Non-vanishing of the Central Derivative of Canonical Hecke L-functions

Stephen D. Miller and Tonghai Yang. February 29, 2000

Let K = ℚ(\sqrt{−D}) be an imaginary quadratic field of discriminant −D < −4, O its ring of integers, and h its ideal class number. A Hecke character χ of K of conductor 𝔣 is a called “canonical” ([Ro1]) if

χ(𝔞̄) = \overline{χ(𝔞)} for each ideal 𝔞 relatively prime to 𝔣.  (1.1)
χ(αO) = ±α for principal ideals αO relatively prime to 𝔣. (1.2)
The conductor 𝔣 is divisible only by primes dividing D.  (1.3)

Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratic twist of a canonical Hecke character (see Section 2 for a precise description of these characters and which fields have them).

Let L(s, χ) denote the Hecke L-function of χ, and Λ(s, χ) its completion; Λ(s, χ) satisfies the functional equation Λ(s, χ) = W (χ)Λ(2 − s, χ), where W (χ) = ±1 is the root number. If χ is a canonical Hecke character with W (χ) = 1, then the central value Λ(1, χ) ≠ 0 by a theorem of Montgomery and Rohrlich [MR]. Of course, it automatically vanishes when W (χ) = −1 by the functional equation. The main result of this paper is

Theorem 1.1. Let χ be a canonical Hecke character whose root number W (χ) = −1. Then the central derivative Λ′(1, χ) ≠ 0.

In Theorem 2.2 we also prove that Λ′(1, χ) ≠ 0 when χ is a small quadratic twist of a canonical character with W (χ) = −1.

When D = p is a prime, canonical Hecke characters are closely connected with the elliptic curves A(p) extensively studied by Gross [Gr]. These curves are defined over F = ℚ(j(\frac{1+\sqrt{−p}}{2})), where j is the usual modular j-function, and have complex multiplication by O. Combining Theorem 1.1 and the above result of [MR] with Gross-Zagier [GZ] and Kolyvagin-Logachev [KL], one has

Corollary 1.2. Let p > 3 be a prime congruent to 3 modulo 4. Then (a) The Mordell-Weil rank of A(p) is

rank_{ℤ}A(p)(F) = \{ \begin{matrix}h, & p ≡ 3 (mod 8) \\ 0, & p ≡ 7 (mod 8). \end{matrix}

(b) The Shafarevich-Tate group � (A(p)/F) is finite.

In [Gr], Gross proved part (a) when p ≡ 7 (mod 8) using a 2-descent.

In the next section we will outline the proof of Theorem 1.1 and an analog for quadratic twists (Theorem 2.2). Sections 3, 4, and 5 are devoted to analytic estimates used in the proofs of the theorems. We conclude in Section 6 with the proof of Corollary 1.2 and other arithmetic applications.

Acknowledgements

For very large D, Theorem 1.1 was originally obtained by D. Rohrlich [Ro4]. We are indebted to him for sharing his method with us, as well as for allowing us to mention several of his results here. Also, we thank him for his inspiration and for suggesting this problem to us. We are grateful to M. Baker, B. Conrad, N. Elkies, B. Gross, K. Rubin, Z. Rudnick, D. Zagier, and S. Zhang for discussions, and to Harvard University for their hospitality. S.M. was partially supported by an NSF post-doctoral fellowship as well as a Yale Hellman fellowship. T.Y. was partially supported by an AMS Centennial fellowship and NSF grant DMS-9700777.

2 Notation and Strategy

We first recall some facts about canonical Hecke characters from [Ro2]. They exist if and only if D ≡ 3 (mod 4) or is a multiple of 8. Multiplying a canonical character by an ideal class character always yields another canonical character. This operation preserves the root number and defines natural families of canonical Hecke characters. When D ≡ 3 (mod 4), there is exactly one family and it has root number (\frac{2}{D}); when D is a multiple of 8, there are two families – one has root number 1 and the other has root number -1.

To avoid confusion, we will sometimes write χcan for a canonical Hecke character of K. In this paper we consider Hecke characters χ of K satisfying conditions (1.1) and (1.2), which are always of the form

χ_{D,d} = χcan · (ϵ_{d} ∘ N_{K/ℚ}).  (2.1)

Here d is a fundamental discriminant and ϵ_{d} = (^{\underline{d}}) is the quadratic dirichlet character with conductor d, prime to D. The root number W (χ) is explicitly computed in [Ro2]. In particular, when D is odd

W (χ_{D,d}) = (\frac{2}{D}) sign(d).  (2.2)

Non-vanishing of the central derivative of canonical Hecke L-functions 3

From now on we will assume that W (χ) = −1. Set

B = \sqrt{DN𝔣} = \{ \begin{matrix}D|d|, & D odd \\ 2D|d|, & 8 | D. \end{matrix}  (2.3)

The Hecke L-function is defined as

L(s, χ) = ∑_{𝔞 integral}χ(𝔞)(N𝔞)^{−s}

= ∑_{ideal classes C} L(s, χ, C),

where L(s, χ, C) is the partial L-series summed over integral ideals in C. Their completed L-functions are defined by

Λ(s, χ) = (\frac{B}{2π} )^{s} Γ(s)L(s, χ)

and

Λ(s, χ, C) = (\frac{B}{2π} )^{s} Γ(s)L(s, χ, C).

Lemma 2.1. When W (χ) = −1, Λ′(1, χ) = 0 if and only if Λ′(1, χ, C) = 0 for each ideal class C of K.

Proof: Associated to χ is a cuspidal new form f of weight 2 and level B^{2} such that L(s, f) = L(s, χ). So Corollary 2 of [GZ] implies that Λ′(1, χ) = 0 if and only if Λ′(1, χ^{σ}) = 0 for every σ ∈ Gal(ℚ̄/ℚ). On the other hand, by Theorem 1 of [Ro3],

{χ^{σ} : σ ∈ Gal(ℚ̄/K)} = {χϕ : ϕ is an ideal class character of K},

and L(s, χ) = L(s, χ̄) by (1.1). Thus Λ′(1, χ) = 0 if and only if Λ′(1, χϕ) = 0 for all ideal class characters ϕ of K. The ideal class characters are linearly independent and

L(s, χϕ) = ∑_{C} ϕ(C)L(s, χ, C),

so the lemma follows. □

To prove Λ′(1, χ) ≠ 0, it now suffices to show Λ′(1, χ, c_{1}) ≠ 0 for the trivial class c_{1} (i.e. the class of principal ideals). Since the root number W (χ) = −1, we have the functional equation

Λ(s, χ, c_{1}) = −Λ(2 − s, χ, c_{1}).  (2.4)

By Cauchy’s theorem

Λ′(1, χ, c_{1}) = \frac{1}{2πi} (∫_{2−i∞}^{2+i∞} Λ(s, χ, c_{1})\frac{ds}{(s − 1)^{2}} − ∫_{−i∞}^{+i∞} Λ(s, χ, c_{1})\frac{ds}{(s − 1)^{2}} ) .

Applying (2.4) we arrive at the formula

\frac{1}{2}Λ′(1, χ, c_{1}) = \frac{1}{2πi} ∫_{2−i∞}^{2+i∞} Λ(s, χ, c_{1})\frac{ds}{(s − 1)^{2}} .  (2.5)

It is clear from property (1.2) of χ that there is a quadratic character ϵ of (O/𝔣)^{∗} such that

χ(αO) = ϵ(α)α.  (2.6)

We can express L(s, χ, c_{1}) as a sum over real and complex ideals:

L(s, χ, c_{1}) = ∑_{n=1}^{∞} ϵ(n)n^{1−2s} + ∑_{n=1}^{∞} a_{n}n^{−s},  (2.7)

where

a_{n} = ∑ ; principal, integral^{N𝔞=n, 𝔞≠𝔞̄} χ(𝔞) = ∑_{u,v>0^{u^{2}+Dv^{2}=4n}} ϵ (\frac{u + \sqrt{−D}v}{2} ) u.  (2.8)

Let

f(x) = \frac{Γ(0, x)}{x} = \frac{1}{x} ∫_{x}^{∞} e^{−t} \frac{dt}{t}  (2.9)

be the inverse Mellin transform of \frac{Γ(s)}{(s−1)^{2}} . Indeed

∫_{0}^{∞} f(x)x^{s} \frac{dx}{x} = ∫_{0}^{∞} ∫_{x}^{∞} x^{s−1}e^{−t} \frac{dt}{t} \frac{dx}{x}
= ∫_{0}^{∞} ∫_{0}^{t} \frac{dx}{x}x^{s−1}e^{−t} \frac{dt}{t}
= \frac{1}{s−1} ∫_{0}^{∞} t^{s−1}e^{−t} \frac{dt}{t} = \frac{Γ(s − 1)}{(s − 1)} = \frac{Γ(s)}{(s − 1)^{2}} ,

so

f(y) = \frac{1}{2πi} ∫_{Re(s)=2} \frac{Γ(s)}{(s − 1)^{2}} y^{−s}ds.  (2.10)

Combining (2.5), (2.7), and (2.10) we obtain

\frac{1}{2}Λ′(1, χ, c_{1}) = \overbrace{∑_{n=1}^{∞} ϵ(n)n · f(2πn^{2}/B)}^{R} + \overbrace{∑_{n=1}^{∞} a_{n}f (\frac{2πn}{B})}^{C} .  (2.11)

Formula (2.11) is essentially due to Rohrlich ([Ro4]), except that he expressed R in terms of dirichlet L-functions.

Examples: D = 8 and 11

We will now illustrate (2.11) with the first two discriminants which occur. Since both ℚ(\sqrt{−8}) and ℚ(\sqrt{−11}) have class number 1,

Λ′(1, χ, c_{1}) = Λ′(1, χ).

In order to compute it using (2.11), we must first describe the character ϵ : (O/𝔣)^{∗} → \{±1\}. When D = 8, 𝔣 = 2\sqrt{−D}O = ℤ8 ⊕ ℤ\sqrt{−32}, and (O/𝔣)^{∗} is generated by (ℤ/8)^{∗} and 1 + \sqrt{−2}. The character ϵ(n) must restrict to (\frac{−8}{n}) for n ∈ (ℤ/8)^{∗}, and is thus determined by its value on 1 + \sqrt{−2}. In fact, W (χ) = ϵ(1 + \sqrt{−2}), so in our case the values of ϵ on the relatively-prime residue classes are given in the following chart:

ϵ(u + v\sqrt{−2}) u 1 3 5 7
v
0  1 1 -1 -1
1  -1 -1 1 1
2  -1 -1 1 1
3  -1 -1 1 1

When D = 11, ϵ (\frac{u+\sqrt{−11}v}{2}) = (\frac{2u}{11}). We now compute L′(1, χ) for these canonical characters, and check (2.11) by comparing the known values of L′(1, E) for the associated elliptic curves E.

D = 8  D = 11
Term R with n^{2} ≤ 50  1.82582357875147  0.81497705252487
Term C with n ≤ 50  -0.28596530872740  -0.0600975766040368
L′(1, χ) = (\frac{2π}{B}) Γ(1)Λ′(1, χ) 1.209401857169272  0.862372296690396
= \frac{4π}{B}(R + C)
Associated curve E  y^{2} = x^{3} + 4x^{2} + 2x  y^{2} + y = x^{3} − x^{2} − 7x + 10
([Cr], curve 256A)  ([Cr], curve 121B)
L′(1, E) from [Cr]  1.2094018572  .8623722967

Proof of Theorem 1.1

By Lemma 2.1 and (2.11), it suffices to prove R > |C|. In the next section, we will prove that R is bounded below by

R > ∑_{n=1}^{∞} λ(n)n · f (\frac{2πn^{2}}{B} )
> .5235B − .8458B^{3/4} − .3951B^{1/2}.  (2.12)

Here λ(n) is Liouville’s function – the completely multiplicative function which is -1 at each prime.

In Section 4 we consider the special case d = 1, and bound term C by Proposition 4.1:

|C| < \{ \begin{matrix}.2369D, & D even \\ .0269D, & D odd. \end{matrix}  (2.13)

Having collected these estimates, proving R > |C| is a simple calculation. Indeed, if D ≥ 24 is even, then B = 2D, and

R > .5235(2D) − .8458(2D)(48)^{−1/4} − .3951(2D)(48)^{−1/2} = .2902D,

so

R > .2369D > |C|.

If D ≥ 19 is odd then

R > .5235D − .8458 · D · 19^{−1/4} − .3951 · D · 19^{−1/2} > .0277D,

R > .0269D > |C|.

There are only two values of D not covered by this argument: D = 8 and 11, which were dealt with in the examples.

□

Quadratic twists

To prove non-vanishing for quadratic twists of canonical characters, the bound (2.13) is not useful. In Section 5 we apply Rohrlich’s method to obtain the following bound on C for χ_{D,d} (Proposition 5.1):^{1}

|C| ≪ D^{15/16+δ}|d|^{51/16+δ},  (2.14)

where δ > 0 is arbitrary and the implied constant depends only on it. Combining (2.11), (2.12), and (2.14) we conclude

Theorem 2.2. For any fixed δ > 0,

Λ′(1, χ_{D,d}) ≠ 0

for |d| ≪ D^{1/35−δ} and W (χ_{D,d}) = −1.

1The notation A ≪ B means A = O(B), i.e. there exists a positive constant C such that |A| ≤ CB.

Remarks. When the root number W (χ) = 1, similar non-vanishing results for twists were obtained in [Ro1], [RVY], and [Ya] for the central L-value.

For canonical Hecke characters, Rohrlich ([Ro4]) computed R as a contour integral of dirichlet L-functions. By shifting contours, R can be expressed as the sum of a residue and a remainder integral. He showed the residue is of size ≫ D, and used Burgess’ sub-convexity estimate to bound the remainder integral by ≪ D^{α}, α < 1. Also, he used the method in Section 5 to show C ≪ D^{α}. The power of the main term is larger than that of the other two terms, and positivity follows for large D. However, the implied constants one gets for these estimates are quite unfavorable. In our proof of Theorem 1.1 we sacrifice the gain in the powers of D for a tie – in favor of better constants.

3 The Main Term R

The purpose of this section is to prove (2.12). We will show term R is large and positive by eventually bounding it from below by the following sum.

Proposition 3.1.

∑_{n=1}^{∞} λ(n)n · f (\frac{2πn^{2}}{x} )

is always positive for x > 0 and in fact

∑_{n=1}^{∞} λ(n)n · f (\frac{2πn^{2}}{x} ) > .5235x − .8458x^{3/4} − .3951x^{1/2}  (3.1)

for x > 1.

Proof: Using (2.10) and the identity

∑_{n=1}^{∞} λ(n)n^{−s} = \frac{ζ(2s)}{ζ(s)},

we write

∑_{n=1}^{∞} λ(n)n · f (\frac{2πn^{2}}{x} ) = \frac{1}{2πi} ∫_{Re s=2} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}ds

= \frac{1}{2πi} ∫_{γ} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}ds

+Res_{s=1} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}

+Res_{s=3/4} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}.

Here γ = C_{1} ∪ C_{2} ∪ C_{3} ∪ C_{4} ∪ C_{5} is the contour consisting of the union of the following five line segments:

C_{1} from 1 − i∞ to 1 − 7i,
C_{2} from 1 − 7i to \frac{1}{2} − 7i,
C_{3} from \frac{1}{2} − 7i to \frac{1}{2} + 7i,
C_{4} from \frac{1}{2} + 7i to 1 + 7i,
C_{5} from 1 + 7i to 1 + i∞

(7 is chosen because the first critical zeroes of ζ(s) are approximately \frac{1}{2} ± 14.13472i). The residue at s = 1 is \frac{πx}{6} ≈ .523599x and the residue at s = 3/4 is

Res_{s=3/4} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)} = \frac{2^{5/4}x^{3/4}Γ(3/4)}{π^{3/4}ζ(1/2)} ≈ −.845767x^{3/4}.

One can easily estimate the integrals over γ as follows.^{2} First,

|∫_{C_{1}} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}ds|

= |∫_{C_{5}} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}ds|

≤ \frac{x}{2π} ∫_{t=7}^{∞} \frac{|Γ(1 + it)|}{t^{2}} \frac{|ζ(2 + 4it)|}{|ζ(1 + 2it)|}dt

≤ x(5 · 10^{−7}).

2All computations were done using Mathematica v4.0 on an Intel Celeron processor under Windows 98.

10 Stephen D. Miller and Tonghai Yang Next

|∫_{C_{2}} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}ds|

= |∫_{C_{4}} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}ds|

≤ x ∫_{σ=1/2}^{1}(2π)^{−σ} \frac{|Γ(σ + 7i)|}{(σ − 1)^{2} + 49} \frac{|ζ(4σ − 2 + 28i)|}{|ζ(2σ − 1 + 14i)|}dσ

≤ x(2 · 10^{−6}).

Finally,

|∫_{C_{3}} (\frac{x}{2π} )^{s} \frac{Γ(s)}{(s − 1)^{2}} \frac{ζ(4s − 2)}{ζ(2s − 1)}ds|

≤ \sqrt{\frac{x}{2π}} ∫_{t=−7}^{7} \frac{|Γ(1/2 + it)|}{1/4 + t^{2}} \frac{|ζ(4it)|}{|ζ(2it)|}dt

≤ 2.48218\sqrt{x}.

Combining these estimates proves (3.1). For x ≥ 20, .5235x − .8458x^{3/4} − .3951x^{1/2}

≥ (.5235 − .8458 · 20^{−1/4} − .3951 · 20^{−1/2})x
≥ .0351x > 0.

The positivity for x < 20 is handled by the next lemma. □ Lemma 3.2. For 0 < x < 20, one has

f(\frac{2π}{x}) > ∑_{n=2}^{∞} n · f (\frac{2πn^{2}}{x} ) .

Proof: It is easy to see that for any 0 < a < 1 and t > \frac{a}{1−a}

f(t) > ae^{−t}/t^{2}.

Take a = \frac{π}{10+π} > .23. Then for 0 < x < 20, one has

f(\frac{2π}{x}) > .23\frac{x^{2}}{4π^{2}} e^{− \frac{2π}{x}} .

One the other hand, clearly, f(x) < e^{−x}/x^{2}, and so

∑_{n=2}^{∞} n · f (\frac{2πn^{2}}{x} ) ≤ ∑_{n=2}^{∞} n\frac{x^{2}}{4π^{2}n^{4}} e^{−2πn^{2}/x}
≤ \frac{x^{2}}{4π^{2}} ∑_{n=2}^{∞} n^{−3}e^{−2πn^{2}/x}.

(3.2)

Since n^{2} ≥ n + 2 for n ≥ 2, this is

≤ \frac{x^{2}}{4π^{2}} \frac{e^{−8π/x}}{8} ∑_{n=0}^{∞} e^{−2πn/x} = \frac{x^{2}e^{−2π/x}}{4π^{2}} \frac{1}{8} \frac{e^{−6π/x}}{(1 − e^{−2π/x})}.  (3.3)

Since \frac{1}{8} \frac{e^{−6π/x}}{(1−e^{−2π/x})} is clearly increasing, it is thus bounded above in 0 < x < 20 by its value ≈ .181 at x = 20. Therefore (3.3) is bounded above by

.19\frac{x^{2}}{4π^{2}} e^{−2π/x} < f(2π/x).

□

Proposition 3.3. (1) If m is any completely multiplicative function with values −1, 0, or 1, then

∑_{n=1}^{∞} m(n)n · f (\frac{2πn^{2}}{x} ) > 0 , x > 0.  (3.4)

(2) If m_{1} and m_{2} are two distinct such functions with m_{1}(p) ≥ m_{2}(p) for every prime p, then

∑_{n=1}^{∞} m_{1}(n)n · f (\frac{2πn^{2}}{x} ) > ∑_{n=1}^{∞} m_{2}(n)n · f (\frac{2πn^{2}}{x} )  (3.5)

for all x > 0.

Proof: We first assume that the functions m, m_{1}, and m_{2} differ from λ at only finitely many primes. Therefore, by Proposition 3.1 and induction, it suffices to prove (3.5) under the following conditions:

(a) m_{2} satisfies (3.4) for all x > 0.

(b) m_{1} and m_{2} differ at exactly one prime, say p, and m_{1}(p) = m_{2}(p) + 1. Under assumption (b), the difference is

∑_{n=1}^{∞} m_{1}(n)n · f (\frac{2πn^{2}}{x} ) − ∑_{n=1}^{∞} m_{2}(n)n · f (\frac{2πn^{2}}{x} )

= ∑_{p|n}(m_{1}(n) − m_{2}(n))n · f (\frac{2πn^{2}}{x} ) .

When m_{2}(p) = −1, m_{1}(p) = 0, the difference is then

− ∑_{p|n} m_{2}(n)n · f (\frac{2πn^{2}}{x} ) = p ∑_{n=1}^{∞} m_{2}(n)n · f (\frac{2πn^{2}}{x/p^{2}} ) > 0

by assumption (a). When m_{2}(p) = 0, m_{1}(p) = 1 and m_{1}(p^{k}n) = m_{2}(n) for p ∤ n. So the difference is

∑_{p|n} m_{1}(n)n · f (\frac{2πn^{2}}{x} ) = ∑_{k=1}^{∞} p^{k} ∑_{n=1}^{∞} m_{2}(n)n · f (\frac{2πn^{2}}{x/p^{2k}} ) > 0

by assumption (a) again.

In the general case, define m^{N}(n) to be the completely multiplicative function derived from m by

m^{N}(p) = \{ \begin{matrix}m(p), & p ≤ N \\ λ(p), & p > N. \end{matrix}  (3.6)

Then m^{N} differs from λ at only a finite number of primes, and thus

∑_{n=1}^{∞} m^{N}(n)n · f (\frac{2πn^{2}}{x} ) ≥ ∑_{n=1}^{∞} λ(n)n · f (\frac{2πn^{2}}{x} ) > 0.

Taking the limit as N → ∞,

∑_{n=1}^{∞} m(n)n · f (\frac{2πn^{2}}{x} ) ≥ ∑_{n=1}^{∞} λ(n)n · f (\frac{2πn^{2}}{x} ) > 0.

This completes the proof of part (1); part (2) can be handled similarly. □

Non-vanishing of the central derivative of canonical Hecke L-functions 13 Corollary 3.4. One has

R ≥ ∑_{n=1}^{∞} λ(n)n · f (\frac{2πn^{2}}{B} ) .

Combining Proposition 3.1 with Corollary 3.4, one obtains (2.12).

4 The Trivial Bound on Remainder Term C

In this section, we will only treat canonical characters, and prove (2.13): Proposition 4.1. When d = 1 and D ≥ 7, term C is bounded by

|C| < \{ \begin{matrix}.0269D, & D odd \\ .2369D, & D even. \end{matrix}  (4.1)

Proof: We first assume D is odd, so B = D. From (2.8) we can bound C term-wise, without appealing to cancellation from the character. To wit,

|C| ≤ ∑_{u≡v (mod 2)^{u,v>0}} uf (\frac{π}{2} (v^{2} + u^{2}/D)) .

Since f(x) < e^{−x}/x^{2},

|C| < ∑_{v≡u (mod 2)^{u,v>0}} ue^{− \frac{πu^{2}}{2D}} \frac{e^{−πv^{2}/2}}{(\frac{π}{2}(v^{2} + u^{2}/D))^{2}}

< ∑_{u=1}^{∞} ue^{− \frac{πu^{2}}{2D}} \frac{4}{π^{2}} ∑_{v≡u (mod 2)^{v=1}}^{∞} v^{−4}e^{−πv^{2}/2}.

The inside sum is bounded by

∑_{v=1, odd}^{∞} v^{−4}e^{−πv^{2}/2} ≈ .20788,

14 Stephen D. Miller and Tonghai Yang so

|C| ≤ .0843 ∑_{u=1}^{∞} ue^{− \frac{πu^{2}}{2D}} .

If D is even, then B = 2D,and the same argument shows that

|C| < 2 (\frac{16}{π^{2}} ∑_{v=1}^{∞} v^{−4}e^{−πv^{2}/4}) (∑_{u=1}^{∞} ue^{−πu^{2}/D})
≤ 1.488 ∑_{u=1}^{∞} ue^{−πu^{2}/D}.

The proof of (4.1) now follows from bound in the Lemma below. □ Lemma 4.2. For a ≥ 1,

∑_{n=1}^{∞} ne^{−n^{2}/a} < a/2.  (4.2)

This conclusion actually holds for all a > 0.

Proof: Using the Poisson summation formula applied to |n|e^{−n^{2}/a},

∑_{n=1}^{∞} ne^{−n^{2}/a} = ∫_{0}^{∞} ne^{−n^{2}/a}dn + 2 ∑_{r=1}^{∞} ∫_{0}^{∞} ne^{−n^{2}/a} cos(2πrn)dn.

The first integral is a/2 and the others are actually negative. This is because

∫_{0}^{∞} ne^{−n^{2}/a} cos(2πrn)dn = \frac{a}{2} (1 − e^{−aπ^{2}r^{2}}2πr\sqrt{a} ∫_{0}^{πr\sqrt{a}} e^{t^{2}}dt)

(cf. [GR], 17.13.27) and

∫_{0}^{πr\sqrt{a}} e^{t^{2}}dt > 1 + ∫_{1}^{aπ^{2}r^{2}} \frac{e^{t}}{2\sqrt{t}}dt = 1 + \frac{1}{2πr\sqrt{a}} (e^{aπ^{2}r^{2}} − e) > \frac{e^{aπ^{2}r^{2}}}{2πr\sqrt{a}}.

□

5 Rohrlich’s Bound on Remainder Term C

We now return to the general case of a canonical character twisted by ϵ_{d} = ^{(\underline{d})}. The method here is adapted from [Ro1] and [Ro4].

Proposition 5.1. For any δ > 0, term C is bounded by

|C| ≪ D^{15/16+δ}|d|^{51/16+δ},

where the implied constant depends only on δ.

Proof: Set A(t) = ∑_{n<t} a_{n}. Integration by parts gives

C = ∫_{D/4}^{∞} f (\frac{2πt}{B}) \frac{d}{dt}A(t)dt = − ∫_{D/4}^{∞} A(t)\frac{d}{dt}f (\frac{2πt}{B}) dt,  (5.1)

because there are no complex ideals of norm < D/4. By [Ro1], p. 553(27), A(t) is bounded above by

|A(t)| ≪ t^{5/4}D^{−5/16+δ}|d|^{19/16+δ}

for D > 8 and t > 0 (the implied constant again depends only on δ). Along with the inequalities

0 < −\frac{d}{dt}f (\frac{2πt}{B}) < \frac{B}{2πt^{2}} e^{−2πt/B} (1 + \frac{B}{2πt}) ,

(5.1) implies

|C| ≪ D^{−5/16+δ}|d|^{19/16+δ} ∫_{D/4}^{∞} [t^{5/4} \frac{B}{2πt^{2}} e^{−2πt/B} (\frac{4}{3} + \frac{B}{2πt})] dt
≪ D^{−5/16+δ}|d|^{19/16+δ} ∫_{D/4}^{∞} \frac{d}{dt} [−\frac{B^{2}}{2π^{2}} t^{−3/4}e^{−2πt/B}] dt
≪ D^{−17/16+δ}|d|^{19/16+δ}B^{2}.

Since either B = D|d| or 2D|d|, this completes the proof. □

6 Arithmetic Applications

Having completed their proofs, we will now give some arithmetic applications of Theorems 1.1 and 2.2, including Corollary 1.2.

Let j be the j-invariant of a fixed isomorphism class of elliptic curves with complex multiplication (CM) by O. Then H = K(j) is the Hilbert class field of K. We can extend any Hecke character χ of K to one on H by

ψ = χ ∘ N_{H/K}.

When χ satisfies (1.1) and (1.2),

ψ(𝔄^{σ}) = ψ(𝔄)^{σ}, σ ∈ Gal(H/ℚ)

for every ideal 𝔄 of H relatively prime to the conductor of ψ. By Theorem 9.1.3 and Lemma 11.1.1 of [Gr], there is a unique elliptic ℚ-curve A over H with

j(A) = j and L(s, A/H) = L(s, ψ)L(s, ψ̄).

(Here we recall that a “ℚ-curve” is an elliptic curve over a number field which is isogenous to all of its Galois conjugates.) Furthermore, A descends to two isogenous elliptic curves over the subfield F = ℚ(j) ([Gr], Theorem 10.2.1). By abuse of notation we will also refer to these curves as A. Let B = Res_{F/ℚ}A be the abelian variety over ℚ obtained from A by restriction of scalars. When D = p is prime, Gross proved ([Gr], Theorem 15.2.5) that T = End_{K}B ⊗ ℚ is a CM number field of degree 2h; thus B is also a CM abelian variety. This result actually extends to composite D via a different argument:

Lemma 6.1. (a) Let T be the subfield of ℂ generated by χ(𝔞), where 𝔞 runs over all ideals of K prime to χ’s conductor. Then T is a CM number field of degree 2h, and Φ = {σ : T → ℂ | σ trivial on K} is a CM type of T.

(b) B is a CM abelian variety of type (T, Φ).

Proof: For each embedding σ : T → ℂ fixing K, σ ∘ χ is another canonical Hecke character of K, and thus it is of the form χϕ, where ϕ is an ideal class character of K. By Theorem 1 of [Ro3], σ ↦ ϕ actually gives a one-to-one correspondence between the complex embeddings of T into ℂ fixing K, and the ideal class characters of K. Thus [T : K] = h and [T : ℚ] = 2h. It is a general fact that T is a CM number field; in this case it can easily be verified using property (1.1) of χ.

By [Sh], Theorem 10, there is a CM abelian variety B′/ℚ of type (T, Φ) associated to χ, and it is unique up to isogeny. In particular

L(s, B′) = ∏_{σ:T^{+}⟶ℂ}L(s, χ^{σ}) = ∏_{ϕ} L(s, χϕ),  (6.1)

where T^{+} is the maximal totally-real subfield of T. On the other hand,

L(s, B) = L(s, A/F) = L(s, ψ) = ∏_{ϕ} L(s, χϕ).

This shows L(s, B) = L(s, B′), so a theorem of Faltings [Fa] guarantees B and B′ are isogenous, proving (b). □

Lemma 6.2. Let χ be a Hecke character of K of the form (2.1). Let A be an associated ℚ-curve over F = ℚ(j) with j-invariant j, and let B = Res_{F/ℚ}A. If ord_{s=1}L(s, χ) ≤ 1 then

(a) The Mordell-Weil ranks of A and B are given by

rank_{ℤ}A(F) = rank_{ℤ}B(ℚ) = h · ord_{s=1}L(s, χ).

(b) The Shafarevich-Tate groups � (A/F) and � (B/ℚ) are finite.

Proof: Since the Mordell-Weil and Shafarevich-Tate groups of A over F are identical to those of B over ℚ, it is sufficient to prove the Lemma for B. Let f be the normalized weight 2 new-form associated to χ as in the proof of Lemma 2.1. The field generated by f’s Fourier coefficients is generated by χ(𝔞) + \overline{χ(𝔞)}, and is thus T^{+}. Equation (6.1) implies

L(s, B) = ∏_{σ:T^{+}⟶ℂ}L(s, f^{σ}).

Now the Lemma follows from a result of Kolyvagin and Logachev ([KL]). □

Combining Lemma 6.2 with the non-vanishing theorems above (Theorems 1.1 and 2.2) and in [MR], one gets the following two corollaries.

Corollary 6.3. Let χ = χcan be a canonical Hecke character of K, and let A and B = Res_{F/ℚ}A respectively be associated ℚ-curves and CM abelian varieties as above. Then

(a) The Mordell-Weil ranks of A and B are given in terms of the root number W (χ) by

rank_{ℤ}A(F) = rank_{ℤ}B(ℚ) = \{ \begin{matrix}h, & W (χ) = −1 \\ 0, & W (χ) = 1. \end{matrix}

In particular, when D is odd, these ranks are h or zero depending on whether D ≡ 3 or 7 mod 8.

(b) The Shafarevich-Tate groups � (A/F) and � (B/ℚ) are finite.

Proof of Corollary 1.2: Take D = p, j = j (\frac{1+\sqrt{−p}}{2}), A = A(p), and apply Corollary 6.3. □

Corollary 6.4. Let χ = χ_{D,d} be a Hecke character of K of the form (2.1). Let A and B = Res_{F/ℚ}A be as above, and fix any δ > 0. If |d| ≪ D^{1/35−δ} (the implied constant depending on δ) and W (χ_{D,d}) = −1, then

(a) The Mordell-Weil ranks of A and B are

rank_{ℤ}A(F) = rank_{ℤ}B(ℚ) = h.

(b) The Shafarevich-Tate groups � (A/F) and � (B/ℚ) are finite.

Finally, we wish to point out that when D is prime, all ℚ-curves over F are associated to Hecke characters of the form (2.1), though this is not true for every composite D. See [Na] for a more-precise description.

Stephen D. Miller  Tonghai Yang
Department of Mathematics  Department of Mathematics
Yale University  State University of New York
P.O. Box 208283  Stony Brook, NY 11794-3651

    New Haven, CT 06520  thyang@math.sunysb.edu

    stephen.miller@yale.edu

References

[Cr] John Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992.

[Fa] G. Faltings, Finiteness theorems for abelian varieties over number fields, pp. 9-27 in Arithmetic Geometry, G. Cornell, J. Silverman, eds., Springer, 1986.

[Gr] Benedict H. Gross, Arithmetic on Elliptic Curves with Complex Mulitplication, Springer Lecture Notes in Mathematics, 776, Berlin, 1980.

[GR] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 5th edition, Alan Jeffrey, editor, Academic Press, New York, 1994.

[GZ] Benedict Gross and Don Zagier Points de Heegner et drives de fonctions L. (French) [Heegner points and derivatives of L-functions] C. R. Acad. Sci. Paris Sr. I Math. 297 (1983), no. 2, 85–87.

[KL] V. A. Kolyvagin and D. Yu. Logachev, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties (Russian), Algebra i Analiz 1 (1989), no. 5, 171-196. Translation in Leningrad Math. J. 1 (1990), no. 5, 1229-1253.

[MR] H. Montgomery and D. Rohrlich, On the L-functions of canonical Hecke characters of imaginary quadratic fields II, Duke Math. J. 49 (1982), 937-942.

[Na] Tetsuo Nakamura, ℚ-curves with complex multiplication, Proc. Amer. Math. Soc., to appear.

[RVY] F. Rodriguez-Villegas and T.H. Yang, Central values of Hecke L-functions of CM number fields, Duke Math. J. 98 (1999), 541-564.

[Ro1] D. Rohrlich, The nonvanishing of certain Hecke functions at the center of the critical strip, Duke Math. J. 47 (1980), 223-232, 547-557.

[Ro2] D. Rohrlich, Root numbers of Hecke L-functions of CM fields, Amer. J. Math 104 (1982), 517-543.

[Ro3] D. Rohrlich, Galois conjugacy of unramified twists of Hecke characters, Duke Math. J. 47 (1980), 695-703.

[Ro4] D. Rohrlich, Non-vanishing of derivatives of L-functions, unpublished manuscript, 1982.

[Sh] Goro Shimura, On the zeta-function of an abelian variety with complex multiplication, Ann. of Math. (2) 94 (1971), 504–533.

[Ya] Tonghai Yang, Nonvanishing of the central value of Hecke characters and the rank of their associated elliptic curves, Compositio Math. 117 (1999), 337-359.