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a splitting in the metaplectic cover Mp(W)A of Sp(W)(A) and has then a Weil .... if v is real and split. U(2) ..... S2 = {v : v is split and χ˜η is ramified at v}. Then Φ¯η ...
THETA LIFTINGS AND HECKE L-FUNCTIONS (J. REINE ANGEW. MATH. 485 (1997), 25–53.)

Tonghai Yang

0 Introduction. In this paper we will prove an explicit formula for the special value at the central point of the L-series of certain Hecke characters of a number field. This formula involves certain theta liftings. Consider a quadratic extension E/F of number fields. If (V, ( , )) is a Hermitian space over E, and (W, < , >) is a skewHermitian space over E, the unitary groups G = G(W ) and G0 = G(V ) form a reductive dual pair in Sp(W), where W = V ⊗ W has the symplectic form 1 2 trE/F ( , )⊗ < , > over F . Associated to such a dual pair are local and global theta liftings. Roughly speaking, for each place v of F , the local theta lifting relates irreducible admissible representations of Gv and G0v . Similarly the global theta lifting relates automorphic forms on the two groups. We consider the very special case in which dimE V = dimE W = 1. In this case, G ∼ = G0 = U (1) = E 1 , 1 ∗ ∗ where E is the kernel of the norm map N : E −→ F , and the representations involved are just characters. In order to define the theta liftings, we must choose (and fix) an idele class character χ of E such that its restriction to the ideles of F is the quadratic character ² = ²E/F of FA∗ associated to E/F ([Ka]). First it is important to observe that the local theta lifting depends on the underlying pair (V, W ) of (skew-)Hermitian spaces, not just on the pair of groups (G, G0 ). Over a local field, there are precisely two isomorphism classes of 1dimensional Hermitian spaces V + and V − . Fix a 1-dimensional space W = E, < x, y >= δx¯ y , where δ is a fixed pure ‘imaginary’ element of E (i.e., ∆ = δ 2 ∈ F ∗ −F ∗2 ). Then local theta and epsilon dichotomies ([HKS], [Har], [Har2], [Moe], [Rog])say that a given character η of E 1 = G occurs in the theta correspondence (lifting) with precisely one of the two groups G(V ± ), and that the sign for which it occurs is determined by the local root number. More precisely, the local theta lifting θ(η, V ² ) 6= 0 if and only if 1 1 ²( , χ˜ η , ψE )χ˜ η (δ) = ². 2 2 Here η˜(z) = η( zz¯ ) so that η˜ is a character of E ∗ = GL1 (E). For a global character η : G(A)/G(F ) ∼ = EA1 /E 1 −→ C1 , we have η = ⊗ηv , and, for a global Hermitian space, the global theta lifting of η to G(V ) is compatible with local theta liftings. Thus this lift automatically vanishes unless all 1 1991 Mathematics Subject Classification. 11F27 11G05 11M20 14H52. Key words and phrases. dichotomy, elliptic curve, L-function, special value, theta lifting. Typeset by A S-T X

the local dichotomy conditions are satisfied. This can occur for at most one global space V . Such a space V exists if and only if the global root number Y 1 1 1 ²( , χ˜ η) = ²v ( , (χ˜ η )v , ψEv ) = 1. 2 2 2 For this V (when it exists), the global theta lifting is non-zero if and only if L( 12 , χ˜ η ) 6= 0. This result was also proved by Rogawski [Rog]. Our formula for the value L( 12 , χ˜ η ) arises by interpreting this value as the square norm of a theta lifting. More precisely, for any Schwartz function φ ∈ S(FA ), there is a theta function θφ (η) associated to φ and η; it may be viewed as an automorphic form of G(V )(A). For a particular good choice of φ = ⊗φv , we have (0.1)

1 L( , χ˜ η ) =< θφ (η), θφ (η) > . 2

We will describe all this in more detail in a moment. Our formula gives a new proof of Rogawski’s result (Theorem 0.4). The formula also gives Theorem 0.1. Let F be a number field and let E be a quadratic extension of F . If χ is an idele class character of E whose restriction to the ideles of F (i.e., FA∗ ) is the quadratic character of FA∗ associated to E/F , then 1 L( , χ) ≥ 0. 2

(0.2)

Moreover, L( 12 , χ) > 0 if and only if some theta lifting θ(1, α) does not vanish. In subsequent papers ([RVY], [Ya2], [Ya3]), we will compute the the theta integral in (0.1), obtain interesting recursive formula for the central Hecke Lvalue in terms of Hilbert modular forms of half integral weight, and prove some nonvanishing results of the central Hecke L-values (and the theta liftings). It should be mentioned that, at about the same time, Jiandong Guo ([Guo]) proved, by means of the relative trace formula, that for any cuspidal representation π on GL(2, F ) with trivial central character, L( 12 , π) ≥ 0. His result is a generalization of (0.2). His result did not tell when the central L-value vanish. We now give a more detailed description of this paper. Let W = E with the skew-Hermitian form < x, y >= δx¯ y , and let W + W− be its doubling. Let 1 G = G(W ) = U (1) = E and H = G(W + W− ) = U (1, 1) be the corresponding isometry groups. Then one has a canonical embedding (0.3)

i : G × G −→ H,

(x1 , x2 )i(g1 , g2 ) = (x1 g1 , x2 g2 ). H(A)

Let P be the standard Siegel parabolic subgroup of H. Let I(s, χ) = IndP (A) χ| |s = ⊗0 I(s, χv ) be the degenerate (induced from a character of a maximal parabolic 2

subgroup) principal series representation. Given a function Φ = one defines the Eisenstein series (0.4)

E(h, s, Φ) =

X

Q

Φv ∈ I(s, χ),

Φ(γh, s)

γ∈P (F )\H(F )

where h ∈ H(A). Applying the Piatetski-Shapiro and Rallis basic identity Q ([GPSR]) to a particular nice function Φη¯ = Φv,¯η ∈ I(s, χ), and computing local L-factors, we obtain the following: Theorem 0.2. (Sketch of Theorem 1.11) (0.5)

L(s + 21 , χ˜ η) = c(s) L(2s + 1, ²)

Z E(i(g1 , g2 ), s, Φη¯)η(g1 )(χη)−1 (g2 ) dg1 dg2 [G×G]

Q where c(s) = cv (s) is an explicit analytic function of s. Here, for an algebraic group G over F , we write [G] = G(F )\G(A). Next, given α ∈ F ∗ /N E ∗ , let Vα = E be the corresponding one-dimensional Hermitian space with the Hermitian form (x, y) = α¯ xy and let Gα be its isometry group, which acts on the left. Let W = Vα ⊗E W ∼ = W with the symplectic form >= 12 trE/F ( , )⊗ < , > over F . One has the reductive dual pair (G, Gα ) in Sp(W). Fix a complete polarization W = F δ ⊕F . G(A)×Gα (A) has a splitting in the metaplectic cover Mp(W)A of Sp(W)(A) and has then a Weil representation ωα,χ on S(F δA ) = S(FA ), which depends on α and χ (the splitting depends on the choice of χ [Ka]). For a fixed character η of G(A), one associates an automorphic form θφ (η) of Gα to a function φ ∈ S(FA ), which is defined via some integral related to the Weil representation. All these automorphic forms of Gα gives rise to an automorphic representation θ(η, α) of Gα —the theta lifting of η on Gα . One of the fundamental questions is whether or not the theta lifting θ(η, α) vanishes. The main purpose of this paper is to prove the following formula. Theorem 0.3. (Main Formula) Assume the root number ²( 12 , χ˜ η ) = 1 and let ∗ ∗ α ∈ F /N E be the unique element such that (0.6)

1 1 ²( , (χ˜ η )v , ψEv )(χ˜ η )v (δ) = ²v (α) 2 2

holds for every place v of F . Then there is a function φ = satisfying (0.7)

< ωα,χ (g, 1)φ, φ >= Φη¯(i(g, 1), 0) 3

Q

φv ∈ S(FA )

for every g ∈ G(A). For any such function φ, one has L( 12 , χ˜ η) = c < θφ (η), θφ (η) > L(1, ²E/F )

(0.8)

for an explicit constant c > 0. Combining Theorem 0.3 with the local epsilon dichotomy for U (1) × U (1) ([HKS]), we obtain the following result of Rogawski. Theorem 0.4. (Rogawski [Rog]) Let E = F (δ) and χ be fixed. Let η be a character of G(A) = EA1 and let α ∈ F ∗ . Then the global theta lifting θ(η, α) does not vanish if and only if the following two conditions are satisfied: (i) Condition (0.6) is satisfied for every place v of F , or equivalently, the corresponding local theta liftings do not vanish. (ii)

L( 12 , χ˜ η ) 6= 0.

Other applications of Theorem 0.3 are indicated at the end of this paper. The Hecke characters considered in this paper has arithmetic nature, and is closely related to CM elliptic curve or more generally to motivic Hecke characters of odd weights. Acknowledgement This paper is part of my thesis at the University of Maryland at College Park. I am indebted to my advisors Stephen Kudla, David Rohrlich, and Lawrence Washington for their advice, inspiration, insight, and help. In particular, I would like to thank my thesis advisor Stephen Kudla for introducing me to the idea to study L-functions by means of representation theory and helping me to do so patiently. I would also like to thank David Rohrlich specially for introducing me to the fascinating subject of L-functions. Finally, I would like to thank the Department of Mathematics at UMCP for providing me the nurturing learning environment. Notation. Let E/F be a quadratic extension of a number fields with associated quadratic character ² = ²E/F . We denote the nontrivial Galois automorphism of E/F by x 7→ x ¯. Let tr and N be the trace and norm maps from E to F . We choose and fix an element δ ∈ E ∗ with δ¯ = −δ. Then ∆ = δ 2 ∈ F ∗ . We also fix an idele class character χ of E such that its restriction to the ideles of F is ². Let E 1 be the kernel of the norm map N . Given a character η of EA1 /E 1 , let η˜ be the character of EA∗ via η˜(z) = η( zz¯ ). Q We will fix an additive character ψ = ψv of FA /F and let ψE = ψ ◦ trE/F . Locally on Fv , we require the Haar measure dx to be self-dual with respect to ψv . On an algebraic group G over F , we only require the Haar measure on G(A) be the product measure of local Haar measures. We also write [G] = G(F )\G(A). 4

Q Given an idele class character λ of E, we write L(s, λ) = v-∞ Lv (s, λ) where ( L(s, λw ) if w|v is nonsplit in E/F (0.9) Lv (s, λ) = L(s, λw )L(s, λw¯ ) if w|v is split in E/F. Here L(s, λw ) is of course the usual local L-factor. Similar convention is used for Tate’s local root number ²( 12 , λw , ψEw ). We write ²( 12 , λ) for the global root number. It is independent of the choice of additive character of E. Let W = E with the skew-Hermitian form < x, y >= δx¯ y , let W− = E with the skew-Hermitian form < x, y >− = −δx¯ y , and let G = G(W ) = G(W− ) = E 1 be the isometry group. As usual, one has the doubling W + W− with the skewHermitian form < (x1 , y1 ), (x2 , y2 ) >= δx1 x¯2 − δy1 y¯2 . Let H = G(W + W− ) = U (1, 1) be its isometry group. Let W d = {(w, w) : w ∈ W } and Wd = {(w, −w) : w ∈ W }. Then W + W− has the standard complete polarization W + W− = Wd ⊕ W d , 1 and the standard E-basis e = 2δ (1, −1) and f = (1, 1). With respect to the standard basis, the map (0.3) is given by ¶ µ 1 1 (g − g ) g1 + g2 1 2 2δ (0.10) i(g1 , g2 ) = . g1 + g2 2 2δ(g1 − g2 )

Let P be the stabilizer of W d in H (the standard Siegel parabolic subgroup of H). Then P has the Levi decomposition P = N M where, with respect to the standard complete polarization, µ ¶ 1 b (0.11) N = {n(b) = : b ∈ F }, 0 1 µ ¶ a 0 M = {m(a) = : a ∈ E ∗ }. 0 a ¯−1 One has (0.12)

H = P × i(G × 1).

There is also the Iwasawa decomposition (0.13)

H(A) = P (A) · KA ,

where KA = ΠKv is the maximal compact subgroup of H(A) defined as follows  H(Ov ) if v is finite     i(Gv × Gv ) if v is real and inert (0.14) Kv =  O(2) if v is real and split    U (2) if v is complex 5

s v When v is finite, let I(s, χv ) = IndH Pv χv | | be the (normalized) induced representation, i.e. the locally constant functions Φv on Hv satisfying

(0.15)

s+ 12

Φ(n(b)m(a)h, s) = χv (a)|a¯ a|v

Φv (h, s).

s v When v is infinite, let I(s, χv ) be the subspace of IndH Pv χv | | of the Kv -finite functions. For a vector space V over a local field Fv , let S(V ) be the space of Schwartz functions on V . When v is infinite, we require the Schwartz functions to be Kv -finite.

Next, given α ∈ F ∗ /N E ∗ , let Vα = E be the corresponding one-dimensional Hermitian space with the Hermitian form (x, y) = α¯ xy, and let Gα = G(Vα ) be its isometry group, which acts on the left. Let W = Vα ⊗E W ∼ = W and ∼ W + W− = Vα ⊗E (W + W− ) = W + W− with the symplectic form over F >=

1 α trE/F ( , ) ⊗ < , > = tr < , > . 2 2

Let Sp(W) and Sp(W + W− ) be the corresponding symplectic groups. One has the reductive dual pair (G(W ), G(Vα )) in Sp(W) via (0.16)

ıα :

G(W ) × G(Vα ) −→ Sp(W) (v ⊗ w)ıα (g1 , g2 ) = g2−1 v ⊗ wg1 ,

and the reductive dual pair (G(W + W− ), G(Vα )) in Sp(W + W− ) via (0.17)

ıα :

G(W + W− ) × G(Vα ) −→ Sp(W + W− ) (v ⊗ w)ıα (h, g) = g −1 v ⊗ wh.

They form a seesaw dual pair in Sp(W + W− )

(0.18)

G(Vα ) × G(Vα ) G(W + W− ) RRR O O RRR lllll lRl i diag lll RRRRRR lll G(W ) × G(W ) G(Vα )

which plays an important role in the doubling method ([Ka4], [KS]). Finally, we say that everything is unramified at v if (a) The field extension E/F is unramified at v, and ψ is unramified at v; (b) The character χ and η˜ is unramified at v; (c) 2∆α ∈ Ov∗ . 6

1. Piatetski-Shapiro and Rallis Basic Identity. Q Q Given a function Φ = Φv ∈ I(s, χ), and a character η = ηv of [G], define the local L-factor as follows: Z (1.1) L(s, ηv , Φv ) = Φv (i(g, 1), s)ηv (g) dg. Gv

Then the Piatetski-Shapiro and Rallis basic identity ([GPSR, page3]) gives (1.2) Z Y E(i(g1 , g2 ), s, Φ)η(g1 )(χη)−1 (g2 ) dg1 dg2 = meas([E 1 ]) L(s, ηv , Φv ). [G×G]

The idea is to construct a good function Φv and to relate the local L-factor L(s, ηv , Φv ) with Lv (s, χ˜ η ). 1.1. Structure of I(s, χv ): the nonsplit case. Let v be a place of F nonsplit in E/F . Let w be the unique place of E above v. Then Ev = E ⊗ Fv = Ew is a quadratic field extension of Fv . We will drop the subscript v in this section. Recall ([HKS]) that, given any character η of G = E 1 , there is a unique section Φη (s) ∈ I(s, χ) defined by 1

Φη (nm(a)i(g, 1), s) = χ(a)|a¯ a|s+ 2 η(g).

(1.3)

By (0.12), it is easy to see that (1.4)

I(s, χ) = ⊕CΦη . η

By (1.1), one has Lemma 1.1. L(s, η, Φη¯(s)) = meas(E 1 ). From now on, assume that F is a finite extension of Qp for some prime number p ( i.e. v is finite). Direct computation gives Lemma 1.2. (i)

If E/F is ramified or χ˜ η is unramified, then

(1.5)

L(s + 12 , χ˜ η) L(s, η, Φη¯) = meas(E 1 ). L(2s + 1, ²E/F )

(ii) (1.6)

If E/F is unramified and χ˜ η is ramified, then η) L(s + 12 , χ˜ L(s, η, Φη¯(s)) = (1 + qF−2s−1 )−1 meas(E 1 ), L(2s + 1, ²E/F )

where qF = #OF /πF OF is the order of the residue field of F . We remark that when χ˜ η is unramified E/F has to be unramified too. Also notice that the L-factors in the numerator of (1.5) and (1.6) are relative to E while the L-factors in the denominator are relative to F . 7

Lemma 1.3. Assume that everything is unramified. Then Φη = Φ0 is the unique spherical element in I(s, χ) such that Φ0 |K = 1. 1.2. Structure of I(s, χv ): the split case. Throughout this subsection, we assume that v is finite and split. So there are two different primes w and w ¯ of E above v. Everything degenerates and we first record some basic facts. First Ev = Ew ⊕ Ew¯ = Fv ⊕ Fv , δ = (x0 , −x0 ) ∈ Ev for some x0 ∈ Fv∗ . The nontrivial automorphism of Ev /Fv is (x , y) = (y , x). As characters of Fv∗ , one has χw¯ = χ−1 w . Secondly, the skew-Hermitian form on Wv = Fv ⊕ Fv is given by < (x1 , x2 ) , (y1 , y2 ) >= δ(x1 , x2 )(y1 , y2 ) = (x0 x1 y2 , −x0 x2 y1 ). The space Wv + Wv,− has a standard basis e = ((1, 1) , (1, 1)).

1 2x0 ((1, 1) ,

(−1, −1)) and f =

Lemma 1.4. With respect to the standard basis {e, f }, one has Gv = {(x, x−1 ) :

x ∈ Fv∗ }

Hv = {(A1 , A2 ) ∈ GL(2, Ev ) : A2 = (detA1 )−1 A1 } ¶ µ (a, d) ∗ ∈ Hv } Pv = { 0 (d−1 , a−1 ) µ ¶ ½µ ¶¾ def ai bi (a1 , a2 ) (b1 , b2 ) where Ai = ∈ GL(2, Fv ) and (A1 , A2 ) = ci di (c1 , c2 ) (d1 , d2 ) Lemma 1.5. One has the following commutative diagram. β

Fv∗ × Fv∗ ↓i

− g →

GL(2, Fv ) ∪

− g →

Bv

− g →

Gv × Gv ↓i

β

Hv ∪

β

Pv

where, with respect to the standard basis, µ 1 x+1 i(x, 1) = 2x 2 0 (x − 1) β(x) = (x, x−1 )

1 2x0 (x

− 1) x+1

for x ∈ Fv∗ ,

β(A) = (A, (detA)−1 A), µ µ ¶ ¶ a b (a, d−1 ) (b, bd−1 a−1 ) β( )= . 0 d 0 (d, a−1 ) 8

¶ ,

We will identify Gv with Fv∗ , Hv with GL(2, Fv ) and Pv with Bv . Under this identification, the quasicharacter χv | |s : Bv −→ C∗ is given by µ a χv,s ( 0

(1.7)

¶ a b ) = χw (ad) · | |s . d d

One has also η˜w = η as characters of F ∗ . Therefore, the L-factors concerned in this case are η) η )w )L(s + 21 , (χ˜ η )−1 Lv (s + 12 , χ˜ L(s + 12 , (χ˜ w ) = . Lv (2s + 1, ²) Lv (2s + 1, trivial) From now on, we drop the subscripts. Let F be a non-archimedean local field, and E = F ⊕ F . Let χ and η are characters of F ∗ . The following lemma is well-known. Lemma 1.6. One has the orbit decomposition µ 1 ∗ (1.8) GL(2, F ) = B × i(F × 1) ∪ B 2x0

0 1



µ ∪B

1 −2x0

0 1

¶ .

Proposition 1.7. (Structure of I(s, χ)) Given a vector f = (f1 , f2 ) ∈ S(F )2 , s there is a unique function Φf ∈ I(s, χ) = IndH P χ| | such that (1.9)

1 1 1 Φf (i(x, 1)) = χ(x)|x|s+ 2 f1 (x) + χ(x)|x|−s− 2 f2 ( ). x

Conversely every function Φ ∈ I(s, χ) has this form. Proof. We first prove the second statement. Let Φ ∈ I(s, χ). Note that µ (1.10)

i(x, 1) =

¶ 1 4x0 (x − 1) 1 2 (x + 1)

2x x+1

0

µ

1

0 1

2x0 (x−1) x+1



When ord x → −∞, µ

1

2x0 (x−1) x+1

0 1



µ →

1 2x0

So when −ord x >> 0, one should have µ µ ¶ 1 0 1 ) = Φ( Φ( 2x0 (x−1) 1 2x0 x+1 9

0 1

¶ .

¶ 0 )=a 1

if x 6= −1.

to be fixed, and so, by (1.7) ¯ ¯ 1 ¯ 4x ¯s+ 2 ¯ Φ(i(x, 1), s) = χ(x) ¯¯ a (x + 1)2 ¯ 1

= χ(x)|x|−s− 2 |2|2s+1 a Similarly,

µ

1

2x0 (x−1) x+1

when x → 0 and one should have ¯ ¯ Φ(i(x, 1), s) = χ(x) ¯¯

0 1



µ →

when − ord x >> 0

1 −2x0

0 1



¯s+ 1 µ 4x ¯¯ 2 1 Φ( ¯ 2 −2x (x + 1) 0

0 1



1

= χ(x)|x|s+ 2 |2|2s+1 b when ord x >> 0 ¶ µ 1 0 where b = Φ( ) is a fixed complex number. We should mention −2x0 1 that the constants a and b might depend on s. Now it is easy to see that Φ(i(x, 1)) has the form (1.9) for some f1 , f2 ∈ S(F ). It also follows from the above argument that Φf is uniquely determined by Φ|i(G×1) . The uniqueness of the first statement is then proved. Conversely, given f1 , f2 ∈ S(F ), define a function Φ = Φf on H via 1 1 1 Φ(i(x, 1), s) = χ(x)|x|s+ 2 f1 (x) + χ(x)|x|−s− 2 f2 ( ), x µ ¶ 1 0 Φ( ) = f2 (0)|2|−2s−1 , 2x0 1 µ ¶ 1 0 Φ( ) = f1 (0)|2|−2s−1 , −2x0 1 µ ¶ 1 u u v Φ( h) = χ(uw)| |s+ 2 φ(h) h ∈ H. 0 w w

we µ claim that ¶ Φ ∈ I(s, χ). It suffices to check that Φ is locally constant at 1 0 . Recall that ±2x0 1 µ ¶ u v Hn = { ∈ GL(2, F ) : u ≡ z ≡ 1 modπ n , v ≡ w ≡ 0 modπ n } w z form a base of H = GL(2, F ) at 1. For any µ ¶ u v A= ∈ Hn , 2x0 w z 10

one has

µ A

1 2x0

0 1



µ

detA

=

v z

z

0

¶µ

1

0 1

2x0 (w+z) z

¶ .

So, by (1.7), µ Φ(A

¶ µ ¶ detA s+ 1 0 1 0 2 ) = χ(detA)| 2 | Φ( 2x0 (w+z) ) 1 1 z z ¶ µ 1 0 = Φ( 2x0 (w+z) ) since det(A) ≡ z ≡ 1 mod π n . 1 z

1 2x0

On the other hand, let x =

w+2z −w ,

µ i((x, 1)) =

then ord x = −ord w and

2x x+1

¶µ

∗ x+1 2

0

1

2x0 (w+z) z

0 1

¶ .

¶ µ 4x 1 0 s+ 12 Φ(i(x, 1), s) = χ(x)| | Φ( 2x0 (w+z) ) 1 (x + 1)2 z µ ¶ 1 0 −s− 12 2s+1 = χ(x)|x| |2| Φ( 2x0 (w+z) ) 1 z when n >> 0. So one has µ 1 Φ( 2x0 (w+z) z

and

µ

¶ 0 ) = f2 (0)|2|−2s−1 1

¶ 0 Φ(A ) for A ∈ Hn and n >> 0. 1 µ ¶ 1 0 This proves that Φ is locally constant at . The same is true at the 2x0 1 µ ¶ 1 0 point . −2x0 1 1 2x0

¶ µ 0 1 ) = Φ( 1 2x0

Corollary 1.8. With notation as above, one has (1.11)

1

1

L(s, η, Φf ) = ζ(f1 , χη| |s+ 2 ) + ζ(f2 , (χη)−1 | |s+ 2 )

R where ζ(f, c) = F ∗ f (x)c(x)d∗ x is the Tate’s zeta function [Tat] for a locally constant function f on F ∗ and a quasicharacter c. 11

Lemma 1.9. Suppose that χη is unramified. Let f1 = char(OF ) be the characteristic function of OF and f2 = char(πOF ) be the characteristic function of πOF , where π is as usual a uniformizer of F . Let f = (f1 , f2 ). Then (1.12)

L(s + 21 , χη)L(s + 12 , (χη)−1 ) L(s, η, Φf ) = meas(OF∗ ). L(2s + 1, trivial)

Moreover, when everythig is unramified, Φf is the normalized spherical section in I(s, χ). Proof. Following Tate [Tat], one has Z ζ(f1 , χη| |

s+ 12

1

χη(x)|x|s+ 2 d∗ x

)= = =

O−{0} ∞ XZ n=0 ∞ X

1

χη(x)|x|s+ 2 d∗ x

πn O∗

Z n −n(s+ 12 ) O∗

n=0

=

1 1−

d∗ x

χη(π) q

meas(OF∗ )

1 χη(π)q −(s+ 2 )

where q = |π|−1 . For the same reason one has 1

−1

ζ(f2 , (χη)

||

s+ 12

)=

(χη(π))−1 q −(s+ 2 ) 1−

1 (χη(π))−1 q −(s+ 2 )

meas(OF∗ ).

Therefore " L(s, η, Φf ) =

meas(OF∗ )

= meas(OF∗ )

#

1

1 1

1 − χη(π)q −(s+ 2 )

+

(χη(π))−1 q −(s+ 2 ) 1

1 − (χη(π))−1 q −(s+ 2 )

1 − q −(2s+1) 1

1

(1 − χη(π)q −(s+ 2 ) )(1 − (χη(π))−1 q −(s+ 2 ) ) L(s + 12 , χη)L(s + 21 , (χη)−1 ) = meas(OF∗ ). L(2s + 1, trivial)

The second claim is easy to verify and is ommitted. Lemma 1.9 has also been obtained by Brocco ([Bro]) using a slightly different computation. A simpler calculation gives the following: 12

Lemma 1.10. Suppose that χη is ramified. Let n be the least integer such that both χ and η are trivial on 1 + π n OF . Let f = (f1 , 0) where f1 is the characteristic function of 1 + π n OF . Then (1.13) L(s + 12 , χη)L(s + 12 , (χη)−1 ) meas(OF∗ ) L(s, η, Φf ) = (1 − q −2s−1 )−1 . L(2s + 1, trivial) [O∗ : 1 + π n O] 1.3. The first formula. Let ( (1.14)

Φv,¯η =

Φv,¯η

if v is nonsplit in E/F

Φv,f

if v is finite and split in E/F

where ( (1.15)

f=

( char (Ov ), char(πOv )) ( char (1 +

πvn Ov ), 0)

if χw η˜w is unramified if χw η˜w is ramified .

Here n is the smallest integer such that both χw and η˜w are trivial on 1 + πvn Ov . Theorem 1.11. Let E be a CM field with F being its totally real subfield. Let Φη¯ = ⊗Φv,¯η with Φv,¯η as above. Let S1 = {v : v is inert and χ˜ η is ramified at v} S2 = {v : v is split and χ˜ η is ramified at v}. Then Φη¯ ∈ I(s, χ) and

(1.16)

L(s + 12 ), χ˜ η) meas([E 1 ])2 Tam(E 1 )c(s) L(2s + 1, ²E/F ) Z = E(i(g1 , g2 ), s, Φη¯)η(g1 )(χη)−1 (g2 ) dg1 dg2 . [G×G]

where c(s) =

Y

(1 + qv−2s−1 )−1

v∈S1

and

qv−n (1 − qv−1 )−1 (1 − qv−2s−1 )−1 ,

v∈S2

Q Tam(E 1 ) =

Y

v

Q 1 ∗ nonsplit meas(Ev ) v split meas(Ov ) meas([E 1 ]) 13

is the Tamagawa number of G = E 1 . Here the measure on EA1 is the product measure as mentioned in the introduction, and meas([E 1 ]) is the volume of [E 1 ] with respect to the quotient measure. Proof. First Φη¯ ∈ I(s, χ) by Lemma 1.3 and Lemma 1.9. Let cv = Lv (s, ηv , Φv )

Lv (2s + 1, ²E/F ) Lv (s + 12 , χ˜ η)

where the last fraction is omitted when v is real. Then by Lemma 1.2, 1.9 and 1.10, one has  meas(Ev1 )(1 + qv−2s−1 )−1     1    meas(Ev ) cv =

meas(Ov∗ )        (1 − qv−2s−1 )−1

for v ∈ S1 , for v ∈ / S1 and nonsplit, for v split and v ∈ / S2 ,

meas(Ov∗ ) [Ov∗ : 1 + π n Ov ]

for v ∈ S2 .

Putting these together and applying the Piatetski-Shapiro and Rallis basic identity (1.2), we are done. 2. Theta liftings and Main theorems. 2.1. Siegel-Weil Formula and Theta lifting. Corresponding to the seesaw dual pair (0.18), one has Lemma 2.1. Suppose that α ∈ F ∗ and χ are given. (i) (2.1)

There is a unique splitting ˜ıα,χ :

G(W + W− )(A) × G(Vα )(A) −→ Mp(W + W− )A

such that the corresponding Weil representation ωα,χ = ωψ ◦ ˜ıα,χ acts on S(Wd (A)) as follows: (a) by χ, (b) (2.2)

The group G(Vα )(A) acts on S(Wd (A)) via the twist of its linear action The map S(Wd (A)) −→ I(0, χ), φ 7→ {h 7→ ωα,χ (g)φ(0)}

gives rise to an intertwining operator between the Weil representation and the induced representation I(0, χ). 14

(ii)

There is a unique splitting ˜ıα,χ :

G(W )(A) × G(Vα )(A) −→ Mp(W)A

such that the following diagram commutes. H(A) ↑i G(A) × G(A)

˜ ıα,χ

−→ ˜ ıα,χ ⊗χ−1˜ ıα,χ

−→

˜ ıα,χ

Mp(W + W− )A ↑ ˜i

←−

Mp(W)A × M p(W)A

˜ ıα,χ ⊗χ−1˜ ıα,χ

←−

G(Vα )(A) ↓ diag G(Vα )(A) × G(Vα )(A).

The same is true locally. Proof. It is a local problem. When v is nonsplit, it is proved in [Ka], see also [HKS, section 1]. The same trick works even if v is split. In next section, we will give another proof of (ii) in the split case. Let ωψ denote the Weil representations of both Mp(W) and Mp(W + W− ) and let ωα,χ = ωψ ◦ ˜ıα,χ . Given a complete polarization W = X ⊕ Y , there are two complete polarizations of W + W− , W + W− = (X + X) ⊕ (Y + Y ) = Wd ⊕ Wd , and so the Weil representation ωψ has realizations on S((X + X)(A)) and S(Wd (A)). There is an intertwining isometry [Li, page 182] δψ :

S((X + X)(A)) −→ S(Wd (A))

between these two realizations defined as follows. Identify Wd with W via the map (w, −w) 7→ w, and write w ∈ W as w = (x, y) with respect to the decomposition W = X ⊕ Y . Then for φ ∈ S((X + X)(A)), one has Z (2.3) δψ (φ)(w) = ψ(2 < u, y >)φ(u + x, u − x) du. X(A)

In particular, Z (2.4)

δψ (φ1 ⊗ φ¯2 )(0) =< φ1 , φ2 >=

φ1 (x)φ¯2 (x) dx. X(A)

By Lemma 2.1, δψ also intertwines the actions of ωα,χ on those two spaces. Let α ∈ F ∗ /N E ∗ . Given a function φ ∈ S(Wd (A)), there is a unique standard section Φ(h, s) ∈ I(s, χ) such that Φ(h, 0) = ωα,χ (h)φ(0). Write E(h, s, φ) = E(h, s, Φ). By a well-known theorem of Langlands E(h, s, φ) is a holomorphic automorphic form on H(A) at s = 0. 15

Another way to construct automorphic forms is to use theta integrals. Indeed, let φ ∈ S(Wd (A)), g ∈ G(Vα )(A) and h ∈ H(A). One defines the theta kernel X

θ− (h, g, φ) =

ωα,χ (h)φ(g −1 w).

w∈Wd (F )

This function is left H(F ) × G(Vα )(F ) invariant and defines a smooth, slowly increasing function on [H] × [G(Vα )]. Let Z (2.6)

θ− (g, h, φ) dg.

I(h, φ) = [G(Vα )]

Then I(h, φ) is absolutely convergent ([Wei]) and defines an automorphic form on H(A). The Siegel-Weil formula claims that the two methods produce essentially the same result, i.e., Theorem 2.2. ([Bro, Theorem 5]) (Siegel-Weil Formula for U (1, 1)) With notation as above, one has E(h, 0, φ) =

2 I(h, φ) meas([E 1 ])

where φ ∈ S(Wd (A)) and h ∈ H(A). Brocco derived the Siegel-Weil formula for U (1, 1) from the Siegel-Weil formula for ( Sp(1), O(2)) ([KR]). The Siegel-Weil formula is expected to be true for general unitary groups, and Kudla and Rallis’s proof in the symplectic case ([KR], [KR2]) should be able to extend to cover this case without much difficulty. Locally, given α ∈ Fv∗ , let R(α, χv ) be the maximal quotient of S(Wd,v ) = S(Ev ) in which G(Vα ) acts via χv . It only depends on the classes α ∈ Fv∗ /N Ev∗ , or equivalently on ²v (α) = ±1. Recall that the map S(Ev ) −→ I(0, χv ),

φ 7−→ {h 7→ ωα,χ,v (h)φ(0)}

factorizes through R(α, χv ) ,→ I(0, χv ) ([Ra]). Lemma 2.3. One has (i) I(0, χv ) = ⊕α∈Fv∗ /N Ev∗ R(α, χv ). Moreover, R(α, χv ) is always an irreducible subrepresentation of I(0, χv ). (ii) (2.7)

When v is nonsplit, Φη¯(h, 0) ∈ R(α, χv ) if and only if 1 1 η )v , ψEv )(χ˜ η )v (δ) = ²v (α) ²( , (χ˜ 2 2 16

where ²( 12 , (χ˜ η )v , 21 ψEv ) is the Tate’s root number. (iii) When v is split, (2.7) is always true, and I(0, χv ) = R(α, χv ) is independent of α ∈ Fv∗ . Proof. (i) When v is nonsplit, the claim is a special case of [KS, Prop. 4.3]. When v is split in E/F , I(0, χ) is irreducible by GL2 -theory, and so I(0, χv ) = R(α, χv ). Claim (ii) follows from the local epsilon dichotomy for U (1) × U (1) ([HKS, Corollary 8.5], it is also true when v is real, see [Ya]). Indeed, by [HKS, Corollary 8.5], one has (2.8)

S(Fv ) = ⊕Cφη¯v

where the sum runs over all characters of Gv satisfying (2.7), and φη¯v is a unitary eigenfunction of Gv with eigencharacter η¯v . Assume that ηv satisfies (2.7), let φ0v = δψ (φη¯v ⊗ φ¯η¯v ). By (2.4), one has then ωα,χ,v (i(g, 1))φ0v (0) =< ωα,χ,v (g)φη¯v , φη¯v > = η¯v (g) < φη¯v , φη¯v > = Φη¯v (i(g, 1)). Since Φη¯v is determined by its restriction on i(Gv × 1), one has Φη¯v (h, 0) ∈ R(α, χv ). Combining this with (i), one proves (ii). It is easy to check that (2.7) is always true in the split case. By the proof of Lemma 2.3(ii), one has Corollary 2.4. Assume that v is nonsplit in E/F and that (αv , ηv ) satisfies (2.7). Let φη¯v be a unitary eigenfunction of Gv with eigencharacter η¯v . (It exists and is unique up to a scalar in C1 ). Then for every g ∈ Gv , one has (2.9)

< ωα,χ,v (g)φη¯v , φη¯v >= Φη¯v (i(g, 1)).

Theorem 2.5. Assume that the root number ²( 12 , χ˜ η ) = 1. Let α ∈ F ∗ /N E ∗ be the unique element such that (2.7) holds for every place v of F . Then there is a Q function φ = φv ∈ S(Wd (A)) = S(EA ) such that (2.10)

ωα,χ (i(g, 1))φ(0) = Φη¯(i(g, 1), 0)

for every g ∈ G(A) = EA1 , where Φη¯ is the function defined by (1.14) and (1.15). Moreover for such a function φ, one has (2.11)

L( 12 , χ˜ η) 1 meas([E 1 ])3 Tam(E 1 )c(0) 2 L(1, ²E/F ) Z = I(i(g1 , g2 ), φ)η(g1 )(χη)−1 (g2 ) dg1 dg2 . [G×G]

17

Proof. For each place v of F , let αv ∈ Fv∗ /N Ev∗ be the unique element satisfying (2.7). Taking product, one has Y

1 ²v (αv ) = ²( , χ˜ η ) = 1. 2

Applying the classical exact sequence 1 −→ F ∗ /N E ∗ −→ FA∗ /N EA∗ −→ {±1} −→ 1, Y (αv ) 7−→ ²v (αv ), one has a unique element α ∈ F ∗ /N E ∗ satisfying (2.7) at every Q place of F . By Lemma 2.3, there is a function satisfying (2.10). Let φ = φv be such a function. Since a function Φ ∈ I(s, χ) is determined by its restriction to i(G(A) × 1), (2.10) implies Φflat (h, 0) = Φη¯(h, 0) for every h ∈ H(A). This in turn implies E(h, 0, Φη¯) = E(h, 0, φ). Now applying the Siegel-Weil formula and Theorem 1.11, one gets (2.11). Now fix α ∈ F ∗ /N E ∗ and a complete polarization W = X ⊕ Y . On S(X(A)), one defines the theta distribution X (2.12) θ(φ) = φ(x). x∈X(F )

Given a function φ ∈ S(X(A)), one defines the theta kernel on G(W )(A) × G(Vα )(A) (2.13)

θφ (g, g 0 ) = θ(ωα,χ (g, g 0 )φ) =

X

ωα,χ (g, g 0 )φ(x).

x∈X(F )

Given a character η of G(W )(A) = EA1 , one defines an automorphic form θφ (η) on G(Vα )(A) = EA1 (sometimes called a theta lifting of η) via Z (2.14)

0

θφ (g, g 0 )η(g) dg.

θφ (η)(g ) = [G]

Let θ(η, α) = {θφ (η) : φ ∈ S(FA )}. Then θ(η, α) is an automorphic representation of G(Vα ), and is called the theta lifting of η to G(Vα ). In this special case, it is easy to see from Lemma 2.1 that (2.15)

θφ (η)(g 0 ) = η(g 0 )θφ (η)(1) 18

for g 0 ∈ G(Vα ) = EA1 .

So the theta lifting θ(η, α) is either the character η or 0. One of the fundamental problems in theta liftings is whether or not θ(η, α) = 0. Note that θφ (η) 6= 0 if and only if < θφ (η), θφ (η) >6= 0. Take two functions φ1 , φ2 ∈ S(X(A)) and two characters η1 and η2 of EA1 . We shall consider the Rallis inner product Z (2.16)

< θφ1 (η1 ), θφ2 (η2 ) >=

[G(Vα )]

θφ1 (η1 )(g 0 )θφ2 (η2 )(g 0 ) dg 0 .

Unfolding the inner product, one has ([Bro], [Li], [Ra], [Ra2]) (2.17) Z < θφ1 (η1 ), θφ2 (η2 ) >= η1 (g1 )(χη2 )−1 (g2 )I(i(g1 , g2 ), δψ (φ1 ⊗ φ¯2 )) dg1 dg2 . [G×G]

By Theorem 2.5, Formula (2.4) and (2.17), one has immediately Theorem 2.6. Let E be a CM field with totally real subfield F . Assume that the root number ²( 12 , χ˜ η ) = 1 and let α ∈ F ∗ /N E ∗ be the unique Q element such that (2.7) holds for every place v of F . Suppose a section φ = φv ∈ S(X(A)) satisfies (2.18)

< ωα,χ (g, 1)φ, φ >= Φη¯(i(g, 1), 0)

where Φη¯ is the function defined in (1.14) and (1.15). Then (2.19) Tam(E 1 )c(0)

L( 12 , χ˜ η) 2 = < θφ (η), θφ (η) > L(1, ²E/F ) meas([E 1 ])3 2 = |θφ (η)(1)|2 . meas([E 1 ])2

In particular, L( 12 , χ˜ η ) 6= 0 if and only if θφ (η) 6= 0. Q To find functions φ = φv satisfying (2.18), we fix from now on the standard complete polarization (2.20)

W = F δ ⊕ F,

1 }. We will also identify S(F δ) with S(F ) together with the standard basis {δ, ∆α and S(Wd ) with S(E). When v is nonsplit, φv can be taken to be a unitary eigenfunction of Gv with eigencharacter η¯v by Corollary 2.4.

The verification of the following lemma can be found in [Ya] or [Ya4]. 19

Lemma 2.7. Assume that v is nonsplit in E/F and that everything is unramified at v. Then the characteristic function φ0v of Ov is a unitary eigenfunction of Gv with eigencharacter η¯v (= 1). 2.2 Construction of φv : the split case. In this subsection, we assume that v is split and finite in E/F and suppress the subscript v. So, for example, F means Fv and E means Ev = Fv ⊕Fv . Recall that in this case, δ = (x0 , −x0 ). Besides the standard complete polarization, one has also in this case a natural complete polarization, which is easier to deal with. In fact, the space W has a natural basis e1 = x0 (1, 0) and e2 = x22α (0, 1) over F 0 and a natural complete polarization (2.21)

W = X1 ⊕ X2 = F e1 ⊕ F e2 .

The space W + W− has a natural basis e1 + e2 and e1,− − e2,− over E. Finally the space W+W− has a natural basis e1 , e1,− , e2 , −e2,− over F and the natural polarization (2.22)

W + W− = (X1 + X1 ) ⊕ (X2 + X2 ).

By pure calculation, one has Lemma 2.8. With respect to the natural bases, one has the following commutative diagram: β0

GL(2) ↑i

∼ =

GL(1) × GL(1)

∼ =

β0

G(W + W− ) ↑i G(W ) × G(W )

ı

α −→

Sp(W + W− ) ↑i

ıα ×ıα

−→

Sp(W) × Sp(W).

Here µ 0

−1

d b

¶ c ), a

µ

a c

β (A) = (A, det(A ) A= µ ¶ A 0 0 , ıα ◦ β (A) = 0 t A−1 ¶ µ g 0 0 , ıα ◦ β (g) = 0 g −1 µ ¶ g1 0 i(g1 , g2 ) = , for g1 , g2 ∈ GL(1), 0 g2

b d

¶ ∈ GL(2),

where ıα and i in the middle and right are defined in the introduction. 20

Corollary 2.9. There is a commutative diagram of splittings as follows: G(W + W− ) ↑i

(2.23)

G(W ) × G(W )

˜ ıχ

−→ ˜ ıχ ⊗χ−2˜ ıχ

−→

Mp(W + W− ) ↑ ˜i Mp(W) × Mp(W)

where, under the identification β 0 and with respect to the natural complete polarizations of Mp(W) and Mp(W + W− ), (2.24)

˜ıχ (g) = (ıα (g), χ(g))

We remark that χ−2 is used in (2.23) instead of χ−1 in Lemma 2.1 because of the identification G = {(x, x−1 ) : x ∈ F ∗ } with F ∗ . Under this identification, one has χ(x, x−1 ) = χ(x, 1)χ((x, 1))−1 = χ(x)χ(x) = χ2 (x). Let ωχ = ωψ ◦ ˜ıχ be the corresponding Weil representation of G(W ) and noindent G(W + W− ). By Lemma 2.8, Corollary 2.9 and trivial calculation, one has Corollary 2.10. (i) S(F ) = S(X1 ) via

The Weil representation ωχ of G(W ) = GL(1) acts on 1

ωχ (g)φ(x) = χ(g)|g| 2 φ(xg). (ii) via

The Weil representation ωχ of G(W + W− ) acts on S(F 2 ) = S(X1 ⊕ X1 ) 1

ωχ (g)φ(x) = χ( det g)| det g| 2 φ(xg). (iii)

If we identify S(F ) ⊗ S(F ) with S(F ⊕ F ), then ωχ ◦ i = ωχ ⊗ χ2 ωχ∨ .

Proposition 2.11. One has (i)

As splittings of G(W + W− ) in Mp(W + W− ), ˜ıχ = ˜ıα,χ,v .

(ii)

As representations of G(W + W− ), δψ ◦ ωχ = ωα,χ,v ◦ δψ .

(iii)

As representations of G(W ) × G(W ), δψ ◦ (ωχ ⊗ χ2 ωχ∨ ) = ωα,χ,v ◦ δψ . 21

Proof. The three claims are equivalent. There is a character ξ of F ∗ such that ˜ıα,χ,v ◦ i = ξ˜ıχ ◦ i = ˜i ◦ (ξ˜ıχ ⊗ ξχ−2˜ıχ ) on G(W ) × G(W ) and so ωα,χ,v ◦ δψ = δψ ◦ (ξωχ ⊗ ξ −1 χ2 ωχ∨ ). For any φ1 , φ2 ∈ S(F ), g ∈ GL(1) = G(W ), one has ωα,χ,v (i(g, 1))δψ (φ1 ⊗ φ¯2 )(0) = δψ (ξ(g)ωχ (g)φ1 ⊗ φ¯2 )(0) Z = ξ(g)ωχ (g)φ1 (x)φ¯2 (x) dx F Z 1 = ξ(g)χ(g)|g| 2 φ1 (xg)φ¯2 (x) dx. F

By Lemma 2.1, the left hand side determines a function in I(0, χ). So does the right hand side. So, by Proposition 1.7, there are functions f1 , f2 ∈ S(F ) such that Z 1 1 1 1 φ1 (xg)φ¯2 (x) dx = χ(g)|g| 2 f1 (g) + χ(g)|g|− 2 f2 ( ); ξ(g)χ(g)|g| 2 g F ∗ i.e., for any g ∈ F , Z 1 φ1 (xg)φ¯2 (x) dx = f1 (g) + |g|−1 f2 ( ). ξ(g) g F Let φ1 = φ2 = φ be the characteristic function of OF . Then Z Z ¯ φ1 (xg)φ2 (x) = φ1 (xg) dx F

OF

½ 1 if g ∈ OF = meas(OF ) −1 |g| if g ∈ / OF 1 = f10 (g) + |g|−1 f20 ( ) g 0 0 where f1 = meas(OF )φ and f2 is meas(OF ) char(πOF ). So we have 1 1 ξ(g)f10 (g) − f1 (g) = |g|−1 (ξ(g)f20 ( ) − f2 ( )). g g In particular, for g ∈ / OF , one has 1 1 f1 (g) = |g|−1 (ξ(g)f20 ( ) − f2 ( )). g g So f1 ∈ S(F ) implies 1 1 f2 ( ) = ξ(g)f20 ( ) = ξ(g), when ord g > 0. Since f2 ∈ S(F ), f2 (g) is constant when ord g >> 0. Therefore ξ = 1. By the proof of the proposition, one also gets 22

1

Corollary 2.12. Suppose that χ˜ η is unramified. Let φ0 = ( meas(OF ))− 2 char(OF ), and let f be as in Lemma 1.9. Then < ωχ (g)φ0 , φ0 >= Φf (i(g, 1), 0). An easy calculation also gives Corollary 2.13. Suppose χ˜ η is ramified. Let n be the minimal integer such that 1 both χ and η are trivial on 1 + π n OF . Let φ0 = ( meas(π n OF ))− 2 char(1 + π n OF ), and let f = ( char(1 + π n OF ), 0) be as in Theorem 1.11. Then < ωχ (g)φ0 , φ0 >= Φf (i(g, 1), 0). Corollaries 2.12 and 2.13 plus a transferring back to the standard polarization implies the existence of φv in the splitting case. For application, we now make it explicit. The space W has two complete polarizations W = F e1 ⊕ F e 2 = F e ⊕ F f 1 where e = δ = (x0 , −x0 ), f = ∆α = x21α (1, 1). We denote by (ωα,χ,v , S(F )) the 0 realization of the Weil representation ωχ on S(F e). Let ! Ã x3 α 1 − 02 (2.25) M= , 1 1 x30 α

then

2

µ ¶ µ ¶ e e1 =M . f e2

There is an isometry (2.26)

ρM :

S(F e1 ) −→ S(F e),

φ 7−→ (ωψ (M )φ)M −1

intertwining the realizations of Weil representation (ωχ , S(F e1 )) and (ωα,χ,v , S(F e)). Here ωψ is the realization of the Weil representation of Mp(W) with respect to the natural polarizatioin. We will identify both F e1 and F e with F . Then the intertwining isometry ρM has the form (2.27)

ρ = ρM :

S(F ) −→ S(F ),

φ 7−→ ωψ (M )φ.

Using Rao’s standard section formula [Rao,Theorem 3.6], one has the explicit formula for ρ: Z 1 x3 α 1 3 2 3 2 (2.28) ρ(φ)(x) = |x0 α| ψ( x0 αx ) ψ(x30 αxy)ψ( 0 y 2 )φ(y) dy. 2 4 F By Corollaries 2.12 and 2.13 and Formula 2.28, one can prove 23

Corollary 2.14. Let φ = ρ(φ0 ), where φ0 is the function defined in Corollary 2.12 and 2.13. Then (2.18) holds locally at v. Moreover, when everything is unramified, one has ρ(φ0 ) = φ0 the characteristic function of OF . Proof. Since ρ is an isometry, (2.18) holds by Corollaries 2.12 and 2.13. When everything is unramified, which means that ψ, χ, and η are unramified, α, x0 and 2 are units. So φ0 is the characteristic function of OF and meas(OF ) = 1. Therefore one has by (2.28) Z 1 3 2 φ(φ0 )(x) = ψ( x0 αx ) ψ(x30 αxy) dy 2 OF 1 = ψ( x30 αx2 )φ0 (x) meas(OF ) 2 = φ0 (x). 2.3 The Main Theorems. Assume that the root number ²( 12 , χ˜ η ) = 1. Let α ∈ F ∗ /N E ∗Qbe the unique element such that (2.7) holds for every place v of F . Define φη¯ = φv,¯η ∈ S(FA ) as follows: When everything is unramified at v, (2.29)

φv,¯η = φ0v = the characteristic function of Ov .

Otherwise, if v is nonsplit in E/F , choose a unitary eigenfunction φv,¯η of Gv with eigencharacter η¯v according to Lemma 2.4. If v is finite and split in E/F , and χ˜ η is unramified at v, let (2.30) ( ρ( char(Ov )) if χ˜ η is unramified at v, − 12 nv φv,¯η = ( meas(Ov )) qv2 ρ( char(1 + πvnv Ov )) if χ˜ η is ramified at v as in Corollary 2.14. Here nv be the smallest integer such that both χ and η˜ are trivial on 1 + πvn Ov . Theorem 2.15. Let E be a CM field with F being its totally real subfield. Then (i) (ii)

The function φη¯ is a function in S(FA ) and satisfies (2.18). Formula (2.19) holds for the function φη¯ just defined.

Proof. By Lemma 2.7 and Lemma 2.14, φη¯ ∈ S(FA ). The rest is obvious by construction and Theorem 2.6. Remark 2.16. Although we have only dealt with CM fields so far, the results we obtained are true for general quadratic extensions of number fields. For example, one can prove: 24

Theorem 2.150 . Let E/F be a quadratic extension of number fields. Let χ be a fixed idele class character of E whose restriction on F is ²E/F . Let η be a character of [E 1 ] such that ²( 12 , χ˜ η ) = 1. Let α ∈ F ∗ /N E ∗ be the unique element Q satisfying (2.7) for every place v of F . Then there is a function φ = φv ∈ S(FA ) and a positive constant c such that (2.31)

L( 12 , χ˜ η) = c < θφ (η), θφ (η) > . L(1, ²)

Q Proof. Given any function φ = φv ∈ S(FA ), by the doubling method (2.17), the Siegel-Weil formula (Theorem 2.2), and the Piatetski-Shapiro and Rallis identity (1.2), one has meas([E 1 ])2 Y 1 < θφ (η), θφ (η) >= L( , ηv , φv ), 2 2 where Z L(s, ηv , φv ) = < ωα,χ,v (g)φv , φv > a(i(g, 1))s ηv (g)dg. Gv

Here a(h) = |a¯ a|v if h = n(b)m(a)k with a ∈ Ev∗ , b ∈ Fv , and k ∈ Kv . When v is not infinite split, let φv be the function defined in Theorem 2.16. Then we have Y η) L( 1 , χ˜ 1 L( , ηv , φv ) < θφ (η), θφ (η) >= c 2 L(1, ²) 2 v|∞, split for some positive constant c > 0. So it is sufficient to choose a function φv ∈ S(Fv ) for every infinite split place v such that 1 L( , ηv , φv ) > 0. 2 Assume that v is real and split in E/F . Let ρ be the isometry given in (2.27), and let ωχ,v be the Weil representation Gv = R∗ on S(R) with respect to the natural basis (2.21). In fact, ρ is also an isometry on L2 (R). Let ( −πx2 (χη)−1 if x 6= 0 v (x)e hv (x) = 0 otherwise . Let φ0v = ρ(hv ). Then both hv and φ0v are in L2 (R). First, < ωα,χ,v (g)φ0v , φ0v > =< ωχ,v (g)hv , hv > Z 1 = χv (g)|g| 2 hv (gx)hv (x) dx R Z 2 2 1 −1 e−π(1+g )x dx = ηv (g)|g| 2 R

=

1 ηv−1 (g)|g| 2 p 1 + g2

25

.

On the other hand, an easy calculation gives a(i(g, 1)) = where a = 14 + x20 and b = always greater than 0. So Z L(s, ηv , φ0v )

|g| + bg + a

− 2x20 . Since b2 − 4a2 = −4x20 < 0, ag 2 + bg + a is

1

R∗

|g|s ηv−1 (g)|g| 2 p ηv (g) d∗ g 1 + g 2 (ag 2 + bg + a)s

R∗

|x|s− 2 √ dx 1 + x2 (ax2 + bx + a)s

= Z

1

= Z



= 0

Z

=2 0

So

1 2

ag 2

1

1

|x|s− 2 √ dx + 1 + x2 (ax2 + bx + a)s s− 12

Z

√ 0

|x| √ dx + 2 2 1 + x (ax2 + bx + a)s 1 L( , ηv , φ0v ) = 4 2

Z

1

0

Z

1

p

1



x(1 + x2 )

1

0

|x|s− 2 dx 1 + x2 (ax2 − bx + a)s 1

|x|s− 2 √ dx. 1 + x2 (ax2 − bx + a)s

dx > 0.

Since S(R) is dense in L2 (R), there is then a function φv ∈ S(R) such that L( 12 , ηv , φv ) > 0. The case where v is complex can be proved similarly. We leave it to the reader. We remark that L(s, ηv , φ0v ) can be expressed in terms of gamma functions ([Bro]). Proof of Theorem 0.1 If ²( 12 , χ) = −1, the functional equation forces = 0. Now assume ²( 12 , χ) = 1. Applying Theorem 2.150 to this χ and η = 1, one has L( 12 , χ) ≥ 0. L(1, ²E/F ) L( 12 , χ)

It is well-known that L(1, ²E/F ) > 0. So L( 12 , χ) ≥ 0. Proof of Theorem 0.4 First if the global theta lifting θ(η, α) 6= 0, then every local theta lifting θ(ηv , αv ) 6= 0. By [HKS, Corollary 8.5], this means that (0.6) (i.e. (2.7)) holds for every place v of F . Now assume that η and α satisfy (0.6) for every place v of F . If L( 12 , χ˜ η ) 6= 0, then θφ (η) 6= 0 for some function 0 η ) = 0. φ ∈ S(FA ) by Theorem 2.15 . So θ(η, α) 6= 0. Conversely, suppose L( 12 , χ˜ Given any function φ ∈ S(FA ), by the doubling method (2.17), the Siegel-Weil 26

formula (Theorem 2.2), and the Piateski-Shapiro and Rallis basic identity (1.2), one can easily show that θφ (η) = ∗

L( 21 , χ˜ η) =0 L(1, ²)

where ∗ is some constant. This implies θ(η, α) = 0. Theorem 0.4 is proved. Remark 2.17. Finally, we remark that the theta integral in Formula 2.19 is computable. In some cases, the resulting formula is quite interesting ([Ya], [RVY]) and can be used to prove some nonvanishing results on the central Hecke L-value and the theta liftings ([RVY], [Ya2-3]). we are content to quote a therem in [Ya] and its consequence without proof. Qr Theorem 2.18. Assume p ≡ 7 mod 8. Let d = i=1 qi > 0 be the discriminant of a real quadratic √ field such that qi are all odd primes split in E/Q (or d = 1). Here E = Q( −p). Let χcan be a canonical Hecke character of E, and let χd = χcan ( d ) ◦ NE/Q . For each ideal class C of E, let a a primitive ideal (no rational factors) in C relatively prime to 2pd and write √ −p 2 2 −b + ], L > 0 a = [L , 2 such that (always possible) b ≡ r mod 16d3 , where r is a fixed square root of −p mod 32d3 . Let √ −b + −p τa = . 8d3 L2 Then ¯ ¯2 ¯ ¯ X ¯ ¯ π 1 ¯ (2.32) L(1, χd ) = √ √ ¯ θd (τa )¯¯ χd (a¯) d d4p¯ ¯ C∈ CL(E) where θd (τ ) =

X (n,2d)=1

is a modular form of weight

1 2

2 d ( )eπin τ n

independent of p.

Setting d = 1, it recovers a formula of Rodriquez Villegas ([RV]). √ Corollary 2.19. Notation and assumption as in Theorem 2.18. If p ≥ logπ 2 d3 , then the central L-value L(1, χd ) > 0. Therefore Gross’s elliptic curve ([Gro]) A(p)d has Q-rank 0. Also the theta lifting θ(ηd , 1) does not vanish, where ηd is a character of [Ep1 ] such that η˜d = ( d ) ◦ NEp /Q . 27

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 E-mail address: [email protected]

29