRI for its hospitality and financial support. R. Schulze-Pillot was also supported by Deutsche Forschungsgemeinschaft during a visit of one month at MSRI and was a guest of the Max-Planck-Institut für Mathematik in Bonn in the final stage of the preparation of this manuscript.

Notations We use some standard notations from the theory of modular forms, in particular, we denote by H_{n} Siegel’s upper half space of degree n (the subscript n = 1 will be omitted);

for functions f on H and g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) we use (f |_{k} g)(z) = det(g)^{\frac{k}{2}} (cz + d)^{−k}f(g < z >)

and similarly for the action of double cosets (Hecke operators). The operators T (p) and U(p) however will be used in their standard normalisation. The space of cusp forms of weight k

for Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sl_{2}(Z) | c ≡ 0 mod N\} will be denoted by [Γ_{0}(N), k]_{0}.

1. Differential Operators

We have to deal with two types of embeddings of products of upper half spaces into H_{3} namely

ι_{12} : \{ \begin{matrix}H × H_{2} & ⟶ & H_{3} \\ (z, Z) & ⟼ & ( \begin{matrix}z & 0 \\ 0 & Z \end{matrix} ) \end{matrix}

and

ι_{111} : \{ \begin{matrix}H^{3} & ⟶ & H_{3} \\ (z_{1}, z_{2}, z_{3}) & ⟼ & ( \begin{matrix}z_{1} \\ & z_{2} \\ & & z_{3} \end{matrix} ) \end{matrix}

Without any danger of confusion we may denote by the same symbols the corresponding “diagonal” embeddings of groups:

ι_{12} : Sl_{2} × Sp(2) → Sp(3) and ι_{111} : Sl_{2}^{3} → Sp(3)

One might try to apply Ibukiyama-type differential operators [16] in the integral representation of the triple L-functions (equivariant for Sl_{2} × Sl_{2} × Sl_{2} ↪ Sp(3)). However in the actual computation of the integral, it is more convenient to have equivariance for Sl_{2} × Sp(2) ↪ Sp(3). Therefore we use Maaß-type operators (see [20]) and the holomorphic differential operators introduced in [6]; we describe these operators here only for Sp(3), but of course they also make sense for Sp(n).

We start from a natural number r and three (even) weights k_{1}, k_{2}, k_{3} with k_{1} = max{k_{i}} and satisfying the condition

k_{2} + k_{3} − k_{1} ≥ r  (1.1)

Then we define nonnegative integers a, b, ν_{2}, ν_{3} by

\begin{aligned}r + a & = & k_{2} + k_{3} − k_{1} \\ k_{1} & = & r + a + b \\ k_{2} & = & r + a + ν_{2} \\ k_{3} & = & r + a + ν_{3} \end{aligned}  (1.2)

Then we have

b = ν_{2} + ν_{3}  (1.3)

We use two types of differential operators on H_{3}. The first one is the Maaß operator

M_{α} = det(Z − Z̄)^{2−α} det(∂_{ij}) det(Z − Z̄)^{α−1}
= ∑_{μ=0}^{3} \frac{ε_{3}(α)}{ε_{μ}(α)} · tr ((Z − Z̄)^{[μ]} · (∂_{ij})^{[μ]})
= ε_{3}(α) + ⋯ + det(Z − Z̄) · det(∂_{ij})

(1.4)

where (following [20])

ε_{μ}(α) = \{ \begin{matrix}1 & μ = 0 \\ α · (α − \frac{1}{2}) ⋯ (α − \frac{μ−1}{2}) & μ > 0 \end{matrix}

and for a matrix A of size n we denote by A^{[μ]} the matrix of μ × μ-minors. We put

M_{α}^{[ν]} = M_{α+ν−1} ∘ ⋯ ∘ M_{α+1} ∘ M_{α}

We recall from [20] that

M_{α}^{[μ]} (f |_{α,β} g) = (M_{α}^{[μ]}f) |_{α+μ,β−μ} g  (1.5)

for all g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sp(3, R) with (f |_{α,β} g)(Z) = det(cZ+d)^{−α} det(c\overline{Z}+d)^{−β}f(g < Z >).

Here α and β are arbitrary complex numbers, but it would be sufficient for us to take α = r + s, β = s with s ∈ C.

The second type of differential operators was introduced in [6]: It maps scalar-valued functions on H_{3} to vector-valued functions , more precisely to C[X_{2}, X_{3}]_{b} -valued functions on H × H_{2} ↪ H_{3} where C[X_{2}, X_{3}]_{b} denotes the space of homogeneous polynomials of degree b; we realize the symmetric tensor representation σ_{b} of Gl(2, C) on this space in the usual way. The operator L_{α}^{(b)} as defined in [6] satisfies

(L_{α}^{(b)}f) |_{α+b,β}^{w} g_{1} = L_{α}^{(b)} (f |_{α,β} ι_{1,2}(g_{1}, 1_{4})) (ι_{1,2}(w, Z))
(L_{α}^{(b)}f) |_{α,β,σ_{b}}^{Z} g_{2} = L_{α}^{(b)} (f |_{α,β} ι_{1,2}(1_{2}, g_{2})) (ι_{1,2}(w, Z)) (1.6)

for all g_{1} ∈ Sl_{2}(R) and all g_{2} ∈ Sp(2, R), where the upper indices Z and w indicate which variable is relevant at the moment and

((L_{α}^{(b)}f) |_{α,β,σ_{b}}^{Z} g_{2}) (ι_{1,2}(w, Z)) =
det(cZ + d)^{−α} det(c\overline{Z} + d)^{−β}σ_{b}(cZ + d)^{−1} (L_{α}^{(b)}) (ι_{12}(w, g_{2} < Z >))

This differential operator can be described explicitly as follows:

L_{α}^{(b)} = \frac{1}{α^{[b]}} ι^{⋆} ∑_{0≤2ν≤b} \frac{1}{ν!(b − 2ν)!(2 − α − b)^{[ν]}} · (D_{↑}D_{↓})^{ν} · (D − D_{↑} − D_{↓})^{b−2ν}  (1.7)

with ι^{⋆} denoting the restriction to H × H_{2} ↪ H_{3},

\begin{aligned}D_{↑} & = & ∂_{11} \\ D_{↓} & = & ∑_{2≤i,j≤3} ∂_{ij}X_{i}X_{j} \\ D − D_{↑} − D_{↓} & = & 2 (∂_{12}X_{1} + ∂_{13}X_{3}) \end{aligned}

and

α^{[ν]} = \frac{Γ(α + ν)}{Γ(α)} = \{ \begin{matrix}1 & ν = 0 \\ α(α + 1) … (α + ν − 1) & ν > 0 \end{matrix}

We should remark here that L_{α}^{(b)} has coefficients, which are rational functions of α with no poles for ℜ(s) > 0.

We shall use the operators

D_{α}^{(a,b)} := L_{α+a′}^{(b)} ∘ M_{α}^{[a′]}

with 2a′ = a and D_{α}^{⋆(a,b)} defined by

D_{α}^{⋆(a,b)}f = (D_{α}^{(a,b)}f) ι_{111}(z_{1}, z_{2}, z_{3})

Denoting by D_{α}^{⋆(a,ν_{2}.ν_{3})} the operator which picks out of D_{α}^{⋆(a,b)} its X_{2}^{ν_{2}}X_{3}^{ν_{3}}- component, we get a decomposition

D_{α}^{⋆(a,b)}f = ∑_{ν_{2}+ν_{3}=b} (D_{α}^{⋆(a,ν_{2}.ν_{3})}f) X_{2}^{ν_{2}}X_{3}^{ν_{3}}  (1.8)

with

\begin{matrix}(D_{α}^{⋆(a,ν_{2}.ν_{3})}f |_{α,β} ι_{111}(g_{1}, g_{2}, g_{3})) \\ = (D_{α}^{⋆(a,ν_{2}.ν_{3})}f) |_{α+a′+b,β−a′}^{z_{1}} g_{1} |_{α+a′+ν_{2},β−a′}^{z_{2}} g_{2} |_{α+a′+ν_{3},β−a′}^{z_{3}} g_{3} \end{matrix}

for all (g_{1}, g_{2}, g_{3}) ∈ Sl_{2}(R)^{3}.

If f is a holomorphic function on H_{3}, then

(y_{1}y_{2}y_{3})^{−a′} · D_{α}^{⋆(a,ν_{2},ν_{3})}(f)

is a nearly holomorphic function (in the sense of Shimura) of all three variables z_{1}, z_{2}, z_{3} ∈ H.

To apply Shimura’s results on nearly holomorphic functions, it is more convenient to use his differential operators δ_{α}^{μ}, which differ (in the one-dimensional case i.e. on H) from the Maaß operators only by a factor constant ×y^{μ}:

δ_{α} = \frac{1}{2πi} (\frac{α}{2iy} + \frac{∂}{∂z} )

δ_{α}^{μ} = δ_{α+2μ−2} ∘ ⋯ ∘ δ_{α}

By elementary considerations about the degree of nearly holomorphic functions (as polynomials in y^{−1}) Shimura observed that nearly holomorphic functions on H are linear combinations of functions obtained from holomorphic functions by applying the operators δ_{α}^{μ} (at least if α is not in a certain finite set, for details see [24, lemma 7]. By the same kind of reasoning we get an identity

\begin{matrix}(y_{1}y_{2}y_{3})^{−a′} · D_{α}^{⋆(a,ν_{2}.ν_{3})} \\ = ∑_{0≤μ_{1},μ_{2},μ_{3}≤a′} δ_{α+a+b−2μ_{1}}^{μ_{1}}δ_{α+a+ν_{2}−μ_{2}}^{μ_{2}}δ_{α+a+ν_{3}−2μ_{3}}^{μ_{3}}D_{α}(a, ν_{2}, ν_{3}, μ_{1}, μ_{2}, μ_{3}) \end{matrix}  (1.9)

We understand that in (1.9) the operator δ_{...}^{μ_{i}} acts with respect to z_{i}, i = 1, 2, 3; moreover D_{α}(…) is a holomorphic differential operator mapping functions on H_{3} to functions on H × H × H. Following again the same line of reasoning as in lemma 7 [loc.cit], adapted to our situation, we easily get that the D_{α}(...) satisfy

D_{α}(a, ν_{2}, ν_{3}, μ_{1}, μ_{2}, μ_{3}) (f |_{α,β} ι_{111}(g_{1}, g_{2}, g_{3}))

= D_{α}(a, ν_{2}, ν_{3}, μ_{1}, μ_{2}, μ_{3})(f) |_{α+a+b−2μ_{1},β}^{z_{1}} g_{1} |_{α+a+ν_{2}−2μ_{2},β}^{z_{2}} g_{2} |_{α+a+ν_{3}−2μ_{3},β}^{z_{3}} g_{3} (1.10)

6 S. BÖCHERER AND R. SCHULZE-PILLOT

for all (g_{1}, g_{2}, g_{3}) ∈ Sl(2, R)^{3} and all holomorphic functions on H_{3} (and hence also for all C^{∞}-functions). The upper indices z_{i} on the right hand side of (1.10) indicate, on which variable g_{i} operates.

We have to remark here that Shimura‘s condition “k > 2r” in Lemma 7 [loc.cit] is satisfied in our situation as long as α is non-real or

\begin{aligned}α + a + b & > & a \\ α + a + ν_{2} & > & a \\ α + a + ν_{3} & > & a \end{aligned}

However the coefficients of the ∂_{ij} on both sides of (1.9) are easily seen to be rational functions of α, therefore (1.9) (and subsequent equations) are true for all α ∈ C as rational functions of α.

It is crucial for us to see that in the identity (1.10) the “holomorphic part”, i.e.

D_{α}(a, ν_{2}, ν_{3}, 0, 0, 0)

is different from zero. For this purpose we consider (y_{1}y_{2}y_{3})^{−a′} · D_{α}^{⋆(a,b)} as a polynomial in ∂_{12}, ∂_{13}, ∂_{23}. It is easy to see that this is a polynomial of total degree 3a + b, the component of degree 3a + b being given by

(∂_{12}∂_{13}∂_{23})^{a} × \frac{1}{α^{[b]}} · \frac{1}{b!} · 2^{b} · (y_{1}y_{2}y_{3})^{−a′} · D_{α}^{⋆(a,b)}

as a polynomial in ∂_{12}, ∂_{13}, ∂_{23}. It is easy to see that this is a polynomial of total degree 3a + b, the component of degree 3a + b being given by

(∂_{12}∂_{13}∂_{23})^{a} × \frac{1}{α^{[b]}} · \frac{1}{b!} · 2^{b} · (∂_{12}X_{2} + ∂_{13}X_{3})^{b}  (1.11)

In particular, this component is free of y_{i}^{−1} and ∂_{ii}, so it can only come from the

D_{α}(a, ν_{2}, ν_{3}, 0, 0, 0)

with ν_{2} + ν_{3} = b.

Now we define a polynomial Q_{α} of the matrix variable S = S^{t} = (s_{i,j})_{1≤i,j≤3} by

D_{α}(a, b, ν_{2}, ν_{3}, 0, 0, 0)e^{trace(SZ)} = Q_{α}(S)e^{s_{11}Z_{11}+s_{22}Z_{22}+s_{33}Z_{33}}  (1.12)

By the same kind of reasoning as in [1, Satz 15] we see that by

(x_{1}, x_{2}, x_{3}) ⟼ P_{r}(x_{1}, x_{2}, x_{3}) := Q_{r} ( \begin{matrix}x_{1}^{t}x_{1} & x_{1}^{t}x_{2} & x_{1}^{t}x_{3} \\ x_{2}^{t}x_{1} & x_{2}^{t}x_{2} & x_{2}^{t}x_{3} \\ x_{3}^{t}x_{1} & x_{3}^{t}x_{2} & x_{3}^{t}x_{3} \end{matrix} )  (1.13)

we get a polynomial function of (x_{1}, x_{2}, x_{3}) ∈ (C^{2r})^{3} which in each variable is a harmonic form of degree a+b, a+ν_{2} , a+ν_{3} respectively; more precisely, P_{r} defines a non-zero element of

(H_{a+b}(2r) ⊗ H_{a+ν_{2}}(2r) ⊗ H_{a+ν_{3}}(2r))^{O(2r)}  (1.14)

For our investigation of the functional equation of triple L-functions we have to modify D_{α}(a, b, ν_{2}, ν_{3}, 0, 0, 0) still further (we switch notation now from α to r + s) We consider the operator Δ = Δ_{r,s}(a, ν_{2}, ν_{3}) given by

F ⟼ (y_{1}y_{2}y_{3})^{s}D_{r+s}(a, ν_{2}, ν_{3}, 0, 0, 0) (F × det(Y)^{−s})  (1.15)

This operator (acting on functions on H_{3}) is easily seen to satisfy

Δ (F |_{r} ι_{111}(g_{1}, g_{2}, g_{3})) = Δ(F) |_{r+a+b}^{z_{1}} g_{1} |_{r+a+ν_{2}}^{z_{2}} g_{2} |_{r+a+ν_{3}}^{z_{3}} g_{3}  (1.16)

for all g_{1}, g_{2}, g_{3} ∈ Sl_{2}(R). By the same kind of argument about nearly holomorphic functions as above we get

Δ_{r,s}(a, ν_{2}, ν_{3}) = ∑ ; \begin{matrix}0 ≤ μ_{1} ≤ [\frac{a+b}{2}] \\ 0 ≤ μ_{2} ≤ [\frac{a+ν_{2}}{2} ] \\ 0 ≤ μ_{3} ≤ [\frac{a+ν_{3}}{2} ] \end{matrix} δ_{r+a+b−2μ_{1}}^{μ_{1}}δ_{r+a+ν_{2}−2μ_{2}}^{μ_{2}}δ_{r+a+ν_{3}−2μ_{3}}^{μ_{3}}Δ_{r,s}(a, ν_{2}, ν_{3}, μ_{1}, μ_{2}, μ_{3})

(1.17)

with holomorphic differential operators Δ_{r,s}(a, ν_{2}, ν_{3}, μ_{1}, μ_{2}, μ_{3}) mapping functions on H_{3} to functions on H × H × H. We should mention here that the differential operators coming up in (1.17) do not have poles as long as r is positive and s is non-real or Re(s) ≥ 0. Again the “holomorphic part” Δ_{r,s}(a, ν_{2}, ν_{3}, 0, 0, 0) defines (as in (1.12),(1.13) an element of

(H_{a+b}(2r) ⊗ H_{a+ν_{2}}(2r) ⊗ H_{a+ν_{3}}(2r))^{O(2r)}

This space is known to be one-dimensional: By a result of Littelmann ([19], p. 145) the decomposition of H_{a+ν_{2}}(2r) ⊗ H_{a+ν_{3}}(2r) is multiplicity free and contains H_{a+b}(2r)), hence there is a unique invariant line in the threefold tensor product. There exists therefore a function c = c_{r}(s) such that

Δ_{r,s}(a, ν_{2}, ν_{3}, 0, 0, 0) = c_{r}(s)D_{r}(a, ν_{2}, ν_{3}, 0, 0, 0).  (1.18)

By comparing coefficients of (∂_{12}∂_{13}∂_{23})^{a} ∂_{12}^{ν_{2}}∂_{13}^{ν_{3}} on both sides of (1.17) we get

c_{r}(s) = \frac{(r + a′)^{[b]}}{(r + s + a′)^{[b]}} = \frac{Γ(r + a′ + b)}{Γ(r + a′)} · \frac{Γ(r + a′ + s)}{Γ(r + a′ + s + b)}  (1.19)

2.Unfolding the integral

For a squarefree number N > 0 and three cuspforms

f = ∑ a_{f}(n)e^{2πinz} ∈ [Γ_{0}(N), k_{1}]_{0}
ϕ = ∑ a_{ϕ}(n)e^{2πinz} ∈ [Γ_{0}(N), k_{2}]_{0}
ψ = ∑ a_{ψ}(n)e^{2πinz} ∈ [Γ_{0}(N), k_{3}]_{0}

with k_{1}, k_{2}, k_{3} as in section 1 we want to compute the threefold integral A(f, ϕ, ψ, s), defined by

∫ ∫_{(Γ_{0}(N)\H)^{3}} ∫ \overline{f(z_{1})ϕ(z_{2})ψ(z_{3})} (D^{⋆(a,ν_{2},ν_{3})} (G_{r,s}^{3})) (ι_{111}(z_{1}, z_{2}, z_{3}))

× y_{1}^{k_{1}+s−a′}y_{2}^{k_{2}+s−a′}y_{3}^{k_{3}+s−a′} ∏_{i=1}^{3} \frac{dx_{i}dy_{i}}{y_{i}^{2}} (2.1)

where G_{r,s}^{3} is the Eisenstein series on H_{3} defined by

G_{r,s}^{3} = ∑_{M∈Γ_{∞}^{3}\Γ_{0}^{3}(N)} 1 |_{r+s,s} M

= ∑_{( \begin{matrix}∗ & ∗ \\ C & D \end{matrix} )=M∈Γ_{∞}^{3}\Γ_{0}^{3}(N)} det(CZ + D)^{−r−s} det(CZ̄ + D)^{−s} (2.2)

In the applications we shall need modified versions of the integral (2.1); it is appropriate to describe these here: We use the well-known fact (see e.g. [25, equation (2.28)]) that holomorphic cusp forms are orthogonal to (C^{∞}-)automorphic forms in the image of the differential operators δ, therefore we may replace (D^{⋆(a,ν_{2},ν_{3})} (G_{r,s}^{3})) (ι_{111}(z_{1}, z_{2}, z_{3})) y_{1}^{k_{1}+s−a′}y_{2}^{k_{2}+s−a′}y_{3}^{k_{3}+s−a′} in the integrand of (2.1) by

(D_{r+s}(a, ν_{2}, ν_{3}, 0, 0, 0) (G_{r,s}^{3})) (ι_{111}(z_{1}, z_{2}, z_{3})) y_{1}^{k_{1}+s}y_{2}^{k_{2}+s}y_{3}^{k_{3}+s}

or by

c_{r}(s)D_{r}(a, ν_{2}, ν_{3}, 0, 0, 0) (E_{r,s}^{3}) (ι_{111}(z_{1}, z_{2}, z_{3})y_{1}^{k_{1}}y_{2}^{k_{2}}y_{3}^{k_{3}}  (2.3)

where

E_{r,s}^{3}(Z) = det(Y)^{s} · G_{r,s}^{3} = ∑_{M∈Γ_{∞}^{3}\Γ_{0}^{3}(N)} det(Y)^{s} |_{r} M

The actual computation of A(f, ϕ, ψ, s) is however most conveniently done using the integral in the version of (2.1)

2.1.The first integration To understand the integration with respect to z_{1}, it is better to consider first the integral

I(s) = ∫_{Γ_{0}(N)\H} \overline{f(z_{1})} (D_{r+s}^{(a,b)}G_{r,s}^{3}) (ι_{1,2}(z_{1}, Z)) y_{1}^{k_{1}+s−a′} \frac{dx_{1}dy_{1}}{y_{1}^{2}} × det(Y)^{s−a′}  (2.4)

with Z = X + iY ∈ H_{2}. We recall from [2, Thm.1.1] that the double cosets

Γ_{∞}\Γ_{0}^{3}(N)/ι_{1,2} (Γ_{0}(N) × Γ_{0}^{2}(N))

can be parametrized by the following set of representatives

{g_{m} | m ∈ N ∪ {0} , m ≡ 0 mod N}  (2.5)

where

g_{m} = ( \begin{matrix}1_{3} & 0_{3} \\ \begin{matrix}0 & m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{matrix} & 1_{3} \end{matrix} )

We split the integral (2.4) into the contributions of the double cosets (2.5):

I(s) = ∑_{m} I_{m}(s)  (2.6)

It is easy to see that the double coset with m = 0 decomposes into left cosets as follows

{ι_{1,2}(γ, δ) | γ ∈ Γ_{∞}\Γ_{0}(N), δ ∈ Γ_{∞}^{2}\Γ_{0}^{2}(N)}

Therefore its contribution to D^{⋆(a,b)}G_{r+s,s}^{3} is just

\begin{matrix}D_{r+s,s}^{⋆(a,b)} (∑_{γ,δ} 1 |_{r+s,s} ι_{1,2}(γ, δ)) \\ = ∑_{γ,δ} D_{r+s}^{⋆(a,b)}(1) |_{r+s+a′+b,s−a′}^{z_{1}} γ |_{r+s+a′,s−a′,σ_{b}}^{Z} δ \end{matrix}

It is obvious that

D_{r+s}^{⋆(a,b)}(1) = \{ \begin{matrix}0 & b > 0 \\ ε_{3}(r + s)ε_{3}(r + s + 1) … ε_{3}(r + s + a′ − 1) & b = 0 \end{matrix}

Unfolding the integral defining I_{0}(s) we easily get (by the cuspidality of f) that

I_{0}(s) = 0

(we omit the standard calculation) For fixed m > 0, m ≡ 0 mod N the left cosets are given by (see [2, Thm1.2])

{g_{m}ι_{1,2}(γ, l(h)g | γ ∈ Γ_{0}(N), h ∈ Γ[m]\Γ_{0}(N), g ∈ C_{2,1}(N)\Γ_{0}^{2}(N)}  (2.7)

where

Γ[m] = Γ_{0}(N) ∩ ( \begin{matrix}0 & m^{−1} \\ −m & 0 \end{matrix} ) Γ_{0}(N) ( \begin{matrix}0 & −m^{−1} \\ m & 0 \end{matrix} ) ,

l = ι_{11} and C_{2,1} is the standard maximal parabolic subgroup of Sp(2) given by

C_{2,1} = {( _{0 0 0} )}

with

C_{2,1}(N) := C_{2,1}(Z) ∩ Γ_{0}^{2}(N).

The summation over γ unfolds the integral for I_{m}(s) to

\begin{matrix}I_{m}(s) = \\ ∫_{H} \overline{f(z_{1})} ∑_{h,g} D_{r+s}^{⋆(a,b)} (1 |_{r+s,s} g_{m}) |_{r+s+a′,s−a′,σ_{b}}^{Z} l(h)g · y_{1}^{k_{1}+s−a′} \frac{dx_{1}dy_{1}}{y^{2}} × det(Y)^{s−a′} \end{matrix}  (2.8)

By lemma 4.2 of [6] and (1.4) we have D^{⋆(a,b)} (1 |_{r+s,s} g_{m})

= 2^{−a}(2r + 2s − 2)^{[a]}(r + s)^{[a′]}L_{r+s+a′}^{(b)} (1 |_{r+s+a′,s−a′} g_{m})
= A(r + s, b) · (1 − m^{2}z_{1}Z^{∗})^{−r−s−a′−b} · (1 − m^{2}z̄_{1}Z̄^{∗})^{−s+a′} · (mX_{2})^{b}

(2.9)

where for Z ∈ H_{2} we denote by Z^{∗} the entry in the upper left corner of Z and

A(s, b) = (\frac{−1}{2πi})^{b} · \frac{2^{−a}(2s − 2)^{[a]}(s)^{[a′]}(2s + a − 2)^{[b]}}{b!(s + a′ − 1)^{[b]}}

This implies

I_{m}(s) = A(r + s, b) ∑_{g,h} (∫_{H} \overline{f(z_{1})} (1 − m^{2}z_{1}Z^{∗})^{−r−s−a′−b} (1 − m^{2}z̄_{1}Z̄^{∗})^{−s+a′}

× y_{1}^{k_{1}+s−a′} \frac{dx_{1}dy_{1}}{y^{2}} × (mX_{2})^{b}) |_{r+s+a′,s−a′,σ_{b}}^{Z} l(h)g det(Y)^{s−a′} (2.10)

The integral in (2.10) is exactly of the same type as in [2, (1.4),(1.5)]. Using the same notation

μ(k, s) = (−1)^{\frac{k}{2}} · 2^{3−k−2s} \frac{π}{k + s − 1}

as in [2] we get by the same reasoning as there

I_{m}(s) = A(r + s, b) · μ(r + a + b, s − a′) · m^{−r−2s}

× ∑_{g} (f^{ρ} |_{r+b+a} Γ_{0}(N) ( \begin{matrix}0 & −m^{−1} \\ m & 0 \end{matrix} ) Γ_{0}(N)) (g < Z >^{∗}>

× (σ_{b} ⊗ det^{k+a} (j(g, Z)))^{−1} X_{2}^{b} (\frac{det ℑ(g < Z >)}{ℑ(g < Z >^{∗})})^{s−a′}

(2.11)

with j(( \begin{matrix}a & b \\ c & d \end{matrix} ) , z) = cz + d. This is (essentially) a vector-valued Klingen-type Eisenstein series attached to the modular form

f^{ρ} |_{k_{1}} Γ_{0}(N) ( \begin{matrix}0 & −m^{−1} \\ m & 0 \end{matrix} ) Γ_{0}(N)

where

f^{ρ}(z) = \overline{f(−z̄)}.

From now on we assume that f is a normalized newform (eigenform) of level N_{f} | N; we write N = N_{f} · N^{f} whenever it is convenient. The Fourier coefficients of f are then totally real and we have an Euler product expansion of type

∑ a_{f}(n)n^{−s} = ( ∏_{p|N_{f}} \frac{1}{(1 − a_{f}(p)p^{−s}} ) · ( ∏_{p|̸N_{f}} \frac{1}{(1 − α_{p}p^{−s})(1 − α′_{p}p^{−s})})  (2.12)

Moreover for any prime q dividing N_{f} we have

a_{f}(q)^{2} = q^{k_{1}−2}

and

f|_{k_{1}}V_{q}^{N} = f|_{k_{1}}V_{q}^{N_{f}} = −a_{f}(q)q^{1− \frac{k_{1}}{2}} f

where V_{q}^{N} denotes the “Atkin-Lehner-involution” given by

V_{q}^{N} = ( \begin{matrix}x & y \\ N & q \end{matrix} )

with xq − Ny = q and q|x (for details we refer to [18]).

Actually we have to work not with the newform f itself, but with f|_{k_{1}} ( \begin{matrix}D^{f} & 0 \\ 0 & 1 \end{matrix} ) where

D^{f} is a fixed divisor of N^{f}

To simplify (2.11) further (for f|_{k_{1}} ( \begin{matrix}D^{f} & 0 \\ 0 & 1 \end{matrix} )), we have to study

\begin{matrix}∑_{m≡0(N)} f |_{k_{1}} ( \begin{matrix}D^{f} & 0 \\ 0 & 1 \end{matrix} ) |_{k_{1}} Γ_{0}(N) ( \begin{matrix}0 & −m^{−1} \\ m & 0 \end{matrix} ) Γ_{0}(N) · m^{−s} \\ = N^{−s} ∑_{m′=1}^{∞} f|_{k_{1}} ( \begin{matrix}D^{f} & 0 \\ 0 & 1 \end{matrix} ) |_{k_{1}} Γ_{0}(N) ( \begin{matrix}1 & 0 \\ 0 & m′^{2}N \end{matrix} ) Γ_{0}(N) |_{k_{1}} ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ) · m′^{−s} \end{matrix} (2.13)

Now we are essentially in a “local” situation, because we may decompose the “Fricke involution” into Atkin-Lehner involutions:

( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ) = γ ∘ ∏_{q|N} V_{q}^{N}

with γ ∈ Γ_{0}(N).

We use the following formal identities:

\begin{matrix}p ∤ N : \\ ∑_{l=0}^{∞} f | Γ_{0}(N) ( \begin{matrix}1 & 0 \\ 0 & p^{2l} \end{matrix} ) Γ_{0}(N)X^{l} = \frac{(1 − X)(1 − p^{2}X^{2})}{(1 − pX)(1 − α_{p}^{2}p^{−k_{1}+2}X)(1 − α′_{p}^{2}p^{−k_{1}+2}X)} \end{matrix} (2.14)

Proof: Standard

12 S. BÖCHERER AND R. SCHULZE-PILLOT

p|N_{f} :

∑_{l=0}^{∞} f | Γ_{0}(N) ( \begin{matrix}1 & 0 \\ 0 & p^{2l+1} \end{matrix} ) Γ_{0}(N) ∘ V_{p}^{N} · X^{l} = p^{1− \frac{k_{1}}{2}} · \frac{a_{f}(p)}{1 − X} · f|_{k_{1}}V_{p}^{N_{f}}

= \frac{−1}{1 − X} f

(2.15)

Proof: Standard, using a_{f}(p^{n}) = a_{f}(p)^{n} and a_{f}(p^{2}) = p^{k_{1}−2}

p|N^{f} :

∑_{l=0}^{∞} f | Γ_{0}(N) ( \begin{matrix}1 & 0 \\ 0 & p^{2l+1} \end{matrix} ) Γ_{0}(N) ∘ V_{p}^{N} × X^{l}

= p^{1− \frac{k_{1}}{2}} · \frac{(f|U(p) − p^{\frac{k}{2}} · f|_{k_{1}} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) · X)}{(1 − α_{p}^{2}p^{−k_{1}+2}X)(1 − α′_{p}^{2}p^{−k_{1}+2}X)}|_{k_{1}}V_{p}^{N} (2.16)

Proof: Standard. Using

f|T (p) = f|U(p) + p^{\frac{k_{1}}{2} −1} · f|_{k_{1}} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )

and

f|_{k_{1}} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) |_{k_{1}}V_{p}^{N} = f

we get for (2.16)

\frac{p^{1− \frac{k_{1}}{2}} a_{f}(p)f|_{k_{1}} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) − (1 + pX)f}{(1 − α_{p}^{2}p^{−k_{1}+2}X)(1 − α′_{p}^{2}p^{−k_{1}+2}X)}  (2.17)

In quite the same way we get (still for the case p|N^{f}):

∑_{l=0}^{∞} f|_{k_{1}} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) | Γ_{0}(N) ( \begin{matrix}1 & 0 \\ 0 & p^{2l+1} \end{matrix} ) Γ_{0}(N)|_{k_{1}}V_{p}^{N} · X^{l}

= \frac{p(1 + pX) · f − a_{f}(p)p^{− \frac{k_{1}}{2} +2}X · f |_{k_{1}} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}{(1 − α_{p}^{2}p^{−k_{1}+2}X)(1 − α′_{p}^{2}p^{−k_{1}+2}X)}|_{k_{1}}V_{p}^{N}  (2.18)

= \frac{p(1 + pX)f|_{k_{1}} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) − a_{f}(p)p^{2− \frac{k_{1}}{2}} f · X}{(1 − α_{p}^{2}p^{−k_{1}+2}X)(1 − α′_{p}^{2}p^{−k_{1}+2}X)}

The usual procedure ( X ⟼ p^{−s} )yields for (2.13)

\begin{matrix}∑_{m≡0(N)} f |_{k_{1}} ( \begin{matrix}D^{f} & 0 \\ 0 & 1 \end{matrix} ) |_{k_{1}} Γ_{0}(N) ( \begin{matrix}0 & −m^{−1} \\ m & 0 \end{matrix} ) Γ_{0}(N) ∘ ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ) · m^{−s} \\ = N^{−s} ∏_{p∤N} \frac{(1 − p^{−s})(1 − p^{2−2s})}{1 − p^{−s+1}} ∏_{p∤N_{f}} \frac{1}{(1 − α_{p}^{2}p^{−s−k_{1}+2})(1 − α_{p}^{2}p^{−s−k_{1}+2})} ∏_{p|N_{f}} \frac{−1}{(1 − p^{−k_{1}−s})} × f̃_{s} \end{matrix}

(2.19)

with

f̃_{s} = ∑_{d|N^{f}} α(d, D, s)f|_{k_{1}} ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} )  (2.20)

where

α(d, D, s) = ∏_{p|N^{f}} α(d_{p}, D_{p}, s)  (2.21)

is a multiplicative function given by (2.14)-(2.18). Here we denote by t_{p} the p-part of a positive rational number t. In the sequel we write α_{p}(d, D, s) instead of α(d_{p}, D_{p}, s).

2.2 Second Unfolding

To continue the computation of the integral (2.1) we first need to find a good parametrization of C_{2,1}(N)\Γ_{0}^{2}(N); we shall follow [22] (with the modifications necessary for level N > 1). We first remark that two elements of Sp_{2}(Z) are equivalent modulo C_{2,1}(Z) iff their last rows are equal up to sign (the same is true for Γ_{0}(N) and C_{2,1}(N)).

For C_{2,1}(Z)\Sp_{2}(Z) the parametrization given in [22] is as follows:

{ι_{1,1}(1_{2}, h) | h ∈ Sl_{2}(Z)_{∞}\Sl_{2}(Z)}  (2.22)
{d(J)ι_{1,1}(h, 1_{2}) | h ∈ Sl_{2}(Z)_{∞}^{+}\Sl_{2}(Z)}  (2.23)
⋃ ; u, v ∈ N ; u, v coprime {d(M) ∘ ι_{1,1}(h, h′) | h ∈ Sl_{2}(Z)_{∞}\Sl_{2}(Z), h′ ∈ Sl_{2}(Z)_{∞}^{+}\Sl_{2}(Z)} (2.24)

where

d : \{ \begin{matrix}Gl_{2}(R) & ⟶ & Sp_{2}(R) \\ A & ⟼ & ( \begin{matrix}(A^{t})^{−1} & 0 \\ 0 & A \end{matrix} ) \end{matrix}

J = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and M is an element of Sl_{2}(Z) with M = ( \begin{matrix}∗ & ∗ \\ u & v \end{matrix} ). Among (2.22),

(2.23),(2.24) precisely the following elements have their last row congruent to (0, 0, ∗, ∗) modulo N:

{ι_{1,1}(1_{2}, h) | h ∈ Γ_{∞}\Γ_{0}(N)}  (2.22′)
{d(J)ι_{1,1}(h, 1_{2}) | h ∈ Γ_{∞}^{+}\Γ_{0}(N)}  (2.23′)

\{d(M) ∘ ι_{1,1}(h, h′) | \begin{matrix}h ∈ Sl_{2}(Z)_{∞}\Sl_{2}(Z), h′ ∈ Sl_{2}(Z)_{∞}^{+}\Sl_{2}(Z) \\ M = ( \begin{matrix}∗ & ∗ \\ u & v \end{matrix} ) ∈ Sl_{2}(Z)_{∞}^{+}\Sl_{2}(Z) \\ u, v ∈ N, uc ≡ 0(N), vc′ ≡ 0(N) \end{matrix} \}  (2.24′)

Here c (and c′) denote the lower left entry of h (and h′).

At this point we should emphasize that (2.22’)-(2.24’) do not give representatives of

C_{2,1}(N)\Γ_{0}^{2}(N),

since these elements are in general not in Γ_{0}^{2}(N), but they are equivalent modulo C_{2,1}(Z) to such representatives (by using a suitable transformation we shall finally transport them into Γ_{0}^{2}(N)).

To describe (2.24’) more appropriately we fix two decompositions

N = N_{1} · N_{2} and N = N′_{1} · N′_{2}

and consider for the moment only those h = ( \begin{matrix}∗ & ∗ \\ c & d \end{matrix} ) and h′ = ( \begin{matrix}∗ & ∗ \\ c′ & d′ \end{matrix} ) with gcd(c, N) =

N_{1} and gcd(c′, N) = N′_{1}. Then the data

u, v, ( \begin{matrix}∗ & ∗ \\ c & d \end{matrix} ) , ( \begin{matrix}∗ & ∗ \\ c′ & d′ \end{matrix} )

describe an element of (2.24’) iff N_{2}|u and N′_{2}|v. These data exist only if N|N_{1} · N′_{1}, because we require u, v to be coprime. It is a standard procedure to translate these considerations into more group theoretic terms: We denote by τ_{N_{1}} = τ_{N_{1}}^{N} an element of Sl_{2}(Z) with τ_{N_{1}} =

( \begin{matrix}α & β \\ N_{1} & N_{2} \end{matrix} ) and N_{2}|α. This implies in particular that (τ_{N_{1}})^{2} ∈ Γ_{o}(N).

Then (2.24’) can also be described by

⋃ ; \begin{matrix}N_{1}N_{2} = N \\ N′_{1}N′_{2} = N \\ N|N_{1}N′_{1} \end{matrix} \{d(M)ι_{1,1}(h, h′) | \begin{matrix}h ∈ (τ_{N_{1}}Γ_{0}(N)τ_{N_{1}}^{−1})_{∞} \τ_{N_{1}}Γ_{0}(N) \\ h ∈ (τ_{N′_{1}}Γ_{0}(N)τ_{N′_{1}}^{−1})_{∞}^{+} \τ_{N′_{1}}Γ_{0}(N) \\ u, v positive, coprime , N_{2}|u, N′_{2}|v \end{matrix} \}  (2.24′′)

By a routine matrix calculation, we see that (with h, h′, M, N_{1}, N′_{1} as in (2.24”))

ι_{1,1}(τ_{(N_{1},N‘_{1})}, 1_{2}) ∘ d(M) ∘ ι_{1,1}(h, h′)  (2.24′′′)

is indeed in Γ_{0}^{2}(N), if we require (as we are allowed to do !) that M is of type M = ( \begin{matrix}r & s \\ u & v \end{matrix} )

with v|r. Now we denote by I_{m,ν_{1},ν_{2}}(z_{2}, z_{3}, s) the X_{2}^{ν_{2}}X_{3}^{ν_{3}}−component of the C[X_{2}, X_{3}]_{b}-valued function I_{m}(s), restricted to (z_{2}, z_{3}) ∈ H × H ↪ H_{2}. With f, ϕ, ψ as before (f now again an arbitrary element in [Γ_{0}(N), k_{1}]_{0}) we consider the double integral

∫ ∫_{(Γ_{0}(N)\H)^{2}} \overline{ϕ(z_{2})ψ(z_{3})}K_{f}^{ν_{2},ν_{3}}(z_{2}, z_{3}, s)y_{2}^{k_{2}}y_{3}^{k_{3}} \frac{dx_{2}dy_{2}}{y_{2}^{2}} \frac{dx_{3}dy_{3}}{y_{3}^{2}}  (2.25)

where

K_{f}(Z, s) =

∑_{g∈C_{2,1}(N)\Γ_{0}^{2}(N)} f(g < Z >^{∗}) (σ_{b} ⊗ det^{k_{1}+a}(j(g, Z)))^{−1} X_{2}^{b} · (\frac{det ℑ(g < Z >)}{ℑ(g < Z >^{∗})})^{s}

is the same Klingen-type Eisenstein series as in (2.11), but with the Hecke operator removed; again K_{f}^{ν_{2},ν_{3}} denotes theX_{2}^{ν_{2}}X_{3}^{ν_{3}}-component of K_{f}, restricted to H × H. We split K_{f}^{ν_{2},ν_{3}} into three parts according to the three types (2.22),(2.23),(2.24) of left cosets:

K_{f}^{ν_{2},ν_{3}} = ∑_{i=1}^{3} K_{f,i}^{ν_{2},ν_{3}}  (2.26)

It is again easy to see that K_{f,1}^{ν_{2},ν_{3}} and K_{f,2}^{ν_{2},ν_{3}} do not contribute to the integral (2.25). Using (2.24”) we further split K_{f,3}^{ν_{2},ν_{3}} as

K_{f,3}^{ν_{2},ν_{3}} = ∑_{N_{1},N′_{1}} K_{f,N_{1},N′_{1}}^{ν_{2},ν_{3}}  (2.27)

We can express K_{f,N_{1},N′_{1}}^{ν_{2},ν_{3}} more explicitely as follows

K_{f,N_{1},N′_{1}}^{ν_{2},ν_{3}} = 2\binom{b}{ν_{2}} ∑_{h,h′,u,v} f |_{k_{1}} τ_{(N_{1},N′_{1})}(v^{2}h(< z_{2} >) + u^{2}h′(< z_{3} >))

× v^{ν_{2}}u^{ν_{3}}j(h, z_{2})^{−k_{1}−a−ν_{2}}j(h′, z_{3})^{−k_{1}−a−ν_{3}} × (\frac{ℑ(h < z_{2} >) · ℑ(h′ < z_{3} >)}{v^{2} · ℑ(h < z_{2} >) + u^{2} · ℑ(h′ < z_{3} >)})^{s} (2.28)

where the summation over h, h′, u, v is given by (2.24”) with N_{1} and N′_{1} fixed.

It is well known how to unfold integrals like

\begin{matrix}I(f, ϕ, ψ, N_{1}, N′_{1}, s) := \\ ∫ ∫_{(Γ_{0}(N)\H)^{2}} \overline{ϕ(z_{2})ψ(z_{3})}K_{f,N_{1},N′_{1}}^{ν_{2},ν_{3}}(z_{2}, z_{3}, s)y_{2}^{k_{2}}y_{3}^{k_{3}} \frac{dx_{2}dy_{2}}{y_{2}^{2}} \frac{dx_{3}dy_{3}}{y_{3}^{2}} \end{matrix}  (2.29)

by applying τ_{N_{1}}^{−1} and τ_{N′_{1}}^{−1}: The result is

I(f, ϕ, ψ, N_{1}, N′_{1}, s)

= 2\binom{b}{ν_{2}}  ∫_{(τ_{N_{1}}Γ_{0}(N)τ_{N_{1}}^{−1})_{∞}\Γ_{0}(N)} ∫_{(τ_{N′_{1}}Γ_{0}(N)τ_{N′_{1}}^{−1})_{∞}\Γ_{0}(N)} \overline{(ϕ |_{k_{2}} τ_{N_{1}}^{−1}) (z_{2}) · (ψ |_{k_{3}} τ_{N′_{1}}^{−1}) (z_{3})}

× ∑_{u,v} f |_{k_{1}} τ_{(N_{1},N′_{1})}(v^{2}z_{2} + u^{2}z_{3})v^{ν_{2}}u^{ν_{3}} (\frac{y_{2}y_{3}}{v^{2}y_{2} + u^{2}y_{3}} )^{s} y_{2}^{k_{2}}y_{3}^{k_{3}} \frac{dx_{2}dy_{2}}{y_{2}^{2}} \frac{dx_{3}dy_{3}}{y_{3}^{2}} (2.30)

We do not want to work with Fourier expansions at several cusps, therefore we assume from now on that f, ϕ, ψ are normalized newforms (eigenforms of all Hecke operators) of levels N_{f}, N_{ϕ} and N_{ψ} (all dividing N).

We decompose N_{1} and N_{2} as

N_{1} = N_{1,f} · N_{1}^{f}, N_{2} = N_{2,f} · N_{2}^{f}  (2.31)

(and the same for N′_{1}, N′_{2} and also for ϕ and ψ)

We mention here the following facts, which we shall use in the sequel: •

R|N ⟹ τ_{N_{1}}^{N} = γ ∘ τ_{(R,N_{1})}^{R} with γ ∈ Γ_{0}(R)

• For any divisor d of N^{f} we have

( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) ∘ τ_{N_{1}}^{N} = γ ∘ ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) ∘ τ_{(dN_{f} ,N_{1})}^{dN_{f}} = γ ∘ τ_{N_{1,f}}^{N_{f}} ∘ ( \begin{matrix}(d, N_{1}^{f}) & 0 \\ 0 & \frac{d}{(d,N_{1}^{f})} \end{matrix} )

with γ ∈ Γ_{0}(N_{f}) •

τ_{N_{1}}^{N} = γ ∘ ( ∏_{q|N_{2}} V_{q}^{N}) ∘ ( \begin{matrix}\frac{1}{N_{2}} & 0 \\ 0 & 1 \end{matrix} )

with γ ∈ Γ_{0}(N) and (using the same notation as in [18]

V_{q}^{N} = ( \begin{matrix}x & y \\ N & q \end{matrix} )

with xq − Ny = q and q|x.

• For any newform g of level N and weight k we have (see [18, Theorem 3])

g |_{k} V_{q}^{N} = −a_{g}(q)q^{1− \frac{k}{2}} · g

At this point we introduce - as we already did for N^{f} in the previous subsection- divisors D^{ϕ} and D^{ψ} of N^{ϕ} and N^{ψ}; as usual we further factorize them (for given N_{1} and N′_{1}) as D^{ϕ} = D_{1}^{ϕ} · D_{2}^{ϕ} and D^{ψ} = D′_{1}^{ψ} · D′_{2}^{ψ}.

Using these facts we get the following Fourier expansions

(ϕ|_{k_{2}} ( \begin{matrix}D^{ϕ} & 0 \\ 0 & 1 \end{matrix} ) |_{k_{2}}τ_{N_{1}}^{−1}) (z_{2})  (2.32)
= ϕ|_{k_{2}} ( \begin{matrix}D^{ϕ} & 0 \\ 0 & 1 \end{matrix} ) |_{k_{2}}τ_{N_{1}}(z_{2})
= ϕ|_{k_{2}}τ_{N_{1,ϕ}}^{N_{ϕ}} |_{k_{2}} ( \begin{matrix}D_{1}^{ϕ} & 0 \\ 0 & D_{2}^{ϕ} \end{matrix} ) (z_{2})

= ϕ|_{k_{2}} ( ∏_{q|N_{2,ϕ}} V_{q}^{N_{ϕ}} ) |_{k_{2}} ( \begin{matrix}\frac{D_{1}^{ϕ}}{N_{2,ϕ}} & 0 \\ 0 & D_{2}^{ϕ} \end{matrix} ) (z_{2})

= ( ∏_{q|N_{2,ϕ}} −a(q)q^{− \frac{k_{2}}{2} +1}) · N_{2,ϕ}^{− \frac{k_{2}}{2}} D_{1}^{ϕ}^{\frac{k}{2}} D_{2}^{ϕ}^{\frac{−k}{2}} · ϕ(\frac{D_{1}^{ϕ}}{D_{2}^{ϕ}N_{2,ϕ}} · z_{2})

= D_{1}^{ϕ}^{\frac{k_{2}}{2}} D_{2}^{ϕ}^{− \frac{k_{2}}{2}} ( ∏_{q|N_{2,ϕ}} −a_{ϕ}(q)q^{−k_{2}+1}) ∑ a_{ϕ}(n′)e^{2πi \frac{D{ϕ \atop 1}}{D{ϕ \atop 2}N_{2,ϕ}} ·n′z_{2}}

(ψ|_{k_{3}} ( \begin{matrix}D^{ψ} & 0 \\ 0 & 1 \end{matrix} ) |_{k_{3}}τ_{N′_{1}}^{−1}) (z_{3})

= D′_{1}^{ψ}^{\frac{k_{3}}{2}} D′_{2}^{ψ}^{− \frac{k_{3}}{2}} ( ∏_{q|N′_{2,ψ}} −a_{ψ}(q)q^{−k_{3}+1}) ∑ a_{ψ}(n′′)e^{2πi \frac{D′_{1}^{ψ}}{D′_{2}^{ψ}N{′ \atop 2,ψ}} n′′z_{3}} (2.33)

18 S. BÖCHERER AND R. SCHULZE-PILLOT

and for a divisor d of N^{f}:

f|_{k_{1}} ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) ∘ τ_{(N_{1},N′_{1})}^{N}(z)

= f|_{k_{1}}τ_{(N_{1},N′_{1})_{f}}^{N_{f}} ∘ ( \begin{matrix}(d, (N_{1}, N′_{1})^{f}) & 0 \\ 0 & \frac{d}{(d,(N_{1},N′_{1})^{f})} \end{matrix} )  (2.34)
= f|_{k_{1}}τ_{(N_{1,f} ,N′_{1,f})}^{N_{f}} ∘ ( \begin{matrix}(d, N_{1}^{f}, N′_{1}^{f}) & 0 \\ 0 & \frac{d}{(d,N_{1}^{f} ,N′_{1}^{f})} \end{matrix} )
= d^{− \frac{k_{1}}{2}} · (d, N_{1}^{f}, N′_{1}^{f})^{k_{1}} ∏_{q|lcm(N_{2,f} ,N′_{2,f})} (−a_{f}(q)q^{−k_{1}+1}) ∑ a_{f}(n)e^{2πiB·nz}

We use here the simple fact that \frac{N^{f}}{(N_{1,f} ,N′_{1,f})} = lcm(N_{2,f}, N′_{2,f}) and

B := \frac{1}{lcm(N_{2,f}, N′_{2,f})} · \frac{(d, N_{1}^{f}, N′_{1}^{f})^{2}}{d}  (2.35)

Now we are ready to plug these Fourier expansions into the expression (2.30) for

I(f|_{k_{1}} ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) , ϕ|_{k_{2}} ( \begin{matrix}D^{ϕ} & 0 \\ 0 & 1 \end{matrix} ) , ψ|_{k_{3}} ( \begin{matrix}D^{ψ} & 0 \\ 0 & 1 \end{matrix} ) , N_{1}, N′_{1}, s)

By integration over x_{2} mod N_{2} and x_{3} mod N′_{2} we see that only those terms a_{f}(n)a_{ϕ}(n′)a_{ψ}(n′′) give non-zero contributions, for which

Bnv^{2} = \frac{n′}{N_{2,ϕ}} · \frac{D_{1}^{ϕ}}{D_{2}^{ϕ}}  (2.36)

and

Bnu^{2} = \frac{n′′}{N′_{2,ψ}} · \begin{matrix}D′ & {ψ \atop 1} \\ D′ & {ψ \atop 2} \end{matrix}  (2.37)

Using

∫_{0}^{∞}∫_{0}^{∞}e^{−4πBnv^{2}y_{2}−4πBnu^{2}y_{3}} (\frac{y_{2}y_{3}}{v^{2}y_{2} + u^{2}y_{3}} )^{s} y_{2}^{k_{2}−2}y_{3}^{k_{3}−2}dy_{2}dy_{3}

= \frac{Γ(s + k_{2} + k_{3} − 2)Γ(s + k_{2} − 1)Γ(s + k_{3} − 1)}{Γ(2s + k_{2} + k_{3} − 2)} ·(4πBn)^{−s−k_{2}−k_{3}+2} ·v^{−2s−2k_{2}+2} ·u^{−2s−2k_{3}+2} (2.38)

we obtain

\begin{matrix}I(f|_{k_{1}} ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) , ϕ|_{k_{2}} ( \begin{matrix}D^{ϕ} & 0 \\ 0 & 1 \end{matrix} ) , ψ|_{k_{3}} ( \begin{matrix}D^{ψ} & 0 \\ 0 & 1 \end{matrix} ) , N_{1}, N′_{1}, s) \\ = 2\binom{b}{ν_{2}} · I_{∞}(s)N_{2}N′_{2}B^{−s−k_{2}−k_{3}+2} · D_{1}^{ϕ}^{\frac{k_{2}}{2}} D_{2}^{ϕ}^{− \frac{k_{2}}{2}} D′_{1}^{ψ}^{\frac{k_{3}}{2}} D′_{2}^{ψ}^{− \frac{k_{3}}{2}} · d^{− \frac{k_{1}}{2}} · (d, N_{1}^{f}, N′_{1}^{f})^{k_{1}} \\ × ∏_{q|lcm(N_{2,f} ,N′_{2,f})}(−a_{f}(q)q^{−k_{1}+1}) ∏_{q|N_{2,ϕ}}(−a_{ϕ}(q)q^{−k_{2}+1}) ∏_{q|N′_{2,ψ}}(−a_{ψ}(q)q^{−k_{3}+1}) \\ × ∑_{u,v,n} a_{f}(n)a_{ϕ}(nv^{2}BN_{2,ϕ} · \frac{D_{2}^{ϕ}}{D_{1}^{ϕ}})a_{ψ}(nu^{2}BN′_{2,ψ} · \begin{matrix}D′ & {ψ \atop 2} \\ D′ & {ψ \atop 1} \end{matrix} ) \\ × v^{−2s−2k_{2}+2+ν_{2}}u^{−2s−2k_{3}+2+ν_{3}}n^{−s−k_{2}−k_{3}+2} \end{matrix} (2.39)

with

I_{∞}(s) = (4π)^{−s−k_{2}−k_{3}+2} · \frac{Γ(s + k_{2} + k_{3} − 2)Γ(s + k_{2} − 1)Γ(s + k_{3} − 1)}{Γ(2s + k_{2} + k_{3} − 2)}  (2.40)

To finish this section, we collect all the information obtained so far; we must take care of the fact that we worked in section (2.2) with the Eisenstein series K_{f}(Z, s) rather than with K_{f}(Z, s − a′) as is required by (2.11). The value of the threefold integral A as defined by (2.1) is

\begin{matrix}A(f|_{k_{1}} ( \begin{matrix}D^{f} & 0 \\ 0 & 1 \end{matrix} ) , ϕ|_{k_{2}} ( \begin{matrix}D^{ϕ} & 0 \\ 0 & 1 \end{matrix} ) , ψ|_{k_{3}} ( \begin{matrix}D^{ψ} & 0 \\ 0 & 1 \end{matrix} ) , s) \\ = 2\binom{b}{ν_{2}} · A(r + s, b) · μ(r + a + b, s − a′) · I_{∞}(s − a′) · N^{−2s−r} \\ × ∏_{p∤N} \frac{(1 − p^{−2s−r})(1 − p^{2−4s−2r})}{1 − p^{−2s−r+1}} ∏_{p∤N_{f}} \frac{1}{(1 − α_{p}^{2}p^{−2s−r−k_{1}+2})(1 − α′_{p}^{2}p^{−2s−r−k_{1}+2})} × ∏_{p|N_{f}} \frac{−1}{1 − p^{−2s−r}} \\ × ∑_{d|N^{f}} α(d, D^{f}, 2s − r) · d^{\frac{−k_{1}}{2}} · (d, N_{1}^{f}, N′_{1}^{f})^{k_{1}}(D_{1}^{ϕ})^{\frac{k_{2}}{2}} · (D_{2}^{ϕ})^{− \frac{k_{2}}{2}} · (D′_{1}^{ψ})^{\frac{k_{3}}{2}} · (D′_{2}^{ψ})^{− \frac{k_{3}}{2}} \\ × ∑_{N_{1}N_{2}=N,N′_{1}N′_{2}=N,N|N_{1}N′_{1}} N_{2} · N′_{2} · B^{−s+a′−k_{2}−k_{3}+2} \\ × (  ∏_{p|lcm(N_{2,f} ,N′_{2,f})}(−a_{f}(p)p^{−k_{1}+1})) ( ∏_{p|N_{2,ϕ}}(−a_{ϕ}(p)p^{−k_{2}+1})) ( ∏_{p|N′_{2,ψ}}(−a_{ψ}(p)p^{−k_{3}+1})) \\ × ∑_{n,u,v} a_{f}(n)a_{ϕ}(nv^{2}BN_{2,ϕ} · \frac{D_{2}^{ϕ}}{D_{1}^{ϕ}})a_{ψ}(nu^{2}BN′_{2,ψ} · \begin{matrix}D′ & {ψ \atop 2} \\ D′ & {ψ \atop 1} \end{matrix} ) \\ × u^{−2s+a−2k_{3}+2+ν_{3}}v^{−2s+a−2k_{2}+2+ν_{2}}n^{−s+a′−k_{2}−k_{3}+2} \end{matrix} (2.41)

We remind the reader that the summation over u and v is subject to the condition N_{2}|u and N′_{2}|v.

3. The Euler factors

Using the multiplicativity properties of the Fourier coefficients of f, ϕ, ψ and that the conditions of summation are of multiplicative type we may now write (2.41) as

2\binom{b}{ν_{2}} · A(r + s, b) · μ(r + a + b, s − a′) · I_{∞}(s − a′) · N^{−2s−r} × ∏_{p} T_{p}(s)  (3.1)

To save notation we write

T_{p} = T_{p}^{0} · T_{p}^{1}  (3.2)

where T_{p}^{1} denotes the last five lines of (2.41). In all cases to be considered, the pair [(N_{1})_{p}, N′_{1}_{p}] can take the values [p, p], [p, 1] and [1, p]; therefore we split T_{p}^{1} as

T_{p}^{1} = C_{pp} + C_{p1} + C_{1p}  (3.3)

If d_{p} can take both values 1 and p, we further decompose C_{∗∗} as

C_{∗∗} = C_{∗∗}^{1} + C_{∗∗}^{p}  (3.4)

according to the cases d_{p} = 1 and d_{p} = p.

Part I: D_{p}^{f} = D_{p}^{ϕ} = D_{p}^{ψ} = 1

IA : The case p|N_{f}, p|N_{ϕ}, p|N_{ψ} (This is the case also considered by Gross/Kudla[11]) The conditions imply d_{p} = N_{p}^{f} = N_{p}^{ϕ} = N_{p}^{ψ} = 1, α_{p}(d, D^{f}, s) = 1 and

B_{p} = \{ \begin{matrix}1 for the case [p, p] \\ \frac{1}{p} & otherwise \end{matrix}

C_{pp}(s) = (∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l+2t})a_{ψ}(p^{l}) (p^{l}))^{−s+a′−k_{2}−k_{3}+2} (p^{t}))^{−2s+a−2k_{2}+2+ν_{2}}

+ ∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l})a_{ψ}(p^{l+2t}) (p^{l}))^{−s+a′−k_{2}−k_{3}+2} (p^{t}))^{−2s+a−2k_{3}+2+ν_{3}}) (3.5)

ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION 21

This equals (we use the fact that a_{ϕ}(p^{2}) and a_{ψ}(p^{2}) are powers of p)

\frac{1}{1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s+a′−k_{2}−k_{3}+2}} · \{\frac{1}{1 − a_{ϕ}(p^{2})p^{−2s+a−2k_{2}+2+ν_{2}}} + \frac{a_{ψ}(p^{2})p^{−2s+a−2k_{3}+2+ν_{3}}}{1 − a_{ψ}(p)p^{−2s+a−k_{3}+2+ν_{3}}} \}

= \frac{1}{1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s+a′−k_{2}−k_{3}+2}} · \{\frac{1}{p^{−2s+a−k_{2}+ν_{2}}} + \frac{p^{−2s+a−k_{3}+ν_{3}}}{1 − p^{−2s+a−k_{3}+ν_{3}}} \}
= \frac{1}{1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2}} · \frac{1 + p^{−2s−r}}{1 − p^{−2s−r}} .
C_{1,p} = p · p^{s−a′+k_{2}+k_{3}−2} · (−a_{f}(p)p^{−k_{1}+1}) · (−a_{ϕ}p^{−k_{2}+1})×  (3.6)

× ∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l} · \frac{1}{p} · p)a_{ψ}(p^{l+2t} · \frac{1}{p})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}} · (p^{l})^{−s+a′−k_{2}−k_{3}+2}

= \frac{1}{1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s+a′−k_{2}−k_{3}+2}} · \frac{a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{3−s−3a′−b−2r}}{1 − a_{ψ}(p^{2})p^{−2s+a−2k_{3}+2+ν_{3}}}

= \frac{1}{1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2}} · \frac{a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{3−s−3a′−b−2r}}{1 − p^{−2s−r}}

Quite the same computation shows that

C_{p,1} = C_{1,p}  (3.7)

Hence

T_{p}^{1} = \frac{1}{1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2}}

× \frac{1 + 2a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+3} + p^{−2s−r}}{1 − p^{−2s−r}} (3.8)

Now we write the numerator as

1 + 2a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+3} + p^{−2s−r} = (1 + a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s+3−3a′−b−2r})^{2}
= \frac{(1 − p^{−2s−r})^{2}}{(1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s+3−3a′−b−2r})^{2}} (3.9)

Therefore we obtain

T_{p}(s) = \frac{−1}{1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2}} · \frac{1}{(1 − a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s+3−3a′−b−2r})^{2}} (3.10)

IB :The case p|N^{f}, p|N_{ϕ}, p|N_{ψ} These conditions imply (N_{f})_{p} = N_{p}^{ϕ} = N_{p}^{ψ} = D_{p}^{ϕ} = D_{p}^{ψ} = 1 and d_{p} ∈ {1, p}

22 S. BÖCHERER AND R. SCHULZE-PILLOT

α_{p}(d, D^{f}, s) = \{ \begin{matrix}−(1 + p^{1−s}) if d_{p} = 1 \\ p^{1− \frac{k_{1}}{2}} · a_{f}(p) if d_{p} = p \end{matrix}

and

B_{p} = \{ \begin{matrix}p if d_{p} = (N_{1})_{p} = N′_{1}_{p} = p \\ \frac{1}{d_{p}} & otherwise \end{matrix}

What we get is this:

C_{pp}^{1}(s) = \frac{−(1 + p^{1−2s−r})(1 + p^{−2s−r})}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}

(3.11)

(similar calculation as in (3.5)

C_{1p}^{1}(s) = −(1 + p^{1−2s−r}) · p · (−a_{ϕ}(p)p^{−k_{2}+1})×

∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l} · p)a_{ψ}(p^{l+2t})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}} · (p^{l})^{−s+a′−k_{2}−k_{3}+2}

=  \frac{(1 + p^{1−2s−r}) · p^{−2s−r}}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}

(3.12)

C_{p1}^{1} = C_{1p}^{1}  (3.13)
C_{pp}^{p}(s) = p^{1− \frac{k_{1}}{2}} a_{f}(p)p^{− \frac{k_{1}}{2}} p^{k_{1}}p^{−s+a′−k_{2}−k_{3}+2}×

{ ∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l+2t}p)a_{ψ}(p^{l}p)(p^{t})^{−2s+a−2k_{2}+2+ν_{2}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}+

∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l}p)a_{ψ}(p^{l+2t}p)(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}}

= \frac{a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+3}(1 + p^{−2s−r})}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}

(3.14)

C_{1,p}^{p} = p^{1− \frac{k_{1}}{2}} a_{f}(p)p^{− \frac{k_{1}}{2}} · p · p^{s−a′+k_{1}+k_{2}−2}(−a_{ϕ}(p)p^{−k_{2}+1})×

∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l} \frac{1}{p} · p)a_{ψ}(p^{l+2t} \frac{1}{p})(p^{t})^{−2s+a−k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}

= \frac{−a_{f}(p)a_{ϕ}(p)a_{ψ}(p)p^{−s−2r−3a′−b+3}}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}

(3.15)

C_{p1}^{p} = C_{1p}^{p}  (3.16)

All the C_{∗∗}^{∗} have the same denominator, the numerator of ∑ C_{∗∗}^{∗} being equal to 1 − p^{−2s−r} + p^{1−2s−r} − a_{f}a_{ϕ}a_{ψ} {p^{−3s−3r−a−b+3} − p^{−s−a−2r−b+3}}

ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION 23

This can be factorized as

−(1 − p^{−2s−r})(1 + α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−2r−3a′−b+3})(1 + α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−2r−3a′−b+3}) Now we also write the denominator of T_{p}^{0} as product of linear factors in p^{−s} using

(1 − α_{p}^{2}p^{−k_{1}+2−2s−r}) = (1 − α_{p}a_{ϕ}a_{ψ}p^{−s−3a′−b−2r+3})(1 + α_{p}a_{ϕ}a_{ψ}p^{−s−3a′−b−2r+3}) (and the same with α′_{p} instead of α_{p}); therefore we get some cancellations and arrive at

T_{p} = \frac{−1}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+2})}
× \frac{1}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+3})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+3})} (3.17)
IC The case p|N_{f}, p|N^{ϕ}, p|N^{ψ}

We have

(B)_{p} = \{ \begin{matrix}1 if N_{1} = N′_{1} = p \\ \frac{1}{p} & otherwise \end{matrix}

and we get

\begin{aligned}C_{1p} & = & p · p^{s−a′+k_{2}+k_{3}−2} · (−a_{f}(p)p^{−k_{1}+1})× \\ & & ∑_{l=1}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l} \frac{1}{p})a_{ψ}(p^{l+2t} \frac{1}{p})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} \\ & = & − ∑_{l′=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l′})a_{ϕ}(p^{l′})a_{ψ}(p^{l′+2t})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l′})^{−s+a′−k_{2}−k_{3}+2} \end{aligned}  (3.18)

C_{p1} = − ∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l+2t})a_{ψ}(p^{l})(p^{t})^{−2s+a−2k_{2}+2+ν_{2}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}  (3.19)

C_{pp}(s) = (∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l+2t})a_{ψ}(p^{l}) (p^{l}))^{−s+a′−k_{2}−k_{3}+2} (p^{t}))^{−2s+a−2k_{2}+2+ν_{2}} + (3.20)

+ ∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l})a_{ψ}(p^{l+2t}) (p^{l}))^{−s+a′−k_{2}−k_{3}+2} (p^{t}))^{−2s+a−2k_{3}+2+ν_{3}})

Summing up the C_{∗∗} we see that the summands in (3.20) with t ≠ 0 cancel against (3.18) and (3.19) and we get

T_{p}(s) = −(1 − p^{−2s−r})^{−1} ∑_{l=0}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l})a_{ψ}(p^{l})(p^{l})^{−s−3a′−b−2r+2}
= \frac{−1}{(1 − a_{f}(p)β_{p}γ_{p}p^{−s−3a′−b−2r+2})(1 − a_{f}(p)β_{p}γ′_{p}p^{−s−3a′−b−2r+2})} × (3.21)
× \frac{1}{(1 − a_{f}(p)β′_{p}γ_{p}p^{−s−3a′−b−2r+2})(1 − a_{f}(p)β′_{p}γ′_{p}p^{−s−3a′−b−2r+2})}

I D: The case p ∤ N

This case (which is in some sense the most difficult one) was previously considered in [9] for the case of equal weights. It was already noticed in [21] and [22] that the result from [9] carries over to the case of arbitrary weights. We just state the result here:

T_{p} = (1 − p^{−2s−r})(1 − p^{2−4s−2r})L_{p}(f ⊗ ϕ ⊗ ψ, s + 3a′ + 2r + b − 2)  (3.22)
Part II: D_{p}^{f} = N_{p}^{f} , D_{p}^{ϕ} = N_{p}^{ϕ}, D_{p}^{ψ} = N_{p}^{ψ}

Only the cases B and C are to be investigated, the case B being the most complicated:

II B : The case p|N^{f}, p|N_{ϕ}, p|N_{ψ}

These conditions imply

α_{p}(d, D^{f}, s) = \{ \begin{matrix}−a_{f}(p)p^{2− \frac{k_{1}}{2} −s} & if d_{p} = 1 \\ p(1 + p^{1−s}) & if d_{p} = p \end{matrix}

B_{p} = \{ \begin{matrix}p & if d_{p} = (N_{1})_{p} = (N′_{1})_{p} = p \\ \frac{1}{d} & otherwise \end{matrix}

C_{pp}^{1} = α_{p}(1, D^{f}, 2s + r)×

{ ∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l+2t})a_{ψ}(p^{l})(p^{t})^{−2s+a−2k_{2}+2+ν_{2}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} +

∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l})a_{ψ}(p^{l+2t})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} }

= \frac{α_{p}(1, D^{f}, s)(1 + p^{−2s−r})}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}
(3.23)

(computation similar to (3.5))

C_{1p}^{1} = α_{p}(1, D^{f}, s) · p · (−a_{ϕ}(p)p^{−k_{2}+1})×

∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l}p)a_{ψ}(p^{l+2t})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}

= \frac{−p^{−2s−r}α_{p}(1, D^{f}, 2s + r)}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}

(3.24)

C_{p,1}^{1} = C_{1,p}^{1}  (3.25)

and

∑ C_{∗∗}^{1} = \frac{−a_{f}(p)p^{2− \frac{k_{1}}{2} −s}}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})}  (3.26)

C_{pp}^{p}(s) = α_{p}(p, D^{f}, s + 2r) · p^{− \frac{k_{1}}{2}} p^{k_{1}}p^{−s+a′−k_{2}−k_{3}+2}×

∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l+2t}p)a_{ψ}(p^{l}p)(p^{t})^{−2s+a−2k_{2}+2+ν_{2}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}+

∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l}p)a_{ψ}(p^{l+2t}p)(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}

= \frac{α_{p}(p, D^{f}, 2s + r)a_{ϕ}(p)a_{ψ}(p)p^{\frac{k_{1}}{2} −s−3a′−2r−b+2}(1 + p^{−2s−r})}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}

(3.27)

(similar computation as in (3.5))

C_{1p}^{p}(s) = α_{p}(p, D^{f}, 2s + r) · p^{− \frac{k_{1}}{2}} · p · p^{s−a′+k_{2}+k_{3}−2}(−a_{ϕ}p^{−k_{2}+1})×

∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l} \frac{1}{p}p)a − ψ(p^{l+2t} \frac{1}{p})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}

= \frac{−α_{p}(p, D^{f}, 2s + r)a_{ϕ}(p)a_{ψ}(p)p^{− \frac{k_{1}}{2} −s−a′−r+2}}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − p^{−2s−r})}

(3.28)

C_{p1}^{p} = C_{1p}^{p}  (3.29)

and hence

∑ C_{∗∗}^{p} = \frac{−a_{ϕ}a_{ψ}(1 + p^{1−2s−r})p^{− \frac{k_{1}}{2} −s−a′−r+3}}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−2r−b+2})}  (3.30)

The numerator of ∑ C_{∗∗}^{∗} is equal to

−\frac{1}{a_{ϕ}a_{ψ}} p^{− \frac{k_{1}}{2} +r+3a′+b−1−s}(1 + a_{f}(p)a_{ϕ}(p)a_{ψ}p^{−s−2r+3−b−3a′} + p^{1−2s−r})  (3.31)

We can therefore apply the same kind of trick as in case I B; denoting by ϵ_{p}(ϕ) the eigenvalue of the Atkin-Lehner-involution V_{p}^{N} acting on ϕ (and similarly for ψ) we end up with

T_{p}(s) = \frac{−ϵ_{p}(ϕ)ϵ_{p}(ψ)p^{1− \frac{r}{2} −s}}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+2})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+2})}
× \frac{1}{(1 − α_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+3})(1 − α′_{p}a_{ϕ}(p)a_{ψ}(p)p^{−s−3a′−b−2r+3})} (3.32)

II C: The case p|N_{f}, p|N^{ϕ}, p|N^{ψ}

\begin{aligned}C_{pp}(s) & = & p^{\frac{k_{2}}{2}} · p^{\frac{k_{3}}{2}} × \\ & & \{ ∑_{l=1}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l+2t} \frac{1}{p})a_{ψ}(p^{l} \frac{1}{p})(p^{t})^{−2s+a−2k_{2}+2+ν_{2}}(p^{l})^{−s+a′−k_{2}−k_{3}+2}+ \\ & & + ∑_{l=1}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l} \frac{1}{p})a_{ψ}(p^{l+2t} \frac{1}{p})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} \} \\ & = & p^{−s+a′+2− \frac{k_{2}+k_{3}}{2}} × \\ & & \{ ∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l+1})a_{ϕ}(p^{l+2t})a_{ψ}(p^{l})(p^{t})^{−2s+a−2k_{2}+2+ν_{2}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} + \\ & & ∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l+1})a_{ϕ}(p^{l})a_{ψ}(p^{l+2t})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} \} \end{aligned} (3.33)

\begin{aligned}C_{1p} & = & p^{− \frac{k_{2}}{2} + \frac{k_{3}}{2}} · p · p^{s−a′+k_{2}+k_{3}−2} · (−a_{f}(p)p^{−k_{1}+1})× \\ & & ∑_{l=0}^{∞} ∑_{t=1}^{∞} a_{f}(p^{l})a_{ϕ}(p^{l} \frac{1}{p}p)a_{ψ}(p^{l+2t} \frac{1}{p} \frac{1}{p})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} \\ & = & −p^{−s+a′+2− \frac{k_{2}+k_{3}}{2}} × \\ & & ∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l+1})a_{ϕ}(p^{l})a_{ψ}(p^{l+2t})(p^{t})^{−2s+a−2k_{3}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{3}+2} \end{aligned}  (3.34)

\begin{aligned}C_{p1} & = & −p^{−s+a′+2− \frac{k_{2}+k_{3}}{2}} × \\ & & ∑_{l=0}^{∞} ∑_{t=0}^{∞} a_{f}(p^{l+1})a_{ϕ}(p^{l+2t})a_{ψ}(p^{l})(p^{t})^{−2s+a−2k_{2}+2+ν_{3}}(p^{l})^{−s+a′−k_{2}−k_{2}+2} \end{aligned}  (3.35)

Hence in the sum of the C_{∗∗} only the “t = 0-part” of C_{p1} survives and we obtain

T_{p} = \frac{−ϵ_{p}(f)p^{1−s− \frac{r}{2}}}{(1 − a_{f}(p)β_{p}γ_{p}p^{−s−3a′−b−2r+2})(1 − a_{f}(p)β_{p}γ′_{p}p^{−s−3a′−b−2r+2})}  (3.36)
× \frac{1}{(1 − a_{f}(p)β′_{p}γ_{p}p^{−s−3a′−b−2r+2})(1 − a_{f}(p)β′_{p}γ′_{p}p^{−s−3a′−b−2r+2})}

Remark: Although our list of Euler factors is complete, the reader should be aware of the fact that in our integral representation (2.1) we are free to interchange the roles of ϕ and ψ (interchanging the roles of ν_{2} and ν_{3} at the same time), but f has to be the cusp form of largest weight. Therefore e.g. in case IB, IIB we should also consider the case where p does not divide the level of ϕ or ψ. It will be left to the reader to show by similar computations as above that in those cases the Euler factor will be the same (as should be more or less clear from an adelic point of view).

4. The Functional equation

The factors T_{p} = T_{p}(s) computed in the previous section are for p ∤ N and for p | gcd(N_{f}, N_{ϕ}, N_{ψ}) up to a shift in the argument and an elementary factor the known Euler factors of the triple product L-function L(f, ϕ, ψ, s) associated to f, ϕ, ψ. We define therefore now:

Definition 4.1 The triple product L-function associated to f, ϕ, ψ is for s > \frac{k_{1}+k_{2}+k_{3}−1}{2} defined as L(f, ϕ, ψ, s) = ∏_{p} L_{p}(f, ϕ, ψ, s) where the Euler factor L_{p}(f, ϕ, ψ, s) is given by

L_{p}(f, ϕ, ψ, s + 3a′ + 2r + b − s) = \begin{cases}−T_{p}(s) & if p | N \\ (1 − p^{−2s−r})^{−1}(1 − p^{2−4s−2r})^{−1}T_{p}(s) & if p ∤ N \end{cases} (4.1)

With

L_{∞}(f, ϕ, ψ, s) = Γ_{C}(s)Γ_{C}(s + 1 − k_{1})Γ_{C}(s + 1 − k_{2})Γ_{C}(s + 1 − k_{3})  (4.2)

(where Γ_{C}(s) = (2π)^{−s}Γ(s) as usual) the completed triple product L-function is

Λ(f, ϕ, ψ, s) = L_{∞}(f, ϕ, ψ, s)L(f, ϕ, ψ, s).

With these notations we have for the integral A(f, ϕ, ψ, s) from 2.1 (with c_{2}(s) as in (1.19), ζ_{p}(s) = (1 − p^{−s})^{−1} for finite primes p and ζ_{∞}(s) = π^{−s/2}Γ(s/2)):

Theorem 4.2

\frac{A(f, ϕ, ψ, s)}{c_{2}(s)}

= \binom{b}{ν_{2}}i^{b+k_{1}}N^{−2s−2}2^{3−2b−a} \frac{π^{2+3a′}}{(s + 1)(2s + 1)} \frac{L_{∞}(f, ϕ, ψ, s + \frac{k_{1}+k_{2}+k_{3}}{2} − 1)}{ζ_{∞}(4s + 2)ζ_{∞}(2s + 2)}

× ∏_{p∤N} \frac{L_{p}(s + \frac{k_{1}+k_{2}+k_{3}}{2} − 1)}{ζ_{p}(4s + 2)ζ_{p}(2s + 2)} ∏_{p|N}(−L_{p}(f, ϕ, ψ, s + \frac{k_{1} + k_{2} + k_{3}}{2} − 1)) (4.3)

Proof. This follows from 2.41 and the results of Section 3.

Theorem 4.3 The function Λ(f, ϕ, ψ, s) has a meromorphic continuation to all of C and satisfies the functional equation

Λ(f, ϕ, ψ, s)

= −N^{−4(s+ \frac{k_{1}+k_{2}+k_{3}}{2} −1)}(gcd(N_{f}, N_{ϕ}, N_{ψ}))^{−(s+ \frac{k_{1}+k_{2}+k_{3}}{2} −1)}(∏_{p|N} ϵ_{p})Λ(k_{1} + k_{2} + k_{3} − 2 − s) (4.4)

where for p | N the number ϵ_{p} is defined as the product of the eigenvalues under the p-Atkin-Lehner-involution w_{p} of those forms among f, ϕ, ψ whose level is divisible by p.

Proof. We put r = 2 since for this choice the functional equation of the Eisenstein series is under s ↦ −s. It is then not difficult to read off the functional equation of E from the calculations in [11]; actually things become somewhat simplified since we need the functional equation only up to oldforms. We need the following Lemma:

Lemma 4.4 The Eisenstein series E_{2,s} satisfies the functional equation:

(s + 1)(2s + 1)ζ_{∞}(4s + 2)ζ_{∞}(2s + 2) ∏_{p∤N} ζ_{p}(4s + 2)ζ_{p}(2s + 2)E_{2,s}(Z)

= −N^{−9s}(s − 1)(2s − 1)ζ_{∞}(−4s + 2)ζ_{∞}(−2s + 2)  (4.5)

× ∏_{p∤N} ζ_{p}(−4s + 2)ζ_{p}(−2s + 2)E_{2,−s}| (\begin{matrix}0_{3} & −1_{3} \\ N1_{3} & 0_{3} \end{matrix}) (Z) + Ẽ_{s}

where Ẽ_{s} is a linear combination of Eisenstein series for groups Γ_{0}^{3}(N′) with N′ strictly dividing N or conjugates of these by a matrix ( \begin{matrix}0_{3} & −1_{3} \\ M1_{3} & 0_{3} \end{matrix} ) with M | (N/N′)

Proof. Let F_{r,s}(Z) be the Eisenstein series of degree 3 defined in the same way as E_{r,s} in Section 2, but with the summation running over coprime symmetric pairs (C, D) with gcd(det C, N) = 1; one has

E_{r,s}|_{r} (\begin{matrix}0 & −1 \\ N & 0 \end{matrix}) = N^{−(3r/2)−3s}F_{r,s}  (4.6)

The calculation of the local intertwining operators M(s) = M_{p}(s) in sections 5 and 6 of [11] gives (in the notations of that article) M_{p}(s)Φ_{p}(s) for p = ∞ or p ∤ N and allows for p ∤ N to express M_{p}(s)Φ_{p}^{0}(s) explicitly as a linear combination of the sections Φ_{p}^{0}(−s), Φ_{p}^{3}(−s), (Φ_{p})_{K}(−s), (Φ_{p})_{K′}(−s). An elementary calculation shows in this case that the coefficient at Φ_{p}^{0}(−s) is zero and that the coefficient at Φ_{p}^{3}(−s) equals

p^{−6s−3} \frac{ζ_{p}(2s − 1)ζ_{p}(4s − 1)}{ζ_{p}(−4s + 2)ζ_{p}(−2s + 2)}.

We let Z = X + iY and g = (g_{∞}, 1, …) ∈ Sp_{3}(A) with g_{∞} = ( \begin{matrix}1_{3} & X \\ 0_{3} & 1_{3} \end{matrix} )( \begin{matrix}Y^{1/2} & 0_{3} \\ 0_{3} & Y^{−1/2} \end{matrix} ). Then it is

well known that with

Φ^{0} = Φ_{∞}^{−2} × ∏_{p|N} Φ_{p}^{0} × ∏_{p∤N}(Φ_{p})_{K}

and analogously defined Φ^{3} one has E_{2,s}(Z̄) = (det Y)^{−1}E(g, s, Φ^{0}) and

F_{2,s}(Z̄) = (det Y)^{−1}E(g, s, Φ^{3})

(see Proposition 7.5 of [11]), and analogous formulae are true for all the global sections Φ for which the p-adic components Φ_{p} are one of the Φ_{p}^{0}, Φ_{p}^{3}, (Φ_{p})_{K}, (Φ_{p})_{K′}. The assertion of the lemma follows upon using the duplication formula for the gamma-function for the contribution from the infinite place and then applying the functional equation of the Riemann zeta function.

We can now finish the proof of Theorem 4.3. From Theorem 4.2, Lemma 4.4 and the fact that the integrand in \frac{A(f,ϕ,ψ,s)}{c_{2}(s)} contains by 1.18 a differential operator independent of s we see (using the results of Part II in Section 3) that

Λ(s + \frac{k_{1} + k_{2} + k_{3}}{2} − 1) = −N^{−4s}(gcd(N_{f}, N_{ϕ}, N_{ψ}))^{−s}(∏_{p|N} ϵ_{p})Λ(−s + \frac{k_{1} + k_{2} + k_{3}}{2} − 1) (4.7)

where for p | N the number ϵ_{p} is defined as in the assertion of Theorem 4.3. Putting s′ = s+ \frac{k_{1}+k_{2}+k_{3}}{2} −1 we obtain the desired functional equation under s′ ↦ k_{1}+k_{2}+k_{3}−2−s′.

5.Computation of the central critical value

Following the strategy of [12] we now evaluate the integral 2.2 at the point s = 0 using a variant of Siegel’s theorem, i. e. , expressing the value at s = 0 of the Eisenstein series G as a linear combination of theta series. The setting for this is basically the same as in [2, 5, 4]. Let M_{1}, M_{2}, M_{3} be relatively prime square free integers such that M_{1} has an odd number of prime divisors. By D = D(M_{1}) we denote the quaternion algebra over Q ramified at ∞ and the primes dividing M_{1} and by R = R(M_{1}, M_{2}) an Eichler order of level M = M_{1}M_{2} in D, i. e. , the completion R_{p} is a maximal order for p ∤ M_{2} and conjugate to \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(Z_{p}) | c ≡ 0 mod p\} for p | M_{2}, where we identify D ⊗ Q_{p} with M_{2}(Q_{p}) for p ∤ M_{1}. By gen(M_{1}, M_{2}, M_{3}) we denote the genus of Z-lattices with quadratic form (quadratic lattices) of R(M_{1}, M_{2}) equipped with the norm form of the quaternion algebra scaled by M_{3}. The genus theta series of degree n of gen(M_{1}, M_{2}, M_{3}) is then

Θ_{M_{1},M_{2},M_{3}}^{gen,(n)}(Z) = ∑_{\{K\}∈gen(M_{1},M_{2},M_{3})} \frac{Θ^{(n)}(K, Z)}{|O(K)|}

where the summation is over a set of representatives of the classes in gen(M_{1}, M_{2}, M_{3}), O(K) is the (finite) group of orthogonal units of the quadratic lattice K, Z is a variable in the Siegel upper half space H_{n} and

Θ^{(n)}(K, Z) = ∑_{x=(x_{1},...,x_{n})∈K^{n}} exp(2πitr(q(x)Z))

with q(x) = (\frac{1}{2}(M_{3}tr(x_{i}x̄_{j})))_{i,j}.

We consider a double coset decomposition

D_{A}^{×} = ⋃_{i=1}^{h=h(M_{1},M_{2})} D_{Q}^{×}y_{i}R_{A}^{×}

of the adelic multiplicative group of D with R_{A}^{×} = D_{∞}^{×} × ∏_{p≠∞} R_{p}^{×} and representatives y_{i} with n(y_{i}) = 1 and (y_{i})_{∞} = 1. Then the lattices I_{ij} = y_{i}Ry_{j}^{−1} (with the norm form scaled by M_{3}) exhaust gen(M_{1}, M_{2}, M_{3}) (with some classes possibly occurring more than once) and it is easily seen that with R_{i} = I_{ii} and e_{i} = |R_{i}^{×}| we have

∑_{i,j=1}^{h} \frac{Θ^{(n)}(I_{ij}, Z)}{e_{i}e_{j}} = 2^{ω}Θ_{M_{1},M_{2},M_{3}}^{gen,(n)}(Z)  (5.1)

where ω = ω(M_{1}, M_{2}) is the number of prime divisors of M_{1}M_{2}. With these notations we have from [2] (Theorem 3.2 and Corollary 3.2) and [5], (p.229):

Lemma 5.1 The value at s = 0 of G_{2,s}(Z) is

∑_{M_{1}M_{2}M_{3}|N} α_{M_{1},M_{2},M_{3}}Θ_{M_{1},M_{2},M_{3}}^{gen,(3)}(Z)

with

α_{M_{1},M_{2},M_{3}} = (−1)^{1+ω(M_{1},M_{2})}(M_{1}M_{2})^{−3}M_{3}^{−6}8π^{4}ζ^{(N)}(2)^{−2}.

In order to compute the value at s = 0 of the differentiated Eisenstein series from Section 2 we have to compute

(D_{2}^{∗(a,ν_{2},ν_{3})}Θ^{(3)}(K, −))(ι_{111}(z_{1}, z_{2}, z_{3}))  (5.2)

for the individual theta series appearing in the sum in Lemma 5.1. We denote by U_{μ} the space of homogenous harmonic polynomials of degree μ in 4 variables and identify an element of U_{μ} with a polynomial on D_{∞} by evaluating it at the component vector of an element of D_{∞} with respect to an orthonormal basis relative to the quaternion norm on D. Similarly, for ν ∈ N let U_{ν}^{(0)} be the space of homogeneous harmonic polynomials of degree ν on R^{3} and view P ∈ U_{ν}^{(0)} as a polynomial on D_{∞}^{(0)} = {x ∈ D_{∞}|tr(x) = 0}. The representations τ_{ν} of D_{∞}^{×}/R^{×} of highest weight (ν) on U_{ν}^{(0)} given by (τ_{ν}(y))(P)(x) = P (y^{−1}xy) for ν ∈ N give all the isomorphism classes of irreducible rational representations of D_{∞}^{×}/R^{×}. By ⟨⟨ , ⟩⟩_{ν}^{(0)} we denote the invariant scalar product in the representation space U_{ν}^{(0)}, by ⟨⟨ , ⟩⟩_{μ} the invariant scalar product in the SO(D_{∞}, norm) =: H_{R}^{+}-space U_{μ}. We notice that the invariant scalar products ⟨⟨, ⟩⟩_{μ_{i}} on the U_{μ_{i}} can be normalized in such a way that they take rational values on the subspaces of polynomials with rational coefficients and that these subspaces generate the U_{μ_{i}}. Indeed, consider the Gegenbauer polynomial C^{(μ_{i})}(x, x′) = obtained from

C_{1}^{(μ_{i})}(t) = 2^{μ_{i}} ∑_{j=0}^{[ \frac{μ_{i}}{2} ]}(−1)^{j} \frac{1}{j!(μ_{i} − 2j)!} \frac{(μ_{i} − j)!}{2^{2j}} t^{μ_{i}−2j}

by

C̃^{(μ_{i})}(x_{1}, x_{2}) = 2_{i}^{μ}(norm(x_{1})norm(x_{2}))^{μ_{i}/2}C_{1}^{(μ_{i})}(\frac{tr(x_{1}\overline{x_{2}})}{2\sqrt{norm(x_{1})norm(x_{2})}})

and normalize the scalar product on U_{μ_{i}} such that C^{(μ_{i})} is a reproducing kernel, i. e.

⟨⟨C^{(μ_{i})}(x, x′), Q(x)⟩⟩_{μ_{i}} = Q(x′)

for all Q ∈ U_{μ_{i}}. Then the C^{(μ_{i})}(·, x′) with rational x′ are rational, generate U_{μ_{i}}, and the reproducing property implies that they have rational scalar products whith each other. The same argument applies to the U_{ν}^{(0)}.

It is well known that the group of proper similitudes of the quadratic space (D, norm) is isomorphic to (D^{×} × D^{×})/Z(D^{×}) via

(x_{1}, x_{2}) ↦ σ_{x_{1},x_{2}} with σ_{x_{1},x_{2}}(y) = x_{1}yx_{2}^{−1}

and that under this isomorphism SO(D, norm) is the image of

{(x_{1}, x_{2}) ∈ D^{×} × D^{×} | n(x_{1}) = n(x_{2})}.

Moreover, the SO(D_{∞}, norm) =: H_{R}^{+}-space U_{ν}^{(0)} ⊗ U_{ν}^{(0)} is isomorphic to the H_{R}^{+}-space U_{2ν} and the isomorphism can be normalized in such a way that it preserves rationality and is compatible with the invariant scalar products on both spaces (which are assumed to be normalized as above).

Denoting by S the Gram matrix of the quadratic lattice K we know from Section 1 that (5.2) is of the form

∑_{x=(x_{1},x_{2},x_{3})∈(Z^{4})^{3}} P (S^{\frac{1}{2}} x_{1}, S^{\frac{1}{2}} x_{2}, S^{\frac{1}{2}} x_{3}) exp(πi(S[x_{1}]z_{1} + S[x_{2}]z_{2} + S[x_{3}]z_{3}))

where P ∈ ⊗_{i=1}^{3}U_{k_{i}−2} is a harmonic polynomial of degree μ_{i} = k_{i} − 2 on R^{4} in each of the variables and is invariant under the (diagonal) action of H_{R} = O(D⊗R, norm). Moreover, P is independent of S and has rational coefficients (up to a factor of π^{3a+2b}). The H_{R}-invariant trilinear form T on U_{μ_{1}} ⊗ U_{μ_{2}} ⊗ U_{μ_{3}} defined by taking the scalar product with the invariant polynomial π^{−3a−2b}P ( as remarked in Section 1 this is up to scalars the unique invariant trilinear form) is hence rational (i. e. takes rational values on tensor products of polynomials with rational coefficients).

If all the μ_{i} are even (as is the case in our situation) then the decomposition of H_{R}^{+} and of U_{μ} as U_{μ/2}^{(0)} ⊗ U_{μ/2}^{(0)} from above gives furthermore that T factors as T^{(0)} ⊗ T^{(0)}, where the unique (up to scalars) D_{∞}^{×}-invariant trilinear form T^{(0)} on U_{μ_{1}/2}^{(0)} ⊗ U_{μ_{2}/2}^{(0)} ⊗ U_{μ_{3}/2}^{(0)} has the same rationality properties as T. Of course both T and T_{0} are just the ordinary multiplication if all the μ_{i} are 0.

For any positive definite symmetric 4 × 4-matrix S we define the U_{μ_{i}}-valued theta series Θ̃_{S}^{(μ_{i})} by

Θ̃_{S}^{(μ_{i})}(z)(x′) = ∑_{x∈Z^{4}} C^{(μ_{i})}(S^{1/2}x, S^{1/2}x′) exp(πiS[x]z)  (5.3)

32 S. BÖCHERER AND R. SCHULZE-PILLOT

We notice that if K is a quaternary quadratic lattice with Gram matrix S the right hand side of 5.3 does not depend on the choice of basis of K with respect to which the Gram matrix is computed (because of the invariance of C^{(μ_{i})} under the (diagonal) action of the orthogonal group); we may therefore write it as Θ̃^{(μ_{i})}(K) as well.

We denote by Θ̃_{M_{1},M_{2},M_{3}}^{(μ_{i})} the weighted average of the Θ̃_{S}^{(μ_{i})} over the Gram matrices of representatives of the classes in the genus gen(M_{1}, M_{2}, M_{3}) as above. We find

(D_{2}^{∗(a,ν_{2},ν_{3})}Θ(S, −))(ι_{111}(z_{1}, z_{2}, z_{3})) = π^{3a+2b}T (Θ̃^{(μ_{1})}(S, z_{1}) ⊗ Θ̃^{(μ_{2})}(S, z_{2}) ⊗ Θ̃^{(μ_{3})}(S, z_{3})) (5.4)

(where T is as above). For the value at s = 0 of (2.1) we obtain therefore

π^{3a+2b} ∑_{M_{1}M_{2}M_{3}|N} α_{M_{1},M_{2},M_{3}}

× T (⟨Θ̃_{M_{1},M_{2},M_{3}}^{(μ_{1})}(z_{1}), f(z_{1})⟩ ⊗ ⟨Θ̃_{M_{1},M_{2},M_{3}}^{(μ_{2})}(z_{2}), ϕ(z_{2})⟩ ⊗ ⟨Θ̃_{M_{1},M_{2},M_{3}}^{(μ_{3})}(z_{3}), ψ(z_{3})⟩) (5.5)

where by ⟨, ⟩ we denote the Petersson product.

In order to evaluate this expression further we use Eichler’s correspondence. We fix M_{1}, M_{2} and an Eichler order R(M_{1}, M_{2}) ⊆ D(M_{1}) as above and set M = M_{1}M_{2}. For an irreducible rational representation (V_{τ}, τ) (with τ = τ_{ν} as above) of D_{R}^{×}/R^{×} we denote by A(D_{A}^{×}, R_{A}^{×}, τ) the space of functions φ : D_{A}^{×} → V_{τ} satisfying φ(γxu) = τ(u_{∞}^{−1})φ(x) for γ ∈ D_{Q}^{×} and u = u_{∞}u_{f} ∈ R_{A}^{×}. It has been discovered by Eichler that these functions are in correspondence with the elliptic modular forms of weight 2 + 2ν and level M = M_{1}M_{2}. This correspondence can be described as follows (using the double coset decomposition D_{A}^{×} = ∪_{i=1}^{h} D^{×}y_{i}R_{A}^{×} from above): Recall from Section 5 of [2] and Section 3 of [4] that for each p | M_{1}M_{2} we have an involution w̃_{p} on the space A(D_{A}^{×}, R_{A}^{×}, τ_{ν}) given by right translation by a suitable element π_{p} ∈ R_{p}^{×} of norm p normalizing R_{p}. This space then splits into common eigenspaces of all these (pairwise commuting) involutions. On A(D_{A}^{×}, R_{A}^{×}, τ_{ν}) we have furthermore for p ∤ N Hecke operators T̃(p) whose action on these functions is expressed by the Brandt-matrices (B_{ij}^{(ν)}(p)) (whose entries are endomorphisms of V_{τ_{ν}}). They commute with the involutions w̃_{p}. On the space A(D_{A}^{×}, R_{A}^{×}, τ_{ν}) we have moreover the natural inner product ⟨ , ⟩_{ν} defined by integration, it is explicitly given by

⟨φ, ρ⟩_{ν} = ∑_{i=1}^{h} \frac{⟨⟨φ(y_{i}), ρ(y_{i})⟩⟩_{ν}^{(0)}}{e_{i}} .

By abuse of language we call (in the case ν = 0) forms cuspidal, if they are orthogonal to the constant functions with respect to this inner product.

We denote for p dividing M_{2} the p-essential part by A_{p,ess}(D_{A}^{×}, R_{A}^{×}, τ) consisting of functions φ that are orthogonal to all ρ ∈ A(D_{A}^{×}, (R′_{A})^{×}, τ) for orders R′ ⊇ R for which the completion R′_{p} strictly contains R_{p}. It is invariant under the T̃(p) for p ∤ M_{1}M_{2} and the w̃_{p} for p | M_{1}M_{2} and hence has a basis of common eigenfunctions of all the T̃(p) for p ∤ N and all the involutions w̃_{p} for p | N. Moreover by the results of [7, 15, 23, 17] we know that in the space A_{ess}(D_{A}^{×}, R_{A}^{×}, τ) of forms that are p-essential for all p dividing M = M_{1}M_{2} strong multiplicity one holds, i.e., each system of eigenvalues of the T̃(p) for p ∤ M occurs at most once, and the eigenfunctions are in one to one correspondence with the newforms in the space S^{2+2ν}(M) of elliptic cusp forms of weight 2 + 2ν for the group Γ_{0}(M) that are eigenfunctions of all Hecke operators (if τ is the trivial representation and R is a maximal order one has to restrict here to cuspidal forms on the quaternion side in order to obtain cusp forms on the modular forms side). This correspondence (Eichler’s correspondence) preserves Hecke eigenvalues for p ∤ M, and if φ corresponds to g ∈ S^{2+2ν}(M) then the eigenvalue of g under the Atkin-Lehner involution w_{p} is equal to that of φ under w̃_{p} if D splits at p and equal to minus that of φ under w̃_{p} if D_{p} is a skew field. From (3.13) of [5] we know that if g having first Fourier coefficient 1 corresponds in this way to φ with ⟨φ, φ⟩_{ν} = 1 then ⟨g, Θ̃^{(μ)}(K)⟩ = ⟨g, g⟩(φ(y_{i}) ⊗ φ(y_{j})) holds.

It is not difficult (see also [13]) to extend this correspondence to not necessarily new forms g ∈ S^{2+2ν}(N) in the following way:

Lemma 5.2 Let N = M_{1}M_{2} be a decomposition as above. Call g̃ ∈ S^{2+2ν}(N) an M′- new form if it is orthogonal to all oldforms coming from g′ ∈ S^{2+2ν}(M′) for M′ | N. Then Eichler’s correspondence from above extends to a one to one correspondence between the set of all M_{1}-new eigenforms of the Hecke operators for p ∤ N in S^{2+2ν}(N) that are eigenfunctions of all the w_{p} for p | M_{2} with the set of all Hecke eigenfunctions in A_{(}D_{A}^{×}, R_{A}^{×}, τ) that are eigenfunctions of all the involutions w̃_{p}. This correspondence is compatible with the Hecke action and the eigenvalues under the respective involutions as above; it maps newforms of level M′ | N (with M_{1} | M′) to forms that are p-essential precisely for the p | (M′/M_{1}). The correspondence can be explicitly given (in a nonlinear way) by the first Yoshida lifting sending φ̃ to the form

∑_{ij}^{h} \frac{⟨⟨φ̃(y_{i}) ⊗ φ̃(y_{j}), Θ̃^{(2ν)}(I_{ij})⟩⟩}{e_{i}e_{j}},

it then sends φ̃ with ⟨φ̃, φ̃⟩_{ν} = 1 to g̃ having first Fourier coefficient one. Moreover, in this normalization it satisfies the scalar product relations from above, i. e. , if g̃ corresponds to φ̃ then ⟨g̃, Θ̃^{(2ν)}(I_{ij})⟩⟨φ̃, φ̃⟩_{ν}^{2} = ⟨g̃, g̃⟩(φ̃(y_{i}) ⊗ φ̃(y_{j})) holds.

Proof. Let M′ be a divisor of N and g ∈ S^{2+2ν}(M′) and let ϵ be a function from the set S of prime divisors of N/M′ (whose cardinality we denote by ω(M/M′)) to {±1}^{ω(M/M′)}. Then the function g^{ϵ} := ∑_{S′⊆S} g| ∏_{p∈S} ϵ(p)w_{p}) in S^{2+2ν}(N) is an eigenfunction of the Atkin-

Lehner involutions w_{p} for the p | (N/M′) with eigenvalues ϵ(p). Similarly, fix a maximal order R̃ ⊆ D and an Eichler order R = R(M_{1}, M_{2}) ⊆ R̃, let M′_{2} | M_{2} and let R(M_{1}, M′_{2}) be the Eichler order of level M_{1}M′_{2} in D containing R and contained in R̃. Let ϵ be a function on the set of prime divisors of M_{2}/M′_{2} as above. Then to φ ∈ A_{ess}(D_{A}^{×}, (R(M_{1}, M′_{2}))_{A}^{×}, τ) we construct as above a unique φ^{ϵ} having the same Hecke eigenvalues for p ∤ N and the same w̃_{p}-eigenvalues for p | M_{1}M′_{2} as φ such that φ^{ϵ} is an eigenfunction of the w̃_{p} for p | (M_{2}/M′_{2}) with eigenvalues ϵ(p).

34 S. BÖCHERER AND R. SCHULZE-PILLOT

Given an M_{1}-new Hecke eigenform g̃ in S^{2+2ν}(N) that is an eigenfunction of all the w_{p} for p | M_{2} with eigenvalues ϵ_{p} we then associate to it the newform g of some level M′ = M_{1}M′_{2} | M_{1}M_{2} that has the same Hecke eigenvalues for p ∤ N so that g̃ = g^{ϵ} with ϵ(p) = ϵ_{p} and apply Eichler’s correspondence to get a φ ∈ A_{ess}(D_{A}^{×}, (R(M_{1}, M′_{2}))_{A}^{×}, τ_{ν}). From [2] we know that this can be normalized such that g is obtained from φ by Yoshida’s lifting. We then pass to the eigenfunction φ^{ϵ} of all w̃_{p} with the same eigenvalues as g for p | M_{2}. The scalar product relation (up to normalization) follows then in the same way as in [5], using the uniqueness of the given set of w̃_{p}-eigenvalues and the fact that application of w_{p} for p | M_{2} transforms Θ̃^{(2ν)}(I_{ij}) to Θ̃^{(2ν)}(I_{i′j}), where y_{i′} represents the double coset of y_{i}π_{p}^{−1} (this is an easy generalization of Lemma 9.1 a) of [2], see also [3]). The same argument shows that Yoshida’s lifting realizes this correspondence (using the well known fact that it gives the right Hecke eigenvalues for the p ∤ N), and using the expression of g^{ϵ} as a Yoshida lifting we find the correct normalization of the scalar product relation as in [5].

We will need a version of the scalar product relation in Lemma 5.2 also for the case of newforms of level strictly dividing the level of the theta series involved.

Lemma 5.3 Let M_{1}, M′_{2} with M = M_{1}M′_{2} dividing N be as before and let g be a normalized newform of level M and weight k = 2+2ν as in the previous Lemma; let S be the set of prime divisors of N/M. Put M_{2} = N/M′_{2} and let R′ = R(M_{1}, M′_{2}) and R = R(M_{1}, M_{2}) ⊆ R′ be Eichler orders of levels M, N respectively in D(M_{1}) and consider a double coset decomposition D_{A}^{×} = ∪_{i=1}^{h}D_{Q}^{×}y_{i}R_{A}^{×} and corresponding quadratic lattices (ideals in D) I_{ij} relative to R as before. Let φ ∈ A_{ess}(D_{A}^{×}, (R′_{A})^{×}, τ_{ν}) be the essential form corresponding to g under Eichler’s correspondence (with ⟨φ, φ⟩_{ν} = 1). Then

⟨g, Θ̃^{(2ν)}(I_{ij})⟩ = ⟨g, g⟩ ∑_{S′⊆S}( ∏_{p∈S′} ˜(w_{p})φ)(y_{i}) ⊗ ( ∏_{p∈S′} ˜(w_{p})φ)(y_{j}))  (5.6)

Proof. Let ϵ be a function from the set of prime divisors of N/M to {±1}^{ω(N/M)} and let g^{ϵ}, φ^{ϵ} be as in the proof of Lemma 5.2. Then g^{ϵ} corresponds to (φ^{ϵ})/\sqrt{⟨φ^{ϵ}, φ^{ϵ}⟩_{ν}}) under Yoshida’s correspondence, so we get

⟨g^{ϵ}, Θ̃^{(2ν)}(I_{ij})⟩ = \frac{⟨g^{ϵ}, g^{ϵ}⟩}{⟨φ^{ϵ}, φ^{ϵ}⟩_{ν}} φ^{ϵ}(y_{i}) ⊗ φ^{ϵ}(y_{j})  (5.7)

by Lemma 5.2. For any S′ ⊆ S the Petersson product ⟨g, g|(∏_{p∈S′} ϵ(p)(w_{p}))⟩ is the same as the product of g with the image of g|(∏_{p∈S′} ϵ(p)(w_{p})) under the trace operator from modular forms for Γ_{0}(N) to modular forms for Γ_{0}(M), hence equal to ∏_{p∈S′} ϵ(p)p^{−ν}a_{g}(p)⟨g, g⟩, since for p ∤ M the map g ↦ g|w_{p} composed with the trace just gives the (renormalized) Hecke operator. The same argument applies to ⟨φ, (∏_{p∈S′} ϵ(p) ˜(w_{p})φ)⟩ and gives the same factor of comparison with ⟨φ, φ⟩ since g and φ have the same (renormalized) Hecke eigenvalues. Thus we have ⟨g^{ϵ}, g^{ϵ}⟩⟨φ^{ϵ}, φ^{ϵ}⟩^{−1} = ⟨g, g⟩⟨φ, φ⟩^{−1}, and summing up the identities (5.7) for all the functions ϵ gives the assertion.

Lemma 5.4 Let f, ϕ, ψ as in Section 2 be newforms of square free levels N_{f}, N_{ϕ}, N_{ψ} with N = lcm(N_{f}, N_{ϕ}, N_{ψ}) and with weights k_{i} = 2 + 2μ_{i}. Then in 5.5 the summand for M_{1}, M_{2}, M_{3} is zero unless M_{3} = 1, N = M_{1}M_{2} and M_{1} | gcd(N_{f}, N_{ϕ}, N_{ψ}) hold.

Proof. Since by Lemma 5.2 we can express f, ϕ, ψ as Yoshida-liftings an easy generalization of Lemma 9.1 b) of [2] shows that f is orthogonal to all Θ̃^{(μ_{1})}(K) for K in gen(M_{1}, M_{2}, M_{3}) for which N_{f} does not divide M_{1}M_{2} and analogously for ϕ, ψ. This establishes the vanishing of all summands for which N ≠ M_{1}M_{2}.

If there is a p | M_{1} not dividing gcd(N_{f}, N_{ϕ}, N_{ψ}) then say p ∤ N_{f}. The Petersson product of f with Θ̃^{(2μ_{1})}(K) for K in gen(M_{1}, M_{2}, 1) is then the same as that of f with the form obtained by applying the trace operator from modular forms on Γ_{0}(N) to modular forms on Γ_{0}(N/p) to the theta series. But it is easily checked that this trace operator annihilates the theta series of K ∈ gen(M_{1}, M_{2}, M_{3}) if p | M_{1}, see [8]; the same argument is applied to ϕ, ψ which shows the last part of the assertion.

Lemma 5.5 Let f̃, ϕ̃, ψ̃ be cusp forms of weights k_{1}, k_{2}, k_{3} for Γ_{0}(N) with square free N as in Section 2, assume them to be eigenforms of the Hecke operators for p ∤ N and of all the Atkin-Lehner involutions w_{p} with eigenvalues ϵ_{f}(p), ϵ_{ϕ}(p), ϵ_{ψ}(p) but not necessarily newforms. Let M_{1}M_{2} = N (with M_{1} as always having an odd number of prime factors) and let R = R(M_{1}, M_{2}) be an Eichler order in D = D(M_{1}). Let φ_{f}, φ_{ϕ}, φ_{ψ} be the forms in A(D_{A}^{×}, R_{A}^{×}, τ_{ν_{i}}) corresponding to f, ϕ, ψ under the correspondence of Lemma 5.2 (with k_{i} = 2 + 2μ_{i} for i = 1, …, 3). Then the summand for M_{1}, M_{2}, M_{3} = 1 in 5.5 is

2^{−ω(N)}α_{M_{1},M_{2},1}⟨f, f⟩⟨ϕ, ϕ⟩⟨ψ, ψ⟩ ∑_{i=1}^{h} (\frac{T_{0}(φ_{f}(y_{i}) ⊗ φ_{ϕ}(y_{i}) ⊗ φ_{ψ}(y_{i}))}{e_{i}})^{2} .

The latter expression is zero unless for all p | N one has p | M_{1} if and only if ϵ_{f}(p)ϵ_{ϕ}(p)ϵ_{ψ}(p) = −1.

Proof. The first part of the assertion is an immediate consequence of Lemma 5.2 and the decomposition of T as T^{(0)} ⊗ T^{(0)}. For the second part we notice that the expression

∑_{i=1}^{h} (\frac{T_{0}(φ_{f} (y_{i})⊗φ_{ϕ}(y_{i})⊗φ_{ψ}(y_{i}))}{e_{i}}) does not change if an involution w̃_{p} is applied to all three func-

tions φ_{f}, φ_{ϕ}, φ_{ψ} since this only permutes the order of summation. On the other hand each summand is multiplied with the product of the eigenvalues of φ_{f}, φ_{ϕ}, φ_{ψ} under w̃_{p}, which in view of the relation between the w_{p} eigenvalues and the w̃_{p}-eigenvalues of corresponding functions proves the assertion.

Although the statement of Lemma 5.2 is not true if one omits the condition that f̃ is an eigenfunction of all the involutions w_{p} the next Lemma shows that by an amusing newforms argument an only slightly changed version of Lemma 5.5 remains true without this condition.

Lemma 5.6 Let f, ϕ, ψ be normalized newforms of levels N_{f}, N_{ϕ}, N_{ψ} of weights k_{i} = 2+2μ_{i} (i = 1, …, 3) as in Section 2 and let Λ_{1}, Λ_{2}, Λ_{3} be pairwise disjoint subsets of the set of primes divisors of N^{f} = N/N_{f}, N^{ϕ}, N^{ψ} respectively such that Λ_{1} ∪ Λ_{2} ∪ Λ_{3} is the set of all primes dividing precisely one of the integers N_{f}, N_{ϕ}, N_{ψ}. For κ = 1, …, 3 let

w̃_{Λ_{κ}} = ∏_{p∈Λ_{κ}} w̃_{p}.

Let a decomposition N = M_{1}M_{2} as before be given, fix a maximal order R̃ in D = D(M_{1}) and an Eichler order R = R(M_{1}, M_{2}) ⊆ R̃ of level M_{1}M_{2} = N in D and consider a double coset decomposition D_{A}^{×} = ∪_{i=1}^{h}D_{Q}^{×}y_{i}R_{A}^{×} and corresponding quadratic lattices (ideals in D) I_{ij} relative to R as before. For each M′_{2} | M_{2} let R(M_{1}M′_{2}) = R(M_{1}, M′_{2}) be the unique Eichler order of level M_{1}M′_{2} contained in R̃ and containing R. Let φ_{1} ∈ A_{ess}(D_{A}^{×}, (R(N_{f})_{A}^{×}, τ_{μ_{1}})) be the form corresponding to f under Eichler’s correspondence and define φ_{2}, φ_{3} analogously with respect to ϕ, ψ. Then

T (⟨f, Θ̃^{(μ_{1})}⟩⟨ϕ, Θ̃^{(μ_{2})}⟩⟨ψ, Θ̃^{(μ_{3})}⟩)

= 2^{−ω(gcd(N_{f} ,N_{ϕ},N_{ψ}))}⟨f, f⟩⟨ϕ, ϕ⟩⟨ψ, ψ⟩(T_{0}(∑_{i=1}^{h} \frac{1}{e_{i}} w̃_{Λ_{1}}φ_{1}(y_{i}) ⊗ w̃_{Λ_{2}}φ_{2}(y_{i}) ⊗ w̃_{Λ_{3}}φ_{3}(y_{i})))^{2} (5.8)

Proof. This is an immediate consequence of Lemmas 5.2 and 5.3: Upon inserting the scalar product relations from these lemmata into the left hand side of (5.8) we obtain a sum of terms of the type

(T_{0}((∑_{i=1}^{h} w̃_{Λ′_{1}}φ_{1}(y_{i}) ⊗ w̃_{Λ′_{2}}φ_{2}(y_{i}) ⊗ w̃_{Λ′_{3}}φ_{3}(y_{i})))^{2}

with arbitrary subsets Λ′_{κ} of the sets of primes dividing N/N_{f}, N/N_{ϕ}, N/N_{ψ}) respectively. Let p be a prime dividing two of the levels, say p | N_{f}, p | N_{ϕ}. Then since applying w̃_{p} to all three of the φ_{κ} only changes the order of summation, the involution w̃_{p} for p ∈ Λ′_{3} may be pulled over to φ_{2}, φ_{3} which are eigenfunctions of w̃_{p}. The terms with the set Λ′_{3} and those with Λ′_{3} \ {p} give therefore the same contribution. Let now p be a prime dividing only one of the levels, say p | N_{f}. If p ∉ Λ′_{2} ∪ Λ′_{3} then the component of φ_{2} ⊗ φ_{3} in the τ_{μ_{1}}-isotypic component of τ_{ν_{2}} ⊗ τ_{ν_{3}} is an oldform with respect to p, hence orthogonal to the p-essential form φ_{1}. Since T_{0}(φ_{1} ⊗ φ_{2} ⊗ φ_{3}) is proportional to the scalar product of φ_{1} with this component of φ_{1} ⊗ φ_{2} such a term gives no contribution; the same argument applies if p ∈ Λ′_{2}∩Λ′_{3} holds. If p is in precisely one of Λ′_{2}, Λ′_{3} then the same argument as in the first case shows that p may be shifted to either one of these sets without changing the contribution of the term. Taking together both cases we find that all terms appearing are of the shape

(T_{0}((∑_{i=1}^{h} w̃_{Λ_{1}}φ_{1}(y_{i}) ⊗ w̃_{Λ_{2}}φ_{2}(y_{i}) ⊗ w̃_{Λ_{3}}φ_{3}(y_{i})))^{2}

with each term appearing 2^{ω(N/ gcd(N_{f} ,N_{ϕ},N_{ψ}))} times, which in view of 5.1 implies the assertion. Collecting all the information obtained we arrive at the main theorem:

Theorem 5.7 The value of the triple product L-function L(f, ϕ, ψ, s) at the central critical value s = \frac{k_{1}+k_{2}+k_{3}}{2} − 1 is

(−1)^{a′}2^{5+4a+3b−ω(gcd(N_{f} ,N_{ϕ},N_{ψ}))}π^{5+9a′+4b} \frac{(a′ + 1)^{[b]}}{2^{[a+b]}2^{[a′]}(ν_{2} + 1)^{[a′]}(ν_{3} + 1)^{[a′]}}

× ⟨f, f⟩⟨ϕ, ϕ⟩⟨ψ, ψ⟩(T_{0}(∑_{i=1}^{h} \frac{1}{e_{i}} w̃_{Λ_{1}}φ_{1}(y_{i}) ⊗ w̃_{Λ_{2}}φ_{2}(y_{i}) ⊗ w̃_{Λ_{3}}φ_{3}(y_{i})))^{2} (5.9)

where the notation is as in Lemma 5.6 and T_{0} is (as explained in the beginning of this section) the up to scalars unique rational invariant trilinear form on the representation space U_{μ_{1}/2}^{(0)} ⊗ U_{μ_{2}/2}^{(0)} ⊗ U_{μ_{3}/2}^{(0)} (with μ_{i} = k_{i} − 2) and takes values in the coefficient fields of f, ϕ, ψ respectively on the polynomials w̃_{Λ_{1}}φ_{κ}(y_{i}) (for κ = 1, …, 3)

It should be noted that the rational quantity on the right hand side can be interpreted as the height pairing of a diagonal cycle with itself in the same way as in [11]. One has just to replace (for κ = 1, …, 3) the group Pic(X) of [11] with the group Pic(V_{κ}) from [14] obtained by attaching to each y_{i} in the double coset decomposition of D_{A}^{×} used above the space of R_{i}^{×}-invariant polynomials in U_{μ_{κ}}^{(0)}. Our functions φ_{κ} may then be interpreted as elements of Pic(V_{κ}). One may then form the tensor product of these three groups and obtain an analogue of the diagonal cycle Δ from [11] by using our Gegenbauer polynomials from above and proceed as in loc. cit.

References

[1] S.Böcherer: Über die Fourier-Jacobientwicklung der Siegelschen Eisensteinreihen II. Mathem.Z.189(1989),81-110

[2] S.Böcherer,R.Schulze-Pillot: Siegel modular forms and theta series attached to quaternion algebras. Nagoya Math.J. 121(1991), 35-96

[3] S.Böcherer,R.Schulze-Pillot: Siegel modular forms and theta series attached to quaternion algebras II, Preprint 1995

[4] S.Böcherer,R.Schulze-Pillot: Mellin transforms of vector valued theta series attached to quaternion algebras. Math. Nachr. 169 (1994), 31-57

[5] S.Böcherer,R.Schulze-Pillot: Vector valued theta series and Waldspurger’s theorem, Abh. Math. Sem. Hamburg 64 (1994), 211-233

[6] S.Böcherer,T.Satoh,T.Yamazaki: On the pullback of a differential operator and its application to vector valued Eisenstein series. Comm.Math.Univ.S.Pauli 41(1992), 1-22

[7] M. Eichler: The basis problem for modular forms and the traces of the Hecke operators, p. 76-151 in Modular functions of one variable I, Lecture Notes Math. 320, Berlin-Heidelberg-New York 1973

[8] J. Funke: Spuroperator und Thetareihen quadratischer Formen, Diplomarbeit Köln 1994

[9] P.Garrett: Decomposition of Eisenstein series: Rankin triple products. Annals of Math. 125(1987), 209-235

[10] P. Garrett, M. Harris: Special values of triple product L-Functions, Am. J. of Math. 115 (1993), 159-238 [11] B.Gross, S.Kudla: Heights and the central critical values of triple product L-functions.Compositio

Math.81(1992), 143-209

[12] M. Harris, S. Kudla: The central critical value of a triple product L-function, Annals of Math. 133 (1991), 605-672 [13] K. Hashimoto: On Brandt matrices of Eichler orders, Preprint 1994

[14] R. Hatcher: Heights and L-series, Can. J. math. 62 (1990), 533-560

[15] H. Hijikata, H. Saito: On the representability of modular forms by theta series, p. 13-21 in Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Tokyo 1973

[16] T.Ibukiyama: On differential operators on automorphic forms and invariant pluriharmonic polynomials. Preprint 1990

[17] H. Jacquet, R. Langlands: Automorphic forms on GL(2), Lect. Notes in Math. 114, Berlin-Heidelberg-New York 1970

[18] W.Li: Newforms and functional equations. Math.Ann.212(1975), 285-315

[19] P. Littelmann: On spherical double cones, J. of Algebra 166 (1994), 142-157

[20] H.Maaß: Siegel’s modular forms and Dirichlet series. Lect. Notes Math. 216 Berlin, Heidelberg, New York: Springer 1971

[21] T.Orloff: Special values and mixed weight triple products (with an appendix by Don Blasius). Invent.math.90(1987), 169-180

[22] T.Satoh: Some remarks on triple L-functions. Math.Ann.276(1987), 687-698

[23] H. Shimizu: Theta series and automorphic forms on GL_{2}. J. of the Math. Soc. of Japan 24 (1972), 638-683

[24] G.Shimura: The special values of the zeta functions associated with cusp forms. Comm.Pure Appl. Math.29(1976), 783-804

[25] G.Shimura: The arithmetic of differential operators on symmetric domains. Duke Math.J.48(1981), 813-843

S. Böcherer, Fakultät für Mathematik und Informatik, Universität Mannheim, Seminargebäude A5, D-68131 Mannheim, Germany

R. Schulze-Pillot, Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany after Oct. 1, 1995: Fachbereich Mathematik, Universität des Saarlandes (Bau 27), Postfach 1150, D-66041 Saarbrücken, Germany