MOTIVES AND SIEGEL MODULAR FORMS

(APPENDIX TO BÖCHERER AND HEIM:

CRITICAL VALUES FOR L-FUNCTIONS ON GSp_{2} × Gl_{2})

By Hiroyuki Yoshida

Introduction

For a Siegel modular form f ∈ S_{k}(Sp(n, Z)), we normalize the Petersson norm so that

⟨f, f⟩ = vol(Sp(n, Z)\ℌ_{n})^{−1} ∫_{Sp(n,Z)\ℌ_{n}} |f(z)|^{2}(det y)^{k−n−1}dxdy.

Let f ∈ S_{k}(Sp(2, Z)) and g ∈ S_{l}(SL(2, Z)) be non-zero Hecke eigenforms. Let L(s, f ⊗ g) be the tensor product L-function, the product taken for the spinor Lfunction of f and Hecke’s L-function of g. We assume the existence of the motive M(f) attached to f (cf. §4) and Deligne’s conjecture. We also assume that the Fourier coefficients of f are contained in a totally real algebraic number field. (This condition can be fulfilled by multiplying f by a non-zero constant.)

Theorem A. If l > 2k − 2, then for n ∈ Z, 2k − 2 ≦ n ≦ l − 1, we have

(\frac{L(n, f ⊗ g)}{π^{4n−4k+8}⟨g, g⟩^{2}} )^{σ} = \frac{L(n, f^{σ} ⊗ g^{σ})}{π^{4n−4k+8}⟨g^{σ}, g^{σ}⟩^{2}} for σ ∈ Aut(C).

Theorem B. If l < 2k − 2, then for n ∈ Z such that k ≦ n ≦ k + l − 3 if k ≧ l, l ≦ n ≦ 2k − 3 if k ≦ l, we have

(\frac{L(n, f ⊗ g)}{π^{4n−2k−l+8}⟨f, f⟩⟨g, g⟩})^{σ} = \frac{L(n, f^{σ} ⊗ g^{σ})}{π^{4n−2k−l+8}⟨f^{σ}, f^{σ}⟩⟨g^{σ}, g^{σ}⟩} for σ ∈ Aut(C).

Let us explain the contents of this paper. In §1, we determine polynomials on M(n, n) which transform according to characters under the left action of a parabolic subgroup of GL(n) and also under the right action of GL(n^{+}) × GL(n^{−}), n^{+} + n^{−} = n. Our method is elementary and we can work over an arbitrary field. In §2, we define “hidden period invariants” of a motive using our results in §1. We shall show that Deligne’s period invariants c^{±} of motives, which are obtained by various algebraic operations of given motives, can be expressed using them. We illustrate the efficacy of this method by taking a motive of rank 4 as an example in §3. In §4, we deduce Theorems A and B applying the results in §3 to the motive M(f).

Typeset by AMS-TEX

§1. On certain relatively invariant polynomials

In this section, we work over an arbitrary base field F. By M(m, n), we denote the set of all m × n matrices and regard M(m, n) as an algebraic variety defined over F. Throughout this section, P_{n} denotes the minimal parabolic subgroup of GL(n) consisting of all lower triangular matrices.

Let P be the lower parabolic subgroup of GL(n) corresponding to the partition n = s_{1} + s_{2} + ⋯ + s_{m}. For (a_{1}, a_{2}, ⋯, a_{m}) ∈ Z^{m}, we define an algebraic character λ_{1} of P by

(1.1) λ_{1} (( \begin{matrix}p_{11} & 0 & … & 0 \\ ∗ & p_{22} & … & 0 \\ ∗ & ∗ & ... & ⋮ \\ ∗ & ∗ & ∗ & p_{mm} \end{matrix} )) = det(p_{11})^{a_{1}} det(p_{22})^{a_{2}} ⋯ det(p_{mm})^{a_{m}},

where p_{ii} ∈ GL(s_{i}). We decompose n as n = n^{+} + n^{−} with non-negative integers n^{+} and n^{−} and embed GL(n^{+}) × GL(n^{−}) as diagonal blocks into GL(n). For k^{+}, k^{−} ∈ Z, we define an algebraic character λ_{2} of GL(n^{+}) × GL(n^{−}) by

(1.2)  λ_{2} (( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )) = (det a)^{k^{+}}(det b)^{k^{−}},  a ∈ GL(n^{+}), b ∈ GL(n^{−}).

We consider a polynomial function f(x) on M(n, n) rational over F which satisfies

(1.3)  f(pxγ) = λ_{1}(p)λ_{2}(γ)f(x)  for all p ∈ P, γ ∈ GL(n^{+}) × GL(n^{−}).

We call f to be of the type (λ_{1}, λ_{2}) or of the type {(a_{1}, a_{2}, ⋯, a_{m}); (k^{+}, k^{−})}. All of such an f generate a graded algebra over F. Our main results in this section assert that every subspace of a given grade of this algebra is at most one dimensional and that this algebra is isomorphic to the polynomial ring with explicitly given generators. We refer the reader to Shimura [Sh2], Theorem 2A and Howe [Ho] among vast literatures on the subjects related to our results.

The restriction of λ_{1} to P_{n} , which we denote by λ_{1}^{∗}, is given by

(1.4)  λ_{1}^{∗} (( \begin{matrix}t_{11} & 0 & … & 0 \\ ∗ & t_{22} & … & 0 \\ ∗ & ∗ & ... & ⋮ \\ ∗ & ∗ & ∗ & t_{nn} \end{matrix} )) = t_{11}^{w_{1}}t_{22}^{w_{2}} ⋯ t_{nn}^{w_{m}},

(1.5)  w_{j} = a_{i}  if s_{1} + s_{2} + ⋯ + s_{i−1} < j ≦ s_{1} + s_{2} + ⋯ + s_{i}.

We call (λ_{1}, λ_{2}) admissible or T = {(a_{1}, a_{2}, ⋯, a_{m}), (k^{+}, k^{−})} an admissible data if the following conditions (1.6) ∼ (1.10) hold.

(1.6)  w_{1} ≧ w_{2} ≧ ⋯ ≧ w_{n^{+}} ≧ k^{+} ≧ 0,
(1.7)  k^{−} ≧ w_{n^{+}+1} ≧ w_{n^{+}+2} ≧ ⋯ ≧ w_{n} ≧ 0.
(1.8)  \begin{matrix}w_{n^{−}+1} = w_{n^{−}+2} = ⋯ = w_{n^{+}} = k^{+}  if n^{+} > n^{−}, \\ w_{n^{+}+1} = w_{n^{+}+2} = ⋯ = w_{n^{−}} = k^{−}  if n^{+} < n^{−}. \end{matrix}
(1.9)  w_{i} + w_{n+1−i} = k^{+} + k^{−},  1 ≦ i ≦ min(n^{+}, n^{−}).

(1.10)

w_{1} ≧ w_{2} ≧ ⋯ ≧ w_{min(n^{+},n^{−})} ≧ max(k^{+}, k^{−})
≧ min(k^{+}, k^{−}) ≧ w_{max(n^{+},n^{−})+1} ≧ ⋯ ≧ w_{n}.

Theorem 1. Let the notation be the same as above. There exists a non-zero polynomial function f(x) on M(n, n) of the type (λ_{1}, λ_{2}) if and only if (λ_{1}, λ_{2}) is admissible. If this is the case, such an f is unique up to the constant multiple and is given by (1.17) up to the constant multiple.

For the proof of Theorem 1, we first show the next lemma.

Lemma 2. Let m and n be natural numbers. Let χ_{1} and χ_{2} be the algebraic characters of P_{m} and P_{n} defined by

χ_{1}(α) = α_{11}^{a_{1}}α_{22}^{a_{2}} ⋯ α_{mm}^{a_{m}}, χ_{2}(β) = β_{11}^{b_{1}}β_{22}^{b_{2}} ⋯ β_{nn}^{b_{m}},

α = ( \begin{matrix}α_{11} & 0 & … & 0 \\ ∗ & α_{22} & … & 0 \\ ∗ & ∗ & ... & ⋮ \\ ∗ & ∗ & ∗ & α_{mm} \end{matrix} ) , β = ( \begin{matrix}β_{11} & 0 & … & 0 \\ ∗ & β_{22} & … & 0 \\ ∗ & ∗ & ... & ⋮ \\ ∗ & ∗ & ∗ & β_{nn} \end{matrix} )

where a_{i}, b_{j} ∈ Z. Put l = min(m, n). For x ∈ M(m, n) and i ≦ l, let det_{i}(x) denote the determinant of the upper right i × i-submatrix of x. Then there exists a non-zero polynomial function g(x) on M(m, n) which satisfies

g(αxβ) = χ_{1}(α)χ_{2}(β)g(x) for all α ∈ P_{m}, β ∈ P_{n}

if and only if the following conditions (1.11) and (1.12) hold.

(1.11)  \begin{matrix}a_{1} ≧ a_{2} ≧ ⋯ ≧ a_{n} ≧ 0,  a_{n+1} = a_{n+2} = ⋯ = a_{m} = 0, \\ b_{i} = a_{n+1−i}, 1 ≦ i ≦ n  if m ≧ n. \end{matrix}

(1.12)

\begin{matrix}a_{1} ≧ a_{2} ≧ ⋯ ≧ a_{m} ≧ 0, \\ b_{i} = a_{n+1−i}, n − m + 1 ≦ i ≦ n,  b_{1} = b_{2} = ⋯ = b_{n−m} = 0  if m ≦ n. \end{matrix}

If these conditions hold, g(x) is equal to

∏_{i=1}^{l−1} det_{i}(x)^{a_{i}−a_{i+1}} det_{l}(x)^{a_{l}}

up to the constant multiple.

Proof. Let g(x) be a non-zero polynomial function as in our lemma. We note that every a_{i} is non-negative since χ_{1}(α) = g(αx)/g(x), α ∈ P_{m} is a polynomial of α. Similarly every b_{j} is non-negative. We also note that

(1.13)  det_{i}(αx) = α_{11}α_{22} ⋯ α_{ii} det_{i}(x),

(1.14)  det_{i}(xβ) = β_{nn}β_{n−1 n−1} ⋯ β_{n+1−i n+1−i} det_{i}(x)

for α ∈ P_{m}, β ∈ P_{n} as in our lemma.

First let us consider the case m = n. Put

w = ( \begin{matrix}0 & 0 & … & 0 & 1 \\ 0 & 0 & … & 1 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 1 & … & 0 & 0 \\ 1 & 0 & … & 0 & 0 \end{matrix} ) ∈ GL(n).

Then P_{n}wP_{n} is Zariski dense in M(n, n). Hence g(w) ≠ 0. Put

Z = {x ∈ M(n, n) | det_{i}(x) ≠ 0 for 1 ≦ i ≦ n}.

Let x ∈ Z. We can write xw = pu with p = (p_{ij}) ∈ P_{n}, u ∈ GL(n) is an upper unipotent matrix. We have

p_{ii} = (−1)^{i−1} det_{i}(x)/ det_{i−1}(x), 1 ≦ i ≦ n,

with the convention det_{0}(x) = 1. Since x = pw · wuw, we get

g(x) = (−1)^{δ} ∏_{i=1}^{n−1} det_{i}(x)^{a_{i}−a_{i+1}}(det x)^{a_{n}}g(w), δ = ∑_{i=1}^{n}(i − 1)a_{i}.

Since Z is Zariski dense in M(n, n), this rational function of x must be a polynomial of x. Since det_{1}(x), det_{2}(x), ⋯, det_{n}(x) are mutually prime irreducible polynomials, we see that a_{i} ≧ a_{i+1} for 1 ≦ i ≦ n − 1 and that g(x) is equal to ∏_{i=1}^{n−1} det_{i}(x)^{a_{i}−a_{i+1}} det(x)^{a_{n}} up to the constant multiple. The relation between a_{i} and b_{j} follows from (1.13) and (1.14).

Next assume m > n. Write x ∈ M(m, n) as x = ( \begin{matrix}y \\ z \end{matrix} ), y ∈ M(n, n), z ∈

M(m − n, n) and put h(y) = g(( \begin{matrix}y \\ 0 \end{matrix} )), y ∈ M(n, n). By

(1.15)  ( \begin{matrix}1_{n} & 0 \\ u & 1_{m−n} \end{matrix} ) ( \begin{matrix}y \\ z \end{matrix} ) = ( \begin{matrix}y \\ uy + z \end{matrix} ) ,  u ∈ M(m − n, n),

we see easily that h(y) ≠ 0. (The subset det y ≠ 0 is Zariski dense.) Since h(y) has the transformation property of the type in our lemma, we have

h(y) = c ∏_{i=1}^{n−1} det_{i}(y)^{a_{i}−a_{i+1}}(det y)^{a_{n}}

with a constant c. Then we see that g(x) is equal to c ∏_{i=1}^{n−1} det_{i}(x)^{a_{i}−a_{i+1}}(det_{n}(x))^{a_{n}} by (1.15). Remaining assertions for the case m > n follow immediately. The case m < n can be treated similarly. This completes the proof.

Proof of Theorem 1. We first prove the theorem for the case P = P_{n} and for polynomial functions of the type (λ_{1}, λ_{2}). Thus λ_{1} is λ_{1}^{∗} which is given by (1.4). Suppose that a non-zero polynomial function of the type (λ_{1}, λ_{2}) exists. By the same reason as in Lemma 2, we have w_{i} ≧ 0, 1 ≦ i ≦ n, k^{+}, k^{−} ≧ 0. If n^{−} = 0, f(x) is a constant multiple of (det x)^{k^{+}}. Since the conditions (1.6) ∼ (1.10) are equivalent to w_{1} = w_{2} = ⋯ = w_{n} = k^{+}, we see that the assertion holds. Similarly the case n^{−} = 0 is clear.

We assume n^{+} ≧ 1 and n^{−} ≧ 1. . Define a polynomial function g(x) on M(n^{+}, n^{−}) by

g(x) = f (( \begin{matrix}1_{n^{+}} & x \\ 0 & 1_{n^{−}} \end{matrix} )) , x ∈ M(n^{+}, n^{−}).

We see that

P_{n} · \{( \begin{matrix}1_{n^{+}} & x \\ 0 & 1_{n^{−}} \end{matrix} ) | x ∈ M(n^{+}, n^{−})\} · (GL(n^{+}) × GL(n^{−}))

is Zariski dense in M(n, n). Hence g ≠ 0 and f is uniquely determined by g. For α ∈ P_{n^{+}} and β ∈ P_{n^{−}}, the relation

( \begin{matrix}α & 0 \\ 0 & β^{−1} \end{matrix} ) ( \begin{matrix}1_{n^{+}} & x \\ 0 & 1_{n^{−}} \end{matrix} ) ( \begin{matrix}α^{−1} & 0 \\ 0 & β \end{matrix} ) = ( \begin{matrix}1_{n^{+}} & αxβ \\ 0 & 1_{n^{−}} \end{matrix} )

yields

g(αxβ) = λ_{1} (( \begin{matrix}α & 0 \\ 0 & β^{−1} \end{matrix} )) λ_{2} (( \begin{matrix}α^{−1} & 0 \\ 0 & β \end{matrix} )) g(x).

By Lemma 2, we obtain the conditions (1.6) ∼ (1.9). Lemma 2 also implies that f is unique up to the constant multiple. To see that the condition (1.10) holds, put

f^{∗}(x) = f(xω), ω = ( \begin{matrix}0 & 0 & 1_{n^{−}} \\ 0 & 1_{t} & 0 \\ 1_{n^{−}} & 0 & 0 \end{matrix} ) ∈ GL(n), t = n^{+} − n^{−}

assuming n^{+} ≧ n^{−}. Then f^{∗} satisfies

f^{∗}(pxγ) = λ_{1}(p)λ_{2}^{∗}(γ)f(x)  for all p ∈ P_{n}, γ ∈ GL(n^{−}) × GL(n^{+}),
λ_{2}^{∗} (( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )) = (det a)^{k^{−}}(det b)^{k^{+}},  a ∈ GL(n^{−}), b ∈ GL(n^{+}).

By the conditions (1.6) and (1.7) satisfied by f^{∗}, we get (1.10). The case n^{+} ≦ n^{−} can be handled similarly.

Let us show the existence of f ≠ 0 of the type (λ_{1}, λ_{2}) when (λ_{1}, λ_{2}) is admissible. Put k = k^{+} + k^{−}, N = n^{+}k^{+} + n^{−}k^{−}. We first note that

(1.16)  w_{1} + w_{2} + ⋯ + w_{n} = N

which follows from (1.8) and (1.9). Let K be an algebraically closed field containing F. For row vectors x_{i}^{±} ∈ K^{n^{±}} (1 ≦ i ≦ n), y_{j}^{±} ∈ K^{k^{±}} (1 ≦ j ≦ k), we define a square matrix Z of size N as follows. Divide the rows of Z according as the partition (1.16). In the i-th block, we put

( \begin{matrix}x_{i}^{+} ⊗ y_{1}^{+} & x_{i}^{−} ⊗ y_{1}^{−} \\ x_{i}^{+} ⊗ y_{2}^{+} & x_{i}^{−} ⊗ y_{2}^{−} \\ ⋮ & ⋮ \\ x_{i}^{+} ⊗ y_{w_{i}}^{+} & x_{i}^{−} ⊗ y_{w_{i}}^{−} \end{matrix} ) ,

regarding x_{i}^{±} ⊗ y_{j}^{±} ∈ K^{d^{±}k^{±}}. Put

x = ( \begin{matrix}x_{1}^{+} & x_{1}^{−} \\ x_{2}^{+} & x_{2}^{−} \\ ⋮ & ⋮ \\ x_{n}^{+} & x_{n}^{−} \end{matrix} ) ,  y = ( \begin{matrix}y_{1}^{+} & y_{1}^{−} \\ y_{2}^{+} & y_{2}^{−} \\ ⋮ & ⋮ \\ y_{k}^{+} & y_{k}^{−} \end{matrix} ) ,  h(x, y) = det Z.

We regard x (resp. y) as a variable on M(n, n) (resp. M(k, k)). When y is fixed, we see easily that the polynomial h(x, y) of x is of the type {(w_{1}, w_{2}, ⋯, w_{n}); (k^{+}, k^{−})}. By the uniqueness assertion already established, we see that

(1.17)  h(x, y) = f(x)g(y)

with polynomials f(x) and g(y) with coefficients in F. Our task is to show h ≠ 0. We note the following “duality” between f and g. Put

w′_{i} = |{j | 1 ≦ j ≦ n, w_{j} ≧ i}|

for 1 ≦ i ≦ k. Then it can be verified that U = {(w′_{1}, w′_{2}, ⋯, w′_{k}); (n^{+}, n^{−})} is an admissible data and that g(y) is of the type U.

Now we proceed to prove h ≠ 0 by induction on N. The case n^{+}n^{−} = 0 or k^{+}k^{−} = 0 being easy, we assume n^{+}, n^{−}, k^{+}, k^{−} ≧ 1. We first consider the case w_{n} ≧ 1. For x ∈ GL(n, K), y ∈ M(k, k, K), suppose that

(1.18)  ∑_{i=1}^{n} ∑_{j=1}^{w_{i}} c_{ij}(x_{i}^{±} ⊗ y_{j}^{±}) = 0

with c_{ij} ∈ K. We choose y_{1}^{±} ∈ K^{k^{±}} as non-zero vectors and take y_{2}^{±}, ⋯, y_{k}^{±} from complements of the spaces generated by y_{1}^{±}. Then (1.18) is equivalent to

∑_{i=1}^{n} c_{i1}x_{i}^{±} = 0,

(1.19)  ∑_{i=1}^{n} ∑_{j=2}^{w_{i}} c_{ij}(x_{i}^{±} ⊗ y_{j}^{±}) = 0.

We get c_{i1} = 0, 1 ≦ i ≦ n since det x ≠ 0. The data {(w_{1} − 1, w_{2} − 1, ⋯, w_{n} − 1); (k^{+} − 1, k^{−} − 1)} is admissible. Hence, by the induction hypothesis, we can choose x_{i}^{±} (1 ≦ i ≦ n), y_{j}^{±} (2 ≦ j ≦ k) so that (1.19) implies all c_{ij} = 0. Next we consider the case w_{n} = 0. By (1.9), we have w_{1} = k. Therefore w′_{k} ≧ 1 and we are reduced to the previous case by the duality.

Now the assertion for the general P clearly follows from the special case P = P_{n} in view of the definition of h(x, y). This completes the proof.

Theorem 3. Let the notation be the same as in Theorem 1. If there exists p^{+} (resp. p^{−}) such that 1 ≦ p^{+} ≦ m (resp. 1 ≦ p^{−} ≦ m) such that s_{1} + s_{2} + ⋯ + s_{p^{+}} = n^{+} (resp. s_{1} + s_{2} + ⋯ + s_{p^{−}} = n^{−}), let f^{+}(x) (resp. f^{−}(x)) denote the determinant of the upper left n^{+} × n^{+}-submatrix (resp. the upper right n^{−} × n^{−}-submatrix) of x. Let A denote the set of integers a such that 1 ≦ a < min(n^{+}, n^{−}),

s_{1} + s_{2} + ⋯ + s_{p} = a, s_{1} + s_{2} + ⋯ + s_{q} = d − a

for some 1 ≦ p < q ≦ m, and that T_{a} = \{(\overbrace{2, ⋯, 2}^{p}, \overbrace{1, ⋯, 1}^{q−p}, \overbrace{0, ⋯, 0}^{m−q}); (1, 1)\} is

an admissible data. For a ∈ A, let f_{a}(x) be a non-zero polynomial of the type T_{a}. Then every polynomial of the admissible type {(a_{1}, a_{2}, ⋯, a_{m}); (k^{+}, k^{−})} can be expressed uniquely as a monomial of det(x), f^{+}(x), f^{−}(x) and f_{a}(x), a ∈ A.

Proof. For two admissible data

T = {(a_{1}, a_{2}, ⋯, a_{m}); (k^{+}, k^{−})}, U = {(b_{1}, b_{2}, ⋯, b_{m}); (l^{+}, l^{−})},

we define their sum by T +U = {(a_{1} +b_{1}, a_{2} +b_{2}, ⋯, a_{m} +b_{m}); (k^{+} +l^{+}, k^{−} +l^{−})}. Then T + U is admissible. By the existence and the uniqueness proved in Theorem 1, it suffices to show that any admissible data T can be written uniquely as a sum of admissible data of det(x), f^{+}(x), f^{−}(x) and f_{a}(x), a ∈ A. The task is somewhat tedious but can be done without difficulty using induction on ∑_{i=1}^{m} a_{i}. This completes the proof.

§2. On the hidden periods of a motive

Let E be an algebraic number fields of finite degree. Let J_{E} denote the set of all isomorphisms of E into C. We put R = E ⊗_{Q} C ≅ C^{J_{E}}. Let M be a motive over Q with coefficients in E. Then M has (at least) three realizations Betti, de Rham and λ-adic.

First let H_{B}(M) denote the Betti realization of M; H_{B}(M) is a finite dimensional vector space over E, whose dimension d = d(M) is called the rank of M. The complex conjugation F_{∞} acts E-linearly on H_{B}(M). We can decompose

(2.1)  H_{B}(M) = H_{B}^{+}(M) ⊕ H_{B}^{−}(M)

where H_{B}^{±}(M) denotes the eigenspace of H_{B}(M) with the eigenvalue ±1. We denote the dimension of H_{B}^{±}(M) over E by d^{±} = d^{±}(M). Furthermore H_{B}(M) has the Hodge decomposition:

(2.2)  H_{B}(M) ⊗_{Q} C = ⊕_{p,q∈Z}H^{pq}(M),

where H^{pq}(M) is a free E ⊗_{Q} C-module. If H^{pq}(M) = {0} whenever p + q ≠ w, M is called of pure weight w, which we shall assume hereafter.

We skip the description of the λ-adic realization of M. We simply note that the L-function L(M, s) of M taking values in R is defined using it and the gamma factor of the functional equation can be described by the Hodge decomposition (2.2) (see [D]).

Let H_{DR}(M) be the de Rham realization of M which is a d-dimensional vector space over E. Let {F^{p}} be the Hodge filtration on H_{DR}(M); {F^{p}} is a decreasing filtration by vector spaces over E. When it is necessary to emphasize the dependence on M, we write F^{p}(M) for F^{p}. Let

I : H_{B}(M) ⊗_{Q} C ≅ H_{DR}(M) ⊗_{Q} C

be the comparison isomorphism. We have

(2.3)  I(⊕_{p′≧p}H^{p′q}(M)) = F^{p}(M) ⊗_{Q} C.

Now we are going to define a period matrix of M. Let {v_{1}^{+}, v_{2}^{+}, ⋯, v_{d^{+}}^{+}} (resp. {v_{1}^{−}, v_{2}^{−}, ⋯, v_{d^{−}}^{−}}) be a basis of H_{B}^{+}(M) (resp. H_{B}^{−}(M)) over E. We write the Hodge filtration as

H_{DR}(M) = F^{i_{i}} ⫌ F^{i_{2}} ⫌ ⋯ ⫌ F^{i_{m}} ⫌ F^{i_{m+1}} = {0}

so that there are no different filtrations between successive members. The choice of numbers i_{μ} may not be unique for F^{i_{μ}}. Put t_{μ} = dim_{E} F^{i_{μ}}, 1 ≦ μ ≦ m. We take a basis {w_{1}, w_{2}, ⋯, w_{d}} of H_{DR}(M) over E so that {w_{d−t_{μ}+1}, ⋯, w_{d}} is a basis of F^{i_{μ}} for 1 ≦ μ ≦ m. Writing

(2.4)  I(v_{j}^{±}) = ∑_{i=1}^{d} x_{ij}^{±}w_{i},  x_{ij}^{±} ∈ R, 1 ≦ j ≦ d^{±},

we obtain a matrix X^{±} = (x_{ij}^{±}) ∈ M(d, d^{±}, R). We put

s_{i} = t_{i} − t_{i+1}, 1 ≦ i ≦ m − 1, s_{m} = t_{m}.

Let P_{M} be the lower parabolic subgroup of GL(d) which corresponds to the partition d = s_{1} + s_{2} + ⋯ + s_{m}. Then the coset of X^{±} in

P_{M}(E)\M(d, d^{±}, R)/GL(d^{±}, E)

does not depend on the choices of basis. We put X = (X^{+} X^{−}) ∈ M(d, d, R) and call it a period matrix of M. The coset of X in

P_{M}(E)\M(d, d^{±}, R)/(GL(d^{+}, E) × GL(d^{−}, E))

is well defined. Here GL(d^{+}, E) × GL(d^{−}, E) is embedded in GL(d) as diagonal blocks.

We are going to apply our results in §1 taking the base field F as Q, P as P_{M} and n = d, n^{±} = d^{±}. Let f(x) be a non-zero polynomial of the type {(a_{1}, a_{2}, ⋯, a_{m}); (k^{+}, k^{−})}. Let X ∈ M(d, d, R) be a period matrix of M. Then f(X) is uniquely determined up to the multiplication by elements of E^{×}. We call f(X) a period invariant of M of the type {(a_{1}, a_{2}, ⋯, a_{m}); (k^{+}, k^{−})}. Hereafter we understand the equality between period invariants mod E^{×}.

Let f(x) = det(x), x ∈ M(d, d). Then f(x) is of the type {(1, 1, ⋯, 1); (1, 1)} and f(X) is Deligne’s period δ(M). Assume that t_{1} − t_{p^{+}+1} = d^{+} for some p^{+}. Let f^{+}(x) be the determinant of the upper left d^{+} × d^{+}-submatrix of x ∈ M(d, d).

Then f^{+}(x) is of the type \{(\overbrace{1, 1, ⋯, 1}^{p^{+}}, 0, ⋯, 0); (1, 0)\} and f^{+}(X) is Deligne’s period c^{+}(M). Similarly if t_{1} − t_{p^{−}+1} = d^{−} for some p^{−}, let f^{−}(x) be the determinant of the upper right d^{−} × d^{−}-submatrix of x. Then f^{−}(x) is of the type

\{(\overbrace{1, 1, ⋯, 1}^{p^{−}}, 0, ⋯, 0); (0, 1)\} and f^{−}(X) is Deligne’s period c^{−}(M). Either one of the above conditions is equivalent to that F^{±}(M) can be defined (cf. [D], §1, [Y], §2). We have F^{∓}(M) = F^{i_{p^{±}+1}}(M); F^{±}(M) can be defined if M has a critical value.

Deligne’s conjecture states that L(M, 0)/c^{+}(M) ∈ E if 0 is critical for M. Other period invariants are hidden for the relation with L(M, 0) but will appear when we make various algebraic operations on M (tensor products with other motives, exterior powers, etc.). Deligne showed c^{±}(M) ≠ 0. The non-vanishing of other period invariants will be proved at the end of this section.^{1}

We write X in the form

X = ( \begin{matrix}𝔛_{1}^{+} & 𝔛_{1}^{−} \\ 𝔛_{2}^{+} & 𝔛_{2}^{−} \\ ⋮ & ⋮ \\ 𝔛_{d}^{+} & 𝔛_{d}^{−} \end{matrix} ) , 𝔛_{i}^{+} ∈ R^{d^{+}}, 𝔛_{i}^{−} ∈ R^{d^{−}}.

If t_{1} − t_{μ} ≦ i < t_{1} − t_{μ+1}, we call 𝔛_{i}^{±} of weight i_{μ} and write w(𝔛_{i}^{+}) = w(𝔛_{i}^{−}) = i_{μ}.

Let us consider the exterior square ∧^{2}M and the symmetric square Sym^{2}M of the motive M.

Proposition 4. Assume that F^{±}(∧^{2}M) can be defined. Let F^{±}(∧^{2}M) = F^{q^{±}}(∧^{2}M). Then c^{+}(∧^{2}M) = f^{+}(X) with a polynomial f^{+}(x) of the type {(a_{1}, a_{2}, ⋯, a_{m}); (k^{+}, k^{−})}, where, taking 𝔛_{i}^{+} so that w(𝔛_{i}^{+}) = i_{μ},

(2.5)  \begin{aligned}a_{μ} & = |\{l | 1 ≦ l ≦ d, w(𝔛_{l}^{+}) ≠ i_{μ}, i_{μ} + w(𝔛_{l}^{+}) < q^{−}\}| \\ & + \{ \begin{matrix}t_{μ} − t_{μ+1} − 1 if 2i_{μ} < q^{−}, \\ 0 if 2i_{μ} ≧ q^{−}, \end{matrix} \end{aligned}

(2.6)  k^{+} = d^{+}(M) − 1,  k^{−} = d^{−}(M) − 1.

We have c^{−}(∧^{2}M) = f^{−}(X) with a polynomial f^{−}(x) . The type of f^{−}(x) is given by

(2.7)  \begin{aligned}a_{μ} & = |\{l | 1 ≦ l ≦ d, w(𝔛_{l}^{+}) ≠ i_{μ}, i_{μ} + w(𝔛_{l}^{+}) < q^{+}\}| \\ & + \{ \begin{matrix}t_{μ} − t_{μ+1} − 1 if 2i_{μ} < q^{+}, \\ 0 if 2i_{μ} ≧ q^{+}, \end{matrix} \end{aligned}

(2.8)  k^{+} = d^{−}(M),  k^{−} = d^{+}(M).

1It is not difficult to prove that if f(X) is non-zero and if f(X) mod E^{×} is well defined for a polynomial f(x), f must be of the above form.

Proof. We have

H_{B}^{+}(∧^{2}M) = (H_{B}^{+}(M) ∧ H_{B}^{+}(M)) ⊕ (H_{B}^{−}(M) ∧ H_{B}^{−}(M)),
H_{B}^{−}(∧^{2}M) = H_{B}^{+}(M) ∧ H_{B}^{−}(M),

I(v_{j}^{±} ∧ v_{k}^{±}) = ∑_{1≦i<l≦d}(x_{ij}^{±}x_{lk}^{±} − x_{lj}^{±}x_{ik}^{±})(w_{i} ∧ w_{l}).

Let W = (W^{+} W^{−}) be a period matrix of ∧^{2}M. The formula above shows that W^{+} can be given by arranging (𝔛_{i}^{+} ∧ 𝔛_{l}^{+}, 𝔛_{i}^{−} ∧ 𝔛_{l}^{−}), 1 ≦ i < l ≦ d according to the increasing order of w(𝔛_{i}^{+}) + w(𝔛_{l}^{+}), regarding 𝔛_{i}^{±} ∧ 𝔛_{l}^{±} ∈ R^{d^{±}(d^{±}−1)/2}. The assertions for c^{+}(∧^{2}M) follow easily from this observation. The formula for c^{−}(∧^{2}M) can be shown similarly. This completes the proof.

Proposition 5. Assume that F^{±}(Sym^{2}M) can be defined and let F^{±}(Sym^{2}M) = F^{q^{±}}(Sym^{2}M). Then c^{±}(Sym^{2}M) = g^{±}(X) with a polynomial g^{±} . The type of g^{+} is given by

(2.9)  \begin{aligned}a_{μ} & = |\{l | 1 ≦ l ≦ d, w(𝔛_{l}^{+}) ≠ i_{μ}, i_{μ} + w(𝔛_{l}^{+}) < q^{−}\}| \\ & + \{ \begin{matrix}t_{μ} − t_{μ+1} + 1 if 2i_{μ} < q^{−}, \\ 0 if 2i_{μ} ≧ q^{−}, \end{matrix} \end{aligned}

(2.10)  k^{+} = d^{+}(M) + 1,  k^{−} = d^{−}(M) + 1.

The type of g^{−} is given by

(2.11)  \begin{aligned}a_{μ} & = |\{l | 1 ≦ l ≦ d, w(𝔛_{l}^{+}) ≠ i_{μ}, i_{μ} + w(𝔛_{l}^{+}) < q^{+}\}| \\ & + \{ \begin{matrix}t_{μ} − t_{μ+1} + 1 if 2i_{μ} < q^{+}, \\ 0 if 2i_{μ} ≧ q^{+}, \end{matrix} \end{aligned}

(2.12)  k^{+} = d^{−}(M),  k^{−} = d^{+}(M).

The proof is similar to that of Proposition 4, so we omit it.

Let N be a motive over Q with coefficients in E. Let us consider periods of M ⊗ N.

Lemma 6. We have

F^{p}(M ⊗ N) = ∑_{k+l=p} F^{k}(M) ⊗ F^{l}(N).

Proof. By (2.3), we get

F^{p}(M ⊗ N) ⊗_{Q} C = ∑_{k+l=p} F^{k}(M) ⊗ F^{l}(N) ⊗_{Q} C

and the assertion follows from this fact.

Let

H_{DR}(N) = F^{j_{1}}(N) ⫌ F^{j_{2}}(N) ⫌ ⋯ ⫌ F^{j_{v}}(N) ⫌ F^{i_{v+1}}(N) = {0},
H_{DR}(M ⊗ N) = F^{k_{1}}(M ⊗ N) ⫌ ⋯ ⫌ F^{k_{z}}(M ⊗ N) ⫌ F^{i_{z+1}}M ⊗ N) = {0}

be the Hodge filtrations. The next lemma follows easily from Lemma 6.

Lemma 7. Choose v_{1}^{(r)}, ⋯, v_{a_{r}}^{(r)} ∈ F^{i_{r}}(M) so that they make a basis of F^{i_{r}}(M)/F^{i_{r+1}}(M) when taken modulo F^{i_{r+1}}(M). Choose w_{1}^{(s)}, ⋯, w_{b_{s}}^{(s)} ∈ F^{j_{s}}(N) so that they make a basis of F^{j_{s}}(N)/F^{j_{s+1}}(N) when taken modulo F^{i_{s+1}}(N). Then

{v_{k}^{(r)} ⊗ w_{l}^{(s)} mod F^{k_{t+1}}(M ⊗ N) | i_{r} + j_{s} = k_{t}, 1 ≦ k ≦ a_{r}, 1 ≦ l ≦ b_{s}}

makes a basis of F^{k_{t}}(M ⊗ N)/F^{k_{t+1}}(M ⊗ N).

Proposition 8. Let X = (X^{+} X^{−}) ∈ M(d(M), d(M), R) and Y = (Y^{+} Y^{−}) ∈ M(d(N), d(N), R) be period matrices of M and N respectively. Write Y^{±} as

Y^{±} = ( 𝔜_{1}^{±} ; 𝔜_{2}^{±} ; ⋮ ; 𝔜_{d(N)}^{±} ) .

Assume that F^{±}(M ⊗N) can be defined and let F^{±}(M ⊗N) = F^{q^{±}}(M ⊗N). Then we have

c^{±}(M ⊗ N) = f^{±}(X)g^{±}(Y)

where f^{±} (resp. g^{±}) is a polynomial on M(d(M), d(M)) (resp. M(d(N), d(N)) with coefficients in Z. The type of f^{±} is given by

(2.13)  a_{μ} = |{l | 1 ≦ l ≦ d(N), i_{μ} + w(𝔜_{l}^{+}) < q^{∓}}|,

(2.14)  k^{+} = d^{±}(N),  k^{−} = d^{±}(N).

Proof. We have

H_{B}^{+}(M ⊗ N) = (H_{B}^{+}(M) ⊗ H_{B}^{+}(N)) ⊕ (H_{B}^{−}(M) ⊗ H_{B}^{−}(N)).

By Lemma 7, we see that c^{+}(M ⊗ N) is given as the determinant of the square matrix of size d^{+}(M)d^{+}(N) + d^{−}(M)d^{−}(N):

(𝔛_{i}^{+} ⊗ 𝔜_{l}^{+}, 𝔛_{i}^{−} ⊗ 𝔜_{l}^{−} | w(𝔛_{i}^{+}) + w(𝔜_{l}^{+}) < q^{−}).

Therefore c^{+}(M ⊗ N) = h^{+}(X, Y) with a polynomial h^{+}(x, y) with coefficients in Z on M(d(M), d(M)) ⊕ M(d(N), d(N)). When y is fixed, we see that the type of h^{+}(x, y) as a polynomial of x is given by (2.13) and (2.14). By the uniqueness assertion of Theorem 1, we see that h^{+}(x, y) = f^{+}(x)g^{+}(y) with polynomials f^{+}(x), g^{+}(y) with coefficients in Z. Thus we established our assertion for c^{+}(M ⊗N). The assertions for c^{−}(M ⊗ N) can be shown similarly. This completes the proof.

Corollary 1. If d^{+}(M) = d^{−}(M), d^{+}(N) = d^{−}(N) and q^{+} = q^{−}, then we have c^{+}(M ⊗ N) = c^{−}(M ⊗ N).

Corollary 2. Let M be a motive over Q with coefficients in E. Then the period invariant of the type {(a_{1}, a_{2}, ⋯, a_{m}), (k^{+}, k^{−})} is non-zero.

Proof. Let X be a period matrix of M. We use the notation of Theorem 3. It suffices to show that f_{a}(X) ≠ 0 for any a ∈ A. Let N be a motive over Q with coefficients in E of pure weight w′ ≧ 1 such that

H_{B}(N) ⊗ C = H^{0,w′}(N) ⊕ H^{w′,0}(N).

Examples of such an N can be obtained as the motive attached to an elliptic modular form (cf. §4). We choose the numbering of the Hodge filtration of H_{DR}(M) so that i_{μ} is the maximum number. Then we see that

(2.15)  i_{c} + i_{m+1−c} = w, 1 ≦ c ≦ m,  s_{μ} = s_{m+1−μ}, 1 ≦ μ ≦ m.

Let a ∈ A and take w′ = w − 1 − 2i_{p}. The conditions that the type of f^{+}(x) coincides with that of f_{a}(x) are

(2.16)  q^{−} − i_{p+1} ≦ w′ < q^{−} − i_{p},  i_{q} < q^{−} ≦ i_{q+1}.

By (2.15), we obtain p + q = m, w′ ≧ 1. Since q^{−} = (w + w′ + 1)/2 (cf. [Y], §2.2) and i_{p} + i_{q+1} = w, we see that (2.16) holds. This completes the proof.

§3. Motives of rank 4

Let M be a motive over Q with coefficients in E. We assume that M is of rank 4 and of pure weight w ≧ 3. We assume further that

(3.1)  H_{B}(M) ⊗ C = H^{0,w}(M) ⊕ H^{w,0}(M) ⊕ H^{c,w−c}(M) ⊕ H^{w−c,c}(M)

is the Hodge decomposition with

(3.2)  0 < 2c < w.

Then the Hodge filtration is

H_{DR}(M) = F^{0} ⫌ F^{c} ⫌ F^{w−c} ⫌ F^{w} ⫌ {0}

and d^{+}(M) = d^{−}(M) = 2. Since F_{∞} acts on H^{w,w}(∧^{2}M) as −1, we have F^{+}(∧^{2}M) = F^{w+1}(∧^{2}M), F^{−}(∧^{2}M) = F^{w}(∧^{2}M). By Proposition 4, we have

c^{±}(∧^{2}M) = f_{∧^{2}}^{±}(X)

with a polynomial f_{∧^{2}}^{±}(x) on M(4, 4). The type of f_{∧^{2}}^{+}(x) is {(2, 1, 1, 0); (1, 1)} and the type of f_{∧^{2}}^{−}(x) is {(3, 2, 2, 1); (2, 2)}. Therefore we obtain

(3.3)  c^{−}(∧^{2}M) = c^{+}(∧^{2}M)δ(M).

Since F_{∞} acts on H^{w,w}(Sym^{2}M) as 1, we have F^{+}(Sym^{2}M) = F^{w}(Sym^{2}M), F^{−}(Sym^{2}M) = F^{w+1}(Sym^{2}M). By Proposition 5, we have

c^{±}(Sym^{2}(M)) = f_{Sym^{2}}^{±}(X),

where the type of f_{Sym^{2}}^{+}(x) is {(5, 4, 2, 1); (3, 3)} and the type of f_{Sym^{2}}^{−}(x) is {(4, 3, 1, 0); (2, 2)}. Hence

(3.4)  c^{+}(Sym^{2}(M)) = c^{+}(M)c^{−}(M)c^{+}(∧^{2}M)δ(M),

(3.5)  c^{−}(Sym^{2}(M)) = c^{+}(M)c^{−}(M)c^{+}(∧^{2}M).

Next let N be a motive over Q with coefficients in E. We assume that N is of rank 2 and of pure weight w′ ≧ 1. We also assume

(3.6)  H_{B}(N) ⊗ C = H^{0,w′}(N) ⊕ H^{w′,0}(N)

is the Hodge decomposition. The motive M ⊗ N is of pure weight w + w′. We assume that F^{±}(M ⊗ N) can be defined and are going to calculate c^{±}(M ⊗ N). If w = w′ or 2c = w − w′, then F^{±}(M ⊗ N) cannot be defined. We have

(3.7)  w ≠ w′  and  2c ≠ w − w′.

Let Y be a period matrix of N. We have

c^{±}(M ⊗ N) = f^{±}(X)g^{±}(Y)

with a polynomial f^{±} (resp. g^{±}) on M(4, 4) (resp. M(2, 2)). By (3.7), we see that F^{+}(M ⊗ N) = F^{−}(M ⊗ N). By Corollary 1 to Proposition 8, we have

(3.8)  c^{+}(M ⊗ N) = c^{−}(M ⊗ N).

Put F^{+}(M ⊗ N) = F^{q^{+}}(M ⊗ N). We must distinguish three cases.

Case A. The case where w′ > w.

We can take q^{+} = w′. By Proposition 8, the type of f^{+}(x) is {(1, 1, 1, 1); (1, 1)} and the type of g^{+}(x) is {(4, 0); (2, 2)}. Hence

(3.9)  c^{±}(M ⊗ N) = δ(M)c^{+}(N)^{2}c^{−}(N)^{2}.

Case B. The case where w′ + 2c > w > w′.

We can take q^{+} = min(w, w′ + c). By Proposition 8, the type of f^{+}(x) is {(2, 1, 1, 0); (1, 1)} and the type of g^{+}(x) is {(3, 1); (2, 2)}. Hence

(3.10)  c^{±}(M ⊗ N) = c^{+}(∧^{2}M)c^{+}(N)c^{−}(N)δ(N).

Case C. The case where w > w′ + 2c.

We can take q^{+} = w−c. By Proposition 8, the type of f^{+}(x) is {(2, 2, 0, 0); (1, 1)} and the type of g^{+}(x) is {(2, 2); (2, 2)}. Hence

(3.11)  c^{±}(M ⊗ N) = c^{+}(M)c^{−}(M)δ(N)^{2}.

§4. Siegel modular forms of genus 2

Let f ∈ S_{k}(Sp(2, Z)) be a non-zero Hecke eigenform. Let E be an algebraic number field which contains eigenvalues of f of all Hecke operators. Let L(s, f) denote the spinor L-function attached to f. The functional equation proved by Andrianov [A] is

(4.1) R(s, f) = (−1)^{k}R(2k−2−s, f),  R(s, f) = (2π)^{−2s}Γ(s)Γ(s−k+2)L(s, f).

Let M(f) be the (hypothetical) motive over Q with coefficients in E such that L(M(f), s)= (L(s, f^{σ}))_{σ∈J_{E}}. Put M = M(f). The rank of M must be 4 and from the functional equation (4.1), we infer that M is of pure weight 2k − 3 and that

(4.2) H_{B}(M)⊗_{Q} C = H^{0,2k−3}(M)⊕H^{2k−3,0}(M)⊕H^{k−2,k−1}(M)⊕H^{k−1,k−2}(M).

Thus M is a special case w = 2k − 3, c = k − 2 of the motive considered in §3.

Let g ∈ S_{l}(SL(2, Z)) be a normalized Hecke eigenform. We take E as an algebraic number field containing eigenvalues of f and g under all Hecke operators. Let N = N(g) be the motive over Q with coefficients in E such that L(N(g), s) = (L(s, g^{σ}))_{σ∈J_{E}} (see Scholl [Sc]). Then N is of pure weight l − 1 and

(4.3)  H_{B}(N) ⊗_{Q} C = H^{0,l−1}(N) ⊕ H^{l−1,0}(N).

Since l ≧ 2, Case C does not occur and we obtain

(4.4)  c^{±}(M ⊗ N) = δ(M)c^{+}(N)^{2}c^{−}(N)^{2}  if l > 2k − 2,

(4.5)  c^{±}(M ⊗ N) = c^{+}(∧^{2}M)c^{+}(N)c^{−}(N)δ(N)  if l < 2k − 2

from (3.9) and (3.10). We note that M ⊗ N has a critical value if and only if l ≠ 2k − 2. It is well known that (cf. [Sh1], [Y], (4.11))

(4.6)  c^{+}(N)c^{−}(N) = i^{1−l}π(⟨g^{σ}, g^{σ}⟩)_{σ∈J_{E}}.

Since ∧^{2}N ≅ T (1 − l), ∧^{4}M ≅ T (6 − 4k), T (m) being a Tate motive, we have

(4.7)  δ(N) = 1 ⊗ (2π\sqrt{−1})^{1−l},  δ(M) = 1 ⊗ (2π\sqrt{−1})^{6−4k}.

To write down c^{±}(M ⊗ N) in terms of data of modular forms, a crucial step is to identify c^{+}(∧^{2}M) with the normalized Petterson norm of f. To this end, we proceed as follows.

Let L_{st}(s, f) be the standard L-function attached to f. Comparing Euler factors, we see easily that

(4.8)  L(∧^{2}M, s) = ζ(s − 2k + 3)(L_{st}(s − 2k + 3, f^{σ}))_{σ∈J_{E}}.

(The functional equation of L_{st}(s, f) is given with respect to s ⟶ 1 − s in Andrianov-Kalinin [AK] and we can check that the gamma factor given there matches with that predicted by the theory of motives. See also Shimura [Sh3].) Using (3.3), (4.7) and [D], 5.1.8, we obtain

(4.9) L(∧^{2}M, n)/(1 ⊗ 2π\sqrt{−1})^{4n−4k+6}c^{+}(∧^{2}M) ∈ E if n is odd and 2k − 1 ≦ n ≦ 3k − 5

by Deligne’s conjecture. Several authors (Harris [Ha], Sturm [St], Böcherer [Bo], Mizumoto [M]) obtained results on the critical values of the standard L-functions. We find [Bo], Satz 14 is most convenient for our present purpose. Assuming that the Fourier coefficients of f are contained in a totally real algebraic number field, it states that^{2}

(4.10)  (\frac{L_{st}(n, f)}{π^{3n+2k}⟨f, f⟩})^{σ} = \frac{L_{st}(n, f^{σ})}{π^{3n+2k}⟨f^{σ}, f^{σ}⟩}

for σ ∈ Aut(C) if n ∈ Z satisfies n ≡ 0 mod 2, 2 < n ≦ k − 2 . Comparing (4.8), (4.9) with (4.10), we obtain the identification:

(4.11)  c^{+}(∧^{2}M) = π^{6−2k}(⟨f^{σ}, f^{σ}⟩)_{σ∈J_{E}}.

Summing up, we obtain

(4.12) c^{±}(M(f) ⊗ M(g)) = (2π\sqrt{−1})^{8−4k}(⟨g^{σ}, g^{σ}⟩)_{σ∈J_{E}}^{2}  if l > 2k − 2,

(4.13)

c^{±}(M(f) ⊗ M(g)) = (2π\sqrt{−1})^{8−2k−l}(⟨f^{σ}, f^{σ}⟩⟨g^{σ}, g^{σ}⟩)_{σ∈J_{E}} if l < 2k − 2.

Theorems A and B given in the introduction now follow immediately from (4.12), (4.13) and from Deligne’s conjecture. For the convenience in the future, we note the formulas^{3}

(4.14) c^{+}(Sym^{2}(M(f))) = (2π\sqrt{−1})^{12−6k}c^{+}(M(f))c^{−}(M(f))(⟨f^{σ}, f^{σ}⟩)_{σ∈J_{E}},

(4.15) c^{−}(Sym^{2}(M(f))) = (2π\sqrt{−1})^{6−2k}c^{+}(M(f))c^{−}(M(f))(⟨f^{σ}, f^{σ}⟩)_{σ∈J_{E}}

which follow from (3.4), (3.5), (4.7) and (4.11).

Remark. Let V be a 4-dimensional vector space over Q endowed with a nondegenerate alternating form. We let GSp(2) act on V in standard way. The representation of GSp(2) on ∧^{2}V splits into a direct sum of a one dimensional representation and a 5-dimensional irreducible representation ρ such that ρ|Sp(2) is an orthogonal representation (cf. Bourbaki [Bou], Chap. VIII, §13, Exercices, 3)). Since the image of the λ-adic representation is contained in GSp(2), Tate’s conjecture implies that ∧^{2}M splits into a direct sum of motives according as this decomposition:

∧^{2}M = T (3 − 2k) ⊕ ρ(M),

where ρ(M) is a motive over Q of rank 5 with coefficients in E.

References

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2The critical value n = 2 is not covered in [Bo].

3J. Tilouine asked me to calculate periods of Sym^{2}(M(f)). Here I fulfilled my promise.

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Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan email address: yoshida@kusm.kyoto-u.ac.jp