NONVANISHING OF CENTRAL HECKE L-VALUES

AND RANK OF CERTAIN ELLIPTIC CURVES

(COMPOSITIO MATH. 117 (1999), NO. 3, 337–359.)

Tonghai Yang

Abstract. Let D ≡ 7 mod 8 be a positive squarefree integer, and let hD be theideal class number of ED = Q(

√−D). Let d ≡ 1 mod 4 be a squarefree integerrelatively prime to D. Then for any integer k ≥ 0 there is a constant M = M(k),independent of the pair (D, d), such that if (−1)k = sign(d), (2k + 1, hD) = 1, and

√D >

12

πd2(log |d|+ M(k)),

then the central L-value L(k + 1, χ2k+1D,d ) > 0. Furthermore, for k ≤ 1, we can take

M(k) = 0. Finally, If D = p is a prime, and d > 0, then the associated elliptic curve

A(p)d has Mordell-Weil rank 0 (over its definition field) when√

D > 12π

d2 log d.

0. Introduction.

Let D ≡ 3 mod 4 be a positive squarefree integer, and let d ≡ 1 mod 4 bea squarefree integer relatively prime to D. We consider Hecke characters χ ofED = Q(

√−D) of conductor d√−DO satisfying

(1) χ(A) = χ(A) for every ideal of ED relatively prime to the conductor,and

(2) χ(αO) = ±α for every principal ideal relatively prime to the conductor.

Here O is the ring of integers of ED. There are hD such Hecke characters foreach pair (D, d), differing from each other by an ideal class character of ED,where hD is the ideal class number of ED. We denote such a Hecke characterby χD,d. These Hecke characters were studied by Rohrlich ([Roh2-3]), who alsoallowed D or d to be even. In particular, he proved, that for almost all pairs(D, d) such that D > |d|39+ε and the root number of χD,d is one, the centralL-value L(1, χD,d) 6= 0. Here ε is any positive number. He and Montgomery([MR]) also proved a more definite result asserting that L(1, χD,1) 6= 0 if andonly if the root number of χD,1 is one. Rodriquez Villegas further gave a niceformula in [RV1-2] for the central L-value L(1, χD,1) for D ≡ 7 mod 8. From this

1991 Mathematics Subject Classification. 11G05 11M20 14H52.Key words and phrases. central Hecke L-value, elliptic curves, eigenfunction, nonvanishing.Partially supported by the NSF grant DMS-9304580

Typeset by AMS-TEX1

2 TONGHAI YANG

formula the nonvanishing of the central L-value becomes obvious. A formula forthe central L-value L(k + 1, χ2k+1

D,1 ) was obtained by similar technique in [RVZ].Using a different method developed in [Ya1], Rodriquez Villegas and the authordiscovered that similar formula is valid for Hecke characters χ2k+1

D,d , where everyprime divisor of d splits in ED. In fact, such a formula exists for a wholeclass of Hecke characters of a CM number field of any degree ([RVY]). In thispaper, we use the same method to derive a formula for L(k + 1, χ2k+1

D,d ) withoutany condition on d (Theorem 2.1), which enables us to prove the followingnonvanishing result in section 3.

Main Theorem. Let D ≡ 7 mod 8 be a positive squarefree integer, and let hD

be the ideal class number of ED = Q(√−D). Let d ≡ 1 mod 4 be a squarefree

integer relatively prime to D. Then for any integer k ≥ 0 there is a constantM = M(k), independent of the pair (D, d), such that if (−1)k = sign(d),(2k + 1, hD) = 1, and

√D >

12π

d2(log |d|+ M(k)),

then the central L-value L(k + 1, χ2k+1D,d ) > 0. Furthermore, for 0 ≤ k ≤ 1 we

can take M(k) = 0.

Refinements and comments will also be given in section 3. The Hecke char-acters considered here are arithmetic in nature; each such character has anassociated Hecke motive (see for example [Sha]). In particular, when k = 0and D = p is a prime, the character χp,d is very closely related to the ellipticcurve A(p)d over a number field F studied by Gross ([Gro]), using Shimura’stheory on CM abelian varieties ([Sh1-2]). He proved, in particular, by means ofdescent theory that A(p) has Mordell-Weil rank 0 over F . Combining a theoremof Rubin ([Ru, Corollary 2.2]) with the main theorem, one has

Corollary. let p ≡ 7 mod 8 be a prime, and let d ≡ 1 mod 4 be a positivesquarefree integer not divisible by p such that

√p > 12

π d2 log d. Then the ellipticcurve A(p)d has Modell-Weil rank 0 over Q(j). Here j = j( 1+

√−p2 ) is the j-

invariant of A(p).

Acknowledgment The author thanks P. Deligne, K. Murty, and L. Washing-ton for their help during the preparation of this paper. He also thanks S. Kudlaand D. Rohrlich for their continuing encouragement and advice. He thanks thereferee for his/her comments and suggestions, which have influenced this finalversion. Finally, the author thanks the Institute for Advanced Study for itshospitality and financial support.

1. Eigenfunctions of Weil representations.

NONVANISHING OF CENTRAL HECKE L-VALUE 3

In this section, we will explicitly construct eigenfunctions of the local Weilrepresentation of the unitary group of one variable in terms of Shrodinger model.They are needed in the next section to derive an explicit formula for the centralHecke L-value L(k + 1, χ2k+1

D,d ) from the main formula in [Ya1]. We considergeneral local fields instead of just Qp, since it is not much harder. In the realcase, the eigenfunctions are essentially classical Hermite functions as we willsee in Lemma 1.1. For the p-adic case (p 6= 2), eigenfunctions were explicitlyconstructed in [Ya2] by means of a lattice model. So we only need to transferthe results to Schrodinger model. We will state the results in this section andgive in the appendix the proof, which is quite technical and lengthy.

Let F be a local field and let E = F (δ) be a quadratic extension of F . Assumethat δ = −δ and ∆ = δ2 ∈ F . Let ψ be a fixed nontrivial character of F andlet ψE = ψ ◦ trE/F . Given α ∈ F ∗, and a character of E∗ such that χ|F∗ = εis the quadratic character of F ∗ associated to E/F , there is a well-defined Weilrepresentation ωα,χ of G = U(1) = E1 on the space S(F ) of Schwartz functionson F (also depending on δ and ψ) ([Ku], see also the appendix). By the epsilondichotomy ([HKS, Corollary 8.5]), one has

(1.1) S(F ) = ⊕Cφη.

where the sum runs over all characters η of E1 satisfying

(1.2) ε(12, χη,

12ψE)χη(δ) = ε(α),

and φη is an eigenfunction of (G,ωα,χ) with eigencharacter η. Here η(z) =η(z/z). The task is to give an explicit formula for φη. First we consider the caseF = R. Recall that every character of C∗ is of the form χn(z) = (z/|z|)n, andthat every character of C1 is of the form ηl(z) = zl.

Lemma 1.1. ([Ya1, Theorem 2.18]) Let F = R and ψ(x) = e2πix. Assumeχη(z) = (|z|/z)2m+1 and δ ∈ iR>0. Then η occurs in ωα,χ if and only if k =

m sign(α)− 1− sign(α)2 ≥ 0. When k ≥ 0,

φk|δ|,|α|(x) = φk(

√|δ3α|x)

is an eigenfunction of ωα,χ with eigencharacter η. Here

φ0 = e−πx2, and φk(x) =

12k

(x− 12π

d

dx)kφ0(x).

Moreover< φk

|δ|,|α|, φk|δ|,|α| >=

1√2|δ3α|

k!(4π)k

.

4 TONGHAI YANG

Notice that there is a unique polynomial Hk(x) of degree k (the k-th Hermitepolynomial) such that

(1.3) φk(x) = Hk(x)φ0(x).

It is easy to check that H0 = 1 and H1 = x. In general, Hk has the same parityas k.

For the rest of this section, we assume that F is a p-adic local field with p 6= 2and that δ is a uniformizer or a unit of E depending on whether E/F is ramifiedor not. Let ψ′ = αδ

4 ψE , and let n(ψ′) be the conductor of ψ′. Let

(1.4) L =

{πn

EOE if n(ψ′) = 2n,

πnδOF ⊕ πn−1OF if n(ψ′) = 2n− 1.

Then the Weil representation of G has a lattice model realization ω on

(1.5) S(L,ψ) = {φ ∈ S(E) : φ(z + l) = ψ′(zl)f(z) for all l ∈ L}

Decomposition of S(L,ψ) is well-understood in [Ya2]. Define

(1.6) ρ : S(L,ψ) −→ S(F ), ρ(f)(x) =∫

F/F∩L

f(xδ + y)ψ′(−δxy) dy.

Then there is a constant c > 0 such that

< ρ(f), ρ(f) >= c < f, f >, for any f ∈ S(L,ψ).

Through ρ, ω gives a Weil representation of G on S(F ). So there is a uniquecharacter ξ of G such that such that

(1.7) ωα,χ(g) ◦ ρ = ξ(g)ρ ◦ ω(g), g ∈ G.

Proposition 1.2. Write g = x + yδ ∈ G. Let G′ = {x + yδ ∈ G : y ∈ πO} andGk = {g ∈ G : g ≡ 1 mod πk}, where π is a uniformizer of F .

(1) If E/F is ramified. Then

(1.8) ξ(g) =

χ(δ(g − 1))(∆,−y)F if g ∈ G1,

χ(δα)ε(12, εE/F , ψ) if g ∈ G−G1.

In particular, when the conductor n(χ) of χ is equal to 1, one has

NONVANISHING OF CENTRAL HECKE L-VALUE 5

(1.9) ξ =

trivial if ε(12, χ,

12ψE)χ(δ) = ε(α),

sign otherwise ,

where sign is the nontrivial character of G/G1 = {±1}.(2) If E/F is unramified and n(ψ′) = n(ψ)− ordF (α) = 2n is even. Then

(1.10) ξ(g) =

{χ(δ(g − 1))(∆,−y)F if g ∈ G1,

χ(δ(g − 1)) otherwise .

In particular, if χ is unramified, then ξ is trivial.

(3) If E/F is unramified and n(ψ′) = n(ψ) − ordF (α) = 2n − 1 is odd.Then

(1.11) ξ(g) =

χ(δ(g − 1))(∆,−y)F if g ∈ G1,

χ(δ(g − 1))(−∆

F

)if g ∈ G′ −G1,

χ(δ(g − 1))(

2∆(x− 1)F

)if g ∈ G−G′.

In particular, if χ is unramified, then ξ = η0, where η0( zz ) = η0(z) = (π, zz)F .

(4) If n(χ) ≤ 1, then ξ is trivial on G1.

Corollary 1.3. Let φη′ ∈ S(L, ψ) be an eigenfunction of ω with eigencharacterη′. Then ρ(φη′) is an eigenfunction of ωα,χ with eigencharacter ξη′, where ξ isgiven in Proposition 1.2.

Applying this to [Ya2, Corollary 2.5], one gets

Corollary 1.4. (1) Assume E/F is ramified and let n(ψ′) = 2n. Thenchar(π[ n

2 ]OF ) is an eigenfunction of (G,ωα,χ) with eigencharacter ξ.

(2) Assume that E/F is unramified and that n(ψ′) = 2n is even. Thenchar(πnOF ) is an eigenfunction of (G, ωα,χ) with eigencharacter ξ.

Given a character η of G = E1 satisfying (1.2), we denote η′ = ηξ−1.

Proposition 1.5. Assume that E/F is unramified and that n(ψ′) = 2n − 1is odd. Assume further that n(η′) ≤ 1. Then η occurs in ωα,χ if and only ifη′ 6= η0. Write ψ′′ = ∆α

2 π2n−2ψ and view it as a character of the residue fieldF = OF /π.

6 TONGHAI YANG

(1): If η′(−1) = (−1F

), let

φ′η(u) = char(πOF )(u) +1

2G(ψ′′)

·∑

A2−B2≡∆ mod π

η′(A + δ

B)(

B

F)ψ′′(

∆α

2Au2) char(OF )(u).

Then φη(u) = φ′η(π1−nu) is an eigenfunction of (G,ωα,χ) with eigencharacterη. Here G(ψ′′) is the Gauss sum of the character ψ′′ of F .

(2): If η′(−1) = −(−1F

) and η′ 6= η0, let a ∈ O∗F , and

φ′η,a(u) = char(a + πOF )(u)− char(−a + πOF )(u)

+1

G(ψ′′)

∑η′(

A + δ

B)(

B

F)ψ′′(Au2 − 2Bau + Aa2) char(OF )(u).

Here the sum runs over (A,B) ∈ F 2 with A2 −B2 ≡ ∆ mod π. Then φη,a(u) =φ′η,a(π1−nu) 6= 0 is an eigenfunction of (G,ωα,χ) with eigencharacter η.

The following two will not be needed in this paper. However we include themhere without proof for completeness and for their own rights. For z ∈ E, wewrite z = R(z) + I(z)δ with R(z) and I(z) ∈ F .

Proposition 1.6. Assume that n(ψ′) = 2n is even. Then η occurs in ωα,χ ifand only if there is w ∈ π−k+n

E O∗E such that η′(g) = ψ′(−wwg) for every g ∈ Gk.In such a case,

φη(u) =∑

g∈G/Gk

η′(g)ψ(∆α

2R(wg)I(wg))ψ(−∆αR(wg)u)

·{

char(I(wg) + πnOF )(u) if E/F is unramified,

char(I(wg) + ∆[ n2 ]OF )(u) if E/F is ramified.

is an eigenfunction of (G,ωα,χ) with eigencharacter η. In particular, Supp φη ⊂π−k+nOF if E/F is unramified, and Supp φη ⊂ π[−k+n

2 ]OF if E/F is ramified.Here Supp φ denotes the support of the function φ.

Proposition 1.7. Assume that E/F is unramified, n(ψ′) = 2n− 1 is odd, andthat n(η′) > 1. Then η occurs in ωα,χ if and only if n(η′) = 2k−1 is odd. In sucha case, there is w ∈ π−k+nO∗E−π−k+n(δOF +πOF ) such that η′(g) = ψ′(−wwg)

NONVANISHING OF CENTRAL HECKE L-VALUE 7

for g ∈ Gk. Moreover,

φη,w(u) =∑

g∈G′/Gk

(η′(g)λ(g))−1ψ(∆α

2I(wg−1)R(wg−1))ψ(−∆αR(wg−1)u)

· char(I(wg−1) + πnOF )(u)

+1√q

∑

g∈(G−G′)/Gk

(η′(g)λ(g))−1ψ(−∆αR(w)I(w))

· ψ(∆α

2y(xI(w)2 − 2I(w)u + xu2)) char(I(wg−1) + πn−1OF )(u)

is an eigenfunction of (G,ωα,χ) with eigencharacter η.

2. The central L-value.

Let D ≡ 7 mod 8 be a squarefree positive integer and let d ≡ 1 mod 4 be asquarefree integer relatively prime to D. Then there is unique decompositiond = d1d2 such that di are fundamental discriminants and that every primedivisor of d1 (d2) is split (inert) in E = Q(

√−D). It is allowed d or di = 1. Weview E as a subfield of C, and fix δ =

√−D = i√

D ∈ iR>0. Let χD,d be aHecke character of E defined in the introduction. Then there is a decompositionχD,d = χD,1η where χD,1 is a canonical character of E and η = (d ) ◦ NE/Q([Roh2-3]). Since η|Q∗A is trivial, there is a character η of E1\E1

A such that

η(z) = η(z/z). Let χ = χcan| |12A , Then χ|Q∗A = ε =

∏εl is the quadratic Hecke

character of Q associated to the Dirichlet character (−D ). For every integerk ≥ 0, let ηk = ηχk|E1

A, then ηk = χ2kη. We assume that the global root number

of χ2k+1η is ONE, i.e., (−1)k = sign(d). Then there is unique decomposition(up to order) D = D1D2 with Di > 0 and

(2.1) ε(12, (χηk)l,

12ψEl

)(χηk)l(δ) = εl(2d2

D2) = εl(

2d1

D1)

for every prime l ≤ ∞ (see [RVY, Lemma 3.1]). Here ψEl= ψl ◦ trEl/Ql

and ψl

is a ‘canonical’ additive character of Ql given by

ψl(x) =

{e2πix if l∞,

e−2πiλ0(x) if l -∞,

where λ0 : Ql −→ Ql/Zl ↪→ Q/Z. For a prime l|d2 with l ≡ 1 mod 4, we defineφl ∈ S(Zl) ⊂ S(Ql) via(2.2)

φl(u) = char(lZl)(u) +1

2G(ψ′′)

∑

A2−B2≡−D mod l

(B

l)ψ′′l (Au2) char(Zl)(u).

8 TONGHAI YANG

Here ψ′′l = − 2D1d2D2

ψl. For a prime l|d2 with l ≡ −1 mod 4, we defineφl ∈ S(Zl) ⊂ S(Ql) via

φl(u) = char(1 + lZl)(u)− char(−1 + lZl)(u) +1

G(ψ′′)

(2.3)

·∑

A2−B2≡−D mod l

(B

l)ψ′′l (Au2 − 2Bu + A) char(Zl)(u).

For an integer a > 0, we also define a theta function

(2.4) θd,k,a(z) = (Im z)−k2

∑

(x,d1)=1

(d1

x)∏

l|d2

φl(x

4d1D1a)Hk(x

√Im z)eπix2z.

Here Hk is k-th Hermite polynomial defined by (1.3). Notice that θd,k,a is verysimple and independent of a when d2 = 1.

Theorem 2.1. Let CL(E) be the ideal class group of E, and let s = s(d) bethe number of prime factors of d. For every ideal class C ∈ CL(E), choose aprimitive ideal A ∈ C−1 relatively prime to 2d, and write

A2 = [a2,−b +

√−D

2], a > 0,

withb ≡ r mod 8d2

1, b ≡ 0 mod D1d2,

where r is a fixed square root of −D mod 16d21. Then

(2.5) L(k + 1, χ2k+1D,d ) = κ|

∑

C∈ CL(E)

(d1a )

(χD,d)2k+1(A)θd,k,a(τA,D1)|2.

Here

κ =2

12−sπk+1

∏l|d2

(1 + l−1)

k!√

D∏

l|d2< φl, φl >

(√

D2

d21d2

√D1

)2k+1

2 ,

and

τA,D1 =b +

√−D

4d21d2D1a2

∈ E.

Proof. (sketch) The proof is similar to that of [RVY, Theorem 3.2] and is basedon the main formula in [Ya1]. Applying [Ya1, Theorem 2.15] to the datum(χ, ηk, δ, ψ, α = 4

d2D2), one has

L(k + 1, χ2k+1D,d )

L(1, (−D ))=

L(12 , χηk)

L(1, (−D ))= c|θφ(ηk)(1)|2.

NONVANISHING OF CENTRAL HECKE L-VALUE 9

Here c is an explicit constant, φ =∏

φl ∈ S(QA) is a Schwartz function onQA given below, and θφ(ηk)(1) is an integral over E1\E1

A given by theta liftingfrom unitary group of one variable to itself. When l is split, φl is given by [Ya1,(2.29)-(2.30)]. When l = ∞, φl = φk is given by Lemma 1.1. When l is finiteand nonsplit (so l 6= 2), φl is given by Corollary 1.4 and Proposition 1.5. Moreprecisely, when l|D, φl = 1√

lcharl−1Zl, and when l|d2, φl is given by (2.2) or

(2.3). Finally, if l - d2∞ is inert in E, φl = char(Zl). In [RVY, section 1], wegave a method to compute θφ(ηk)(1) in terms of φ ([RVY, Corollary 1.4 andProposition 1.7]). Applying the method to this situation, we obtain the desiredformula (after some computation). The case d2 = 1 was computed in [RVY].

Combining this with a theorem of Shimura and a trick of Rohrlich (see [Roh2]for detail), one has

Theorem 2.2. Notation as in Theorem 2.1. Assume that (2k + 1, hD) = 1.Then the following are equivalent.

(1) The central L-value L(k + 1, (χD,d)2k+1) = 0.

(2) For every ideal class C ∈ CL(E), and a (and any) primitive idealA ∈ C−1, relatively prime to 2d, τA,1 is a root of the theta function θd,k,a,a = NA.

(3) The global theta lifting θα(ηk) (with respect to (α = 2d2D2

, χ, ψ, δ)) van-ishes.

We remark that τA,1 does not depend on the decomposition D = D1D2

associated to formula (2.1). This is because if the central L-value for one choiceof the canonical Hecke character vanishes, it will vanish for any choice of thecanonical Hecke character by a theorem of Rohrlich ([Roh2]).

3. The proof of the main thoerem.

First we notice that the functions φl defined via (2.2) and (2.3) can be viewedas functions on Fl. Indeed, one has

φl(u) = δ0,u +1

2G(ψ′′)

∑

A2−B2=−D

(B

l)ψ′′(Au2)(3.1)

= δ0,u +1

2G( 12ψ′′)

∑

x∈F∗l(x + D

x

l)ψ′′(

12(x− D

x)u2)

10 TONGHAI YANG

for l ≡ 1 mod 4, and

φl(u) = δ1,u − δ−1,u +1

G(ψ′′)

∑

A2−B2=−D

(B

l)ψ′′(Au2 − 2Bu + A)

(3.2)

= δ1,u − δ−1,u +1

G( 12ψ′′)

∑

x∈F∗l(x + D

x

l)ψ′′(

12(x(u− 1)2 − D(u + 1)2

x))

for l ≡ −1 mod 4. Here u ∈ Fl and δa,u is the Kronecker symbol. Also theequality A2 − B2 = −D is in Fl. Recall ψ′′ = − 2

d2Dψl (we take D1 = 1 by theremark in the end of section 2).

Lemma 3.1. Assume l ≡ 1 mod 4 and write l = a2 + b2 with b being a positiveeven integer. Then φl(0) = 1± b√

l. For u ∈ F∗l , one has φl(u) 6= 0.

Proof. By (3.1), one has

φl(0) = 1± 12√

l

∑

x∈F∗l(x3 + Dx

l).

Let AD be the elliptic curve defined by y2 = x3 + Dx. Then

#AD(Fl) = l + 1 +∑

x∈F∗l(x3 + Dx

l).

On the other hand, it is well-known ([Sil, page 185]) that

#AD(Fl) =

{l + 1± 2a if (−D/l) = 1,

l + 1± 2b if (−D/l) = −1.

Therefore φl(0) = 1± b√lin our case. For u ∈ F∗l , ψ′′((x−D

x )u2) ≡ 1 mod (1−ζl).So

±2G(ψ′′)φl(u) ≡∑

x∈F∗l(x + D

x

l) ≡ ±2b 6≡ 0 mod (1− ζl).

In particular, φl(0) 6= 0.

Lemma 3.2. Assume l ≡ −1 mod 4. Then φl(0) = 0 and φl(u) 6= 0 for everyu ∈ F∗l . Moreover, there is a map j : F∗l −→ C∗ such that

(3.3) j(ab) = j(a)j(b)σa2 ,

NONVANISHING OF CENTRAL HECKE L-VALUE 11

and

(3.4) φl(u) = j(a)φl(u

a)σa2

for every a, u ∈ F∗l . Here σa ∈ Gal(Q(ζl)/Q) is given via ζσa

l = ζal .

Proof. Obviously,

φl(0) =1

G(ψ′′)

∑

A2−B2=−D

(B

l)ψ′′(A) = 0

since (−1l ) = −1. For every a ∈ F∗l , define

(3.5) φl,a(u) = δa,u − δ−a,u +1

G(ψ′′)

∑

A2−B2=−D

(B

l)ψ′′(Au2 − 2Bau + Aa2)

for u ∈ Fl. One has φl,1 = φl. One can view φl,a ∈ S(Zl) ⊂ S(Ql) viaφl,a(u) = φl,a(u mod l) for u ∈ Zl. By Proposition 1.7, φl,a 6= 0 is also aneigenfunction of ωα,χ,l with eigencharacter (ηk)l, where α = 4

d2D2is as in the

proof of Theorem 2.1. By the multiplicity one theorem, there is a unique nonzerocomplex number j(a) such that

φl(u) = j(a)φl,a(u)

for every u ∈ Fl. On the other hand, one has

φl,a(u) = φl(u

a)σa2 .

This proves (3.4). It follows easily from (3.4) that φl(u) 6= 0 for every u ∈ F∗l .Applying (3.4) twice, one gets (3.3) (since φl(u) 6= 0).

The following lemma can be checked by standard method in exponential sums(see [Li, chapter 6] for example) and is left to the reader.

Lemma 3.3. Let ai ∈ Fp, i = 1, 2, 3, and a1 ∈ F∗p. Let ψ be a nontrivial additivecharacter of Fp. Then

(3.6)

∣∣∣∣∣∣∑

x∈F∗p(x + a1/x

l)ψ(a2x + a3/x)

∣∣∣∣∣∣< b

√p.

Here

b =

4 if a2a3 6= 0,

2 if a2 = a3 = 0,

3 otherwise .

Now we proceed to prove the main theorem in the introduction. We divideit into two theorems.

12 TONGHAI YANG

Theorem 3.4. Assume that D ≡ 7 mod 8 and d ≡ 1 mod 4 are two positivesquarefree integers such that every positive factor of d is inert in ED and iscongruent to 1 modulo 4. Let k ≥ 0 be an even integer, and let hD be the idealclass number of ED = Q(

√−D). Then there exists a constant M = M(k),independent of the pair (D, d), such that if (hD, 2k + 1) = 1, and

√D ≥ 12

πd(log d + M(k)),

then the central L-value L(k + 1, χ2k+1D,d ) > 0. One can take M(0) = 0.

Proof. By Theorem 2.2 and the assumption, it suffices to show that τ = b+√−D4d

is not a root of the theta function

θd,k,1(z) = (Im z)−k2

∑

x∈Zφ(x)Hk(x

√Im z)eπix2z.

Here we denote φ(x) =∏

l|d φl(x4 ). By Lemma 3.3, one has

(3.7) |φ(x)| ≤∏

l|d2 ≤

√d.

By Lemma 3.1, one has

(3.8) |φ(0)| ≥∏

l|d

12l≥ d−

32 .

Set c = e−π√

D4d d3 and assume c < 1. Since Hk is a polynomial of x of degree k,

there is a constant C1 = C1(k) > 0 such that

|Hk(x)| ≥ C1|x|k for |x| ≥ π−1.

So

(3.9) |Hk(n√

Im τ)| ≥ C1nkπ−

k2 (− log c + 3 log d)

k2 ,

when c ≤ 1e and n is a positive integer. Set C2 = |Hk(0)| > 0 (k is even).

Combining (3.7)− (3.9), one has

( Im τ)k2 |θd,k,1(τ)|

≥ C2d− 3

2 − 2C1π− k

2 d12 (− log c + 3 log d)

k2

∞∑n=1

nkd−3n2cn2

≥ d−32 f(c)

NONVANISHING OF CENTRAL HECKE L-VALUE 13

where

f(x) = C2 − 2C1π− k

2 x(C3 − log x)k2

∞∑n=1

nkxn2−1,

and C3 ≥ 1 is chosen such that 3 log x < x2k for x > C3. Here we have used the

inequality− log c + 3 log d

d2k

< C3 − log c.

Notice that f(x) is independent of D or d. Since f(0) = C2 > 0, there is aconstant 0 < C4 < 1/e such that f(x) > 0 for 0 < x < C4. Therefore, whenc < C4, i.e.,

√D > 12

π d(log d− 13 log C4), one has

θd,k,1(τ) 6= 0,

and so L(k + 1, χ2k+1D,d ) > 0. Taking M(k) = − 1

3 log C4, we have proved thegeneral statement of the theorem. When k = 0, Hk = 1, similar but simplerargument gives

|θd,0,1(τ)| ≥ d−32 − 2d

12

∞∑n=1

cn2

≥ d−32 (1− 2d2 c

1− c)

for c = e−π√

D4d < 1 (different from the c used above). So for c < 1

d3 < 12d2+1 (we

may assume d > 1, the case d = 1 is trivial), or, equivalently,√

D ≥ 12π d log d,

one hasθd,0,1(τ) 6= 0.

So we can take M(0) = 0. This proves the theorem

Theorem 3.5. Let D ≡ 7 mod 8 be a positive squarefree integer, and let d ≡1 mod 4 be a squarefree integer not satisfying the special condition in Theorem3.4. Let k ≥ 0 be an integer. Then there is a constant M(k), independent of thepair (D, d), such that if sign(d) = (−1)k, (2k + 1, hD) = 1, and

√D ≥ 4

πd2(log |d|+ M(k)),

then the central L-value L(k +1, χ2k+1D,d ) > 0. One can take M(k) = 0 for k ≤ 1.

Proof. When d does not satisfy the special condition in Theorem 3.4, the thetafunction θd,k,1 does not have a constant term. As in section 2, we write d = d1d2

such that every prime factor of d1 is split in ED and every prime factor of d2 is

14 TONGHAI YANG

inert in ED. As before, it is sufficient to prove that τ = b+√−D

4d21d2

is not a root ofthe theta function θd,k,1 given by (2.4) with D1 = 1. Set φ(x) =

∏l|d φl( x

4d1).

By Lemma 3.3, one has |φl(x)| ≤ 4 and so

(3.10) |φ(x)| ≤ 4|d2|

(4 < l except for l = 3). Notice that√

lφl(x) is an algebraic integer in the l-thcyclotomic field Q(ζl). It is not difficult to see from this fact and Lemmas 3.2and 3.3 that

|√

lφl(x)| ≥∏

1 6=σ∈Gal(Q(ζl)/Q)

|(√

lφl(x))σ|−1 ≥ (4√

l)1−l

for x ∈ Z∗p. So

(3.11) |φ(x)| ≥ 4|d2|− 32 |d2|,

for x ∈ Z∗p. Set c = e−π√

D4d2 |d|. As in the proof of Theorem 3.4, there is a

constant C1 = C1(k), independent of (D, d), such that

(3.12) |Hk(n√

Im τ)| ≤ C1nkπ−k/2|d2|k/2(− log c + log |d|)k/2

when n is a positive integer and 0 < c < 1. On the other hand, since Hk hasonly finite many roots, there are positive constants C2 and C3 < 1 such that

|Hk(x)| ≥ C2 for |x| >√− 1

πlog C3.

So

(3.13) |Hk(√

Im τ)| ≥ C2 for c ≤ C3.

Combining (3.10)− (3.13), one has then

( Im τ)k2

2|θd,k,1(τ)|

≥ 4C2|d2|− 32 |d2|c− 4C1π

−k/2|d2|k/2+1(− log c + log |d|)k/2∞∑

n=2

nk|d|−n2|d2|c|d2|n2

≥ 4|d2|− 32 |d2|cf1(c, d))

where

f1(c, d) = C2 − π−k2 C1

|d2|k/2+1

|d| 52 |d2|−1(− log c + log |d|

|d| 2k )k2

∞∑n=2

nkcn2−1.

NONVANISHING OF CENTRAL HECKE L-VALUE 15

Choose C4 ≥ 1 so that log x < x2k for x > C4. Then

− log c + log |d||d| 2k ≥ C4 − log c.

Notice that |d2|k/2+1

|d| 52 |d2|−1is bounded above as a function of d. So there is a constant

C5 = C5(k), independent of the pair (D, d), such that f1(c, d) ≥ f(c) where

f(x) = C2 − C5x3(C4 − log x)

k2

∞∑n=2

nkxn2−4.

Now the same argument as in the proof of Theorem 3.4 gives the general state-ment of this theorem. Similar argument to the last part in the proof of Theorem3.4 (together with slightly better lower bound for |φl(x)|) shows that one cantake M(k) = 0 for k ≤ 1. We leave the detail to the reader.

Remark 3.6. When k = 0, the main theorem claims that for all the pairs (√

D, d)in the region above

√D > 12

π d2 log d with D ≡ 7 mod 8 and d ≡ 1 mod 4squarefree, the central L-value L(1, χD,d) > 0. This is strong considering thegeneral belief that whether an L-function vanishes at its center is tricky andhard to tell.

When every prime factor of d is split in ED one can drop log d from thecondition ([RVY, Theorem ]), and when every factor of D is congruent to 1modulo 4 and is inert in ED one can replace d2 by d (Theorem 3.4). A naturalquestion is what, if any, is the ultimate inequality to guarantee the nonvanishingof L(1, χD,d). Can that be

√D > M log d for some constant M?

Remark 3.7. Recall that χD,d = χD,1ηd where ηd = (d ) ◦NED/Q, and χD,1 is acanonical Hecke character of ED. Although χD,d can be viewed as a quadratictwists of χD,1, it might be better to view it as a ‘quadratic’ twist of (d ) by theimaginary quadratic field ED (see [Lie] and [RVY] for similar ideas). The resultof Montgomery and Rohrlich ([MR]) mentioned in the introduction is then thatthe central L-value of a ‘quadratic’ twist χD,1 of the trivial character by ED

does not vanish unless it is forced to by its functional equation. Is the same truefor other ‘small’ d? The following proposition gives a partial affirmative answerto the question for d = 5 (we don’t consider the case when D is even).

Proposition 3.8. Let D ≡ 7 mod 8 be a squarefree positive integer relativeprime to 5. Then the central L-value

L(1, χD,5) 6= 0,

16 TONGHAI YANG

Proof. First we assume that (D/5) = 1, i.e., D ≡ ±1 mod 5. By Theorem 2.2it is sufficient to prove that τ = b+

√−D200 is not a root of

θd(z) =∑

(x,5)=1

(5x

)e2πx2z.

Here b is some integer. Set c = e−π√

D100 . Since D ≡ 7 mod 8 and D ≡ ±1 mod 5,

one has D ≥ 31. This implies c < .84, which is enough to guarantee

12|θ(τ)| > c−

∑n>1

cn2> c− c4 − c9

∞∑n=0

c7n > 0.

So L(1, χD,5) 6= 0 in this case. Now we turn to the case (D/5) = −1, i.e.,D ≡ a mod 5 with a = ±2. one can show by (3.1) that

(3.14) φ5(u) =

{1 + a√

5if u ≡ 0 mod 5,

− a√5

cos π√5

if u 6≡ 0 mod 5.

By Theorem 2.2, it is sufficient to prove that τ = b+√−D20 is not a root of the

theta functionθ5,0,1(z) =

∑

x∈Zφ5(x/4)eπix2z.

Here b is any integer satisfying

b ≡ 0 mod 5, and b2 ≡ −D mod 16.

Set c = e−π√

D20 ≤ e−

π√

720 < .66, and b′ = b/5 ∈ Z. Write Im z for the imaginary

part of the complex number z. By (3.14), φ5(x) depends only on whether x ≡0 mod 5, and |φ5(0)| < 3|φ5(1)|. So

Im (θ5,0,1(τ)) = 2φ5(1)∑

(n,5)=1,n>0

Im (eπin2b′

4 )cn2+2φ5(0)

∑

5|n,n>0

Im (eπin2b′

4 )cn2.

Notice that Im (eπin2b′

4 ) = 0 or ±√

22 depending on whether 2 divides n or not.

Therefore (c > .66)

1√2|φ5(1)| |Im (θ5,0,1(τ))| > c− 3

∞∑n=3

cn2> c− 3c9

∞∑n=0

c7n > 0.

In particular, θ5,0,1(τ) 6= 0, and thus L(1, χD,5) 6= 0.

NONVANISHING OF CENTRAL HECKE L-VALUE 17

Appendix. Proof of Propositions 1.2 and 1.5.

Let the notation be as in section 1. we first recall some basic facts on theWeil representation ωα,χ of G = U(1) on S(F ). First, there is an embedding

(A1) ıα : G −→ Sp(1) = SL2(F ), g = x + yδ 7→(

x ∆2αyy

∆α x

).

Let rS be Rao’s standard section of Sp(1) on S(F ) and let c be the corresponding

standard 2-cocycle ([Rao]). For gi =(

ai bi

ci di

)∈ Sp(1) with g1g2 = g3, one

has ([Rao, Corollary 4.3])

(A2) c(g1, g2) =

1 if c1c2c3 = 0,

γF (12c1c2c3ψ) otherwise .

Here γF is the local Weil index ([Wei], [Rao, Appendix]). For g = x + yδ ∈ G,define

(A3) µ(g) = χ(δ(g − 1))γF (αy(1− x)ψ)(∆,−2y(1− x))F .

Then ωα,χ(g) = µ(g)rS(ıα(g)) defines a Weil representation of G ([Ku, Proposi-tion 4.8]). Finally, when n(ψ′) = 2n− 1 is odd, let ([Ya2, Theorem 3.5])

(A4) λ(g) =

(x

F) if g ∈ G′,

γF (∆αy

2ψ) if g /∈ G′.

Then ω(g) = λ(g)−1rL(g) is a Weil representation of G on S(L,ψ), where rL isthe action of G on S(L,ψ) defined via [Ya2, (3.2)]. When n(ψ′) = 2n is even,the Weil representation ω of G on S(L,ψ) is just the right translation.

Lamma A1. (a) If E/F is ramified, then

ξ(g) = µ(g)c(ıα(g), w).

(b) If E/F is unramified and n(ψ′) is even, then

ξ(g) =

{µ(g)c(ıα(g), w) if g ∈ G′

µ(g) if g /∈ G′.

(c) If E/F is unramifed and n(ψ′) is odd, then

ξ(g) =

{λ(g)µ(g)c(ıα(g), w) if g ∈ G′

λ(g)µ(g) if g /∈ G′.

18 TONGHAI YANG

Proof. We only prove Claim (c). The proof of Claims (a) and (b) is similar(simpler) and is left to the reader. First note L ∩ F = πn−1OF . Let f0 bethe characteristic function of L. An easy calculation shows that ρ(f0) is thecharacteristic function of πnOF .

Assume first that g = x + yδ ∈ G′. Then y ∈ πOF and x ∈ O∗F . Let

w =(

0 −11 0

), and write

(A5) ıα(g)w =(

x−1 ∆2xy0 x

)w

(1 − y

∆αx0 1

)= AwB

where A and B have the obvious meanings. Set

f1(u) = rS(w)ρ(f0)(u) =∫

πnOF

ψ(−uv) dv.

where dv is the self-dual Haar measure on F with respect to ψ. Straightforwardcalculation using [Rao, Theorem 3.6] gives

rS(AwB)f1(u) = ρ(f0)(u).

Therefore

ωα,χ(g)ρ(f0)(u) = µ(g)c(ıα(g), w)rS(ıα(g)w)f1(u)

= µ(g)c(ıα(g), w)rS(AwB)f1(u)

= µ(g)c(ıα(g), w)ρ(f0)(u)

= µ(g)c(ıα(g), w)λ(g)−1ρ(ω(g)f0)(u).

Combining this with (1.7), one has ξ(g) = λ(g)µ(g)c(ıα(g), w).

Next, assume that g = x + yδ /∈ G′, so y ∈ O∗F . In this case

(A6) ıα(g) =(

∆αy x

0 y∆α

)w

(1 ∆αx

y

0 1

).

Direct calculation using [Rao, Theorem 3.6] and (A6) gives

(A7) ωα,χ(g)ρ(f0)(u) =µ(g)√

qψ(

∆αx

2yu2) char(πn−1OF ).

On the other hand, By [Ya2, Lemma 3.2 and Theorem 3.5], one has

ω(g)f0(z) =

0 if z /∈ Ln−1

λ−1(g)√q

ψ′(δuv)ψ′(δu2xy−1) if z = uδ + v ∈ Ln−1.

NONVANISHING OF CENTRAL HECKE L-VALUE 19

So one has by (1.6)

(A8) ρ(ω(g)f0)(u) =1

λ(g)√

qψ(

∆αx

2yu2) char(πn−1OF )(u).

Combining (1.7) with (A7) and (A8), one has ξ(g) = λ(g)µ(g). Claim (c) isproved.

Proof of Proposition 1.2 First, we assume g ∈ G1, so x ≡ 1 mod π,y ≡ 0 mod π, and

x− 1 =∆y2

x + 1.

So

ξ(g) = χ(δ(g − 1))γF (−2αxy∆ψ)γF (2αxy∆ψ)(∆, ∆y)

= χ(δ(g − 1))(∆,−y).

In particular, if n(χ) ≤ 1, then g − 1 = δy(1 + δyx+1 ). So

χ(δ(g − 1)) = χ(∆y) = (∆, ∆y) = (∆,−y).

Therefore ξ(g) = 1. This proves (4) and the first part of the first three claims.

Next, we assume g ∈ G′ − G1, i.e., g ≡ −1 mod πE , or equivalently x ≡−1 mod π, y ≡ 0 mod π. In such a case, one has

(A9) µ(g)c(ıα(g), w) = χ(δ(g − 1))γF (2αyψ)γ(−2αy∆ψ)(∆,−y)F .

When E/F is ramified, we may assume π = ∆. one has by (A9) and LemmaA1,

ξ(g) = χ(δ(g − 1))γF (−∆ψ)γF (ψ)(∆,−2α).

Given a character ψ of F of conductor n, one defines a character

ψ : OF /πOF −→ C∗, x mod πOF 7→ ψ(πn−1x).

Write G(ψ) for the Gauss sum of ψ. If n is odd, then n(∆ψ) is even. By [Rao,A11, A2] one has

γF (−∆ψ)γF (ψ) = (ε(−1))1−n G(ψ)|G(ψ)| = ε(

12, εE/F , ψ).

This proves (1.8). When n(χ) ≤ 1, χ(δ(g − 1)) = χ(−2δ), and so

ξ(g) = χ(δα)ε(12, εE/F , ψ).

20 TONGHAI YANG

Applying [Roh1, Propositions 3 and 8], one has (1.9). The unramified case issimilar and is left to the reader.

Finally, we assume that g ∈ G−G′. So x± 1 ∈ O∗F and y ∈ O∗F . Also E/Fmust be unramified in this case by [Ya2, Lemma 1.1].

If n(ψ′) = n(αψ) is even, then one has by [Rao, appendix]

ξ(g) = µ(g) = χ(δ(g − 1))γF (y(1− x)αψ)(∆,−2y(1− x))F

= χ(δ(g − 1)).

If n(ψ′) = n(αψ) is odd, then one has by [Rao, appendix]

ξ(g) = χ(δ(g − 1))γF (y(1− x)αψ)γF (2y∆αψ)(∆,−2y(1− x))F

= χ(δ(g − 1))γF (y(1− x)αψ)γF (2y∆αψ)

= χ(δ(g − 1))(

2∆(x− 1)F

).

This completes the proof of Proposition 1.2. To prove Proposition 1.5, one needs

Proof of Proposition 1.5. For z ∈ E, we write z = R(z) + I(z)δ withR(z) and I(z) ∈ F . Given w ∈ E, let fw be the unique function in S(L,ψ) suchthat Supp (fw) = w + L and fw(w) = 1. Integrating (1.6) for fw, one has

ρ(fw)(u) = ψ(∆α

2R(w)I(w))ψ(−∆αR(w)u)

·{

char(I(w) + πnOF )(u) if E/F is unramified,

char(I(w) + ∆[ n2 ]OF )(u) if E/F is ramified.

(A11)

By [Ya2, Theorem 0.4] and Corollary 1.3,

ρ(φ′) = ρ(fw) + η′(−1)(−1F

)ρ(f−w)

+∑

g∈G/G′,g 6=1

η(g)−1ωα,χ(g)ρ(fw + η′(−1)(−1F

)f−w)

is an eigenfunction of (G,ωα,χ) with eigencharacter η if it is nonzero, wherew ∈ Ln−1. When g = x + yδ ∈ G − G′, one has by (A6), (A11), and [Rao,Theorem 3.6]

ωα,χ(g)ρ(fw)(u) =µ(g)ψ(−∆α

2 R(w)I(w))√q

ψ(∆α

2y(xI(w)2 − 2I(w)u + xu2)) char(πn−1OF )(u).

NONVANISHING OF CENTRAL HECKE L-VALUE 21

Putting things together, and applying (Lemma A1), one has proved that

φη,w(u) = η′(−1)(−1F

)ψ(∆αR(w)u) char(−I(w) + πnOF )(u)

+ ψ(−∆αR(w)u) char(I(w) + πnOF )(u) +ψ(−∆αR(w)I(w))√

q∑

g∈G/G′,g 6=1

(η′(g)λ(g))−1

{ψ(

∆α

2y(xu2 − 2I(w)u + xI(w)2))

+η′(−1)(−1F

)ψ(∆α

2y(xu2 + 2I(w)u + xI(w)2))

}char(πn−1OF )(u)

is an eigenfunction of (G,ωα,χ) with eigencharacter η if it is nonzero.

When η′(−1) = (−1F

), set w = 0, and applying [Ya2, Theorem 3.5] for λ, onegets

12φη,0(u) = char(πnOF )(u) +

12G(ψ′′)

∑

g∈G/G1,g 6=±1

η′(g)−1(I(g)F

)ψ(∆αR(g)2I(g)

u2) char(πn−1OF )(u).

It is not difficult to see that a 7→ g(a) = δ+aδ−a gives a bijection between the

projective line P 1(F ) and G/G1. Set A = R(g)/I(g), and B = 1/I(g). Thenfor g = g(a)

A =12(a +

∆a

), and B =12(−a +

∆a

).

It is easy to check that a 7→ (A,B) is a bijection between F ∗ and (A,B) ∈ F 2

with A2 −B2 = ∆. Therefore,

12φη,0(u) = char(πnOF )(u) +

12G(ψ′′)

·∑

A2−B2≡∆ mod π

η′(A + δ

B)(

B

F)ψ(

∆α

2Au2) char(πn−1OF )(u)

= φ′η(π1−nu) = φη(u),

is the function sought in proposition 1.5(1). It remains to prove that it isnonzero. But

φη(0) = 1 +1

2G(ψ′′)

∑

A2−B2≡∆ mod π

η′(A + δ

B)(

B

F) 6= 0

22 TONGHAI YANG

since G(ψ′′) /∈ Q(e2πiq+1 ) and the sum is in Q(e

2πiq+1 ). This proves (1).

When η′(−1) = −(−1F

), and η′ 6= η0, φη,w = 0 for every w ∈ L. So there isw = πn−1a ∈ Ln−1 − L with a ∈ O∗F such that φη,w 6= 0. A simple manipu-lation shows that φη,w(u) = φ′η,a(πn−1u) is the function sought in Proposition1.5(2). It remains to prove that φ′η,a 6= 0 for every a ∈ O∗F . We can identifyGal(Q(ζq, ζq+1)/Q(ζq+1)) with F ∗ via b 7→ σb. Here ζσb

q = ζbq for a n-th primi-

tive root ζq of 1. It is easy to check that φ′η,a can be viewed as a function on Fwith values in Q(ζq, ζq+1), and that

(A12) φ′η,a(u) = φη,1(u/a)σa2 .

So one φ′η,a 6= 0 implies every φ′η,a 6= 0. This proves Claim (2).

Proposition 1.6 follows easily from Corollary 1.3 and [Ya2, Theorem 1.1]. Theproof of Proposition 1.7 is similar to that of Proposition 1.5.

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[Ya1] Tonghai Yang, Theta liftings and L-functions of elliptic curves, thesis at the Uni-versity of Maryland (part of it appeared in the Crelle, 485(1997) 25-53) (1995).

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109

E-mail address: thyang@math.lsa.umich.edu