Subconvexity of Hecke L-functions in the Grossencharacter-aspect

Shin-ya Koyama

1. Introduction. Let K be a number field with [K : Q] = n = r_{1} + 2r_{2}. The real and complex conjugates of K are denoted by K^{(q)} (q = 1, ..., r_{1}) and K^{(p)} (p = r_{1}+1, ..., n) with K^{(p+r_{2})} = \overline{K^{(p)}}. The conjugate of μ ∈ K in K^{(j)} (j = 1, ..., n) is denoted by μ^{(j)}. Put N(μ) = μ^{(1)} ⋯ μ^{(n)}.

Let J_{K} and P_{K} be the idele group and the principal idele group of K, respectively. Grossencharacter λ of K is a continuous character of the idele class group C_{K} = J_{K}/P_{K}. We use the same symbol λ for the induced character of J_{K}. We decompose as J_{K} = J_{∞} × J_{0} with the infinite part J_{∞} and the finite part J_{0}. Let U_{0} be the unit group of J_{0}, and put U_{𝔪,0} = {𝔲 ∈ U_{0} | 𝔲 ≡ 1 (mod 𝔪)}, where 𝔪 is an integral ideal of K. The set {U_{𝔪,0}} is a fundamental neighbourhood system of 1 in J_{0}. A map from J_{𝔪,0} = {𝔞 ∈ J_{0} | 𝔞_{𝔭} = 1 (∀𝔭|𝔪)} to an ideal class group G(𝔪) = {𝔞̃ | (𝔞, 𝔪) = 1} defined by

J_{𝔪,0} ∋ 𝔞 ↦ 𝔞̃ = ∏_{𝔭} 𝔭^{ν_{𝔭}(𝔞)} ∈ G(𝔪)

is a homomorphism whose kernel is a subset of U_{𝔪,0}.

By continuity a Grossencharacter λ satisfies that λ(U_{𝔪,0}) = 1 for some 𝔪. The greatest common divisor 𝔣 of such 𝔪’s are called the conductor of λ. For 𝔞 ∈ J_{𝔣,0}, the value λ(𝔞) is determined by the ideal class 𝔞̃ ∈ G(𝔣). Thus we define a character λ̃ of G(𝔣) as λ̃(𝔞̃) := λ(𝔞). The Hecke L-function with respect to the Grossencharacter λ is defined by

L(s, λ) = ∑_{𝔞̃∈G(𝔣)} \frac{λ̃(𝔞̃)}{N(𝔞̃)^{s}} . (Re(s) > 1)

When we restrict λ on J_{∞} = R^{×r_{1}} × C^{×r_{2}}, we have for x = (x^{(1)}, ..., x^{(r_{1}+r_{2})}) ∈ J_{∞}

λ(x) = ∏_{v=1}^{r_{1}}(sgn x^{(v)})^{a_{v}} ∏_{v=1}^{r_{1}+r_{2}} |x^{(v)}|^{ib_{v}} ∏_{v=r_{1}+1}^{r_{1}+r_{2}} (\frac{x^{(v)}}{|x^{(v)}|})^{a_{v}}

with a_{v} ∈ Z and b_{v} ∈ R, and a_{1}, ..., a_{r_{1}}, which may take only the values 0 or 1, depend only on the narrow class character mod 𝔣.

In what follows we fix a_{1}, ..., a_{r_{1}} so that the narrow class character χ mod 𝔣 is fixed. We will make the remaining part in the Grossencharacter vary, which is parametarized by an n-dimensional vector a = (b_{1}, ..., b_{r_{1}+r_{2}}, a_{r_{1}+1}, ..., a_{r_{1}+r_{2}}) with ∑_{v} b_{v} = 0. We denote such Hecke character λ by χλ^{a}. If χλ^{a} is a normalized Hecke character, then χλ^{ma} is also normalized for any m ∈ Z. The chief concern of this

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paper is to estimate the size of L(\frac{1}{2} + it, χλ^{ma}) as |m| → ∞ with χ, a and t being fixed. We may call this problem the m-aspect in the estimate of Hecke L-functions. In what follows we write χλ^{m} := χλ^{ma}.

Duke [D] computed the convexity bound

L(\frac{1}{2} + it, χλ^{m}) ≪_{t,ε} |m|^{\frac{n}{4} +ε}  (1.1)

which can be proved by applying the Phragmén-Lindelöf principle to the lines Re(s) = 1 + δ and Re(s) = −δ with δ > 0. We improve this bound by using the approximate functional equation in the m-aspect, which will be proved in the next section. In Section 3 we express the exponential sum over certain range of norms in terms of the sum over algebraic integers in certain cubes. The main theorem is proved using the van der Corput method in Section 4. It asserts that

Theorem 1.1. In the above notation we have as |m| → ∞

L(\frac{1}{2} + it, λ^{ma}) ≪_{t,ε} |m|^{\frac{n}{4} − \frac{1}{18} +ε}

for any ε > 0.

The Grossencharacter-aspect is recently considered in a different context. In their paper [PS], Petridis and Sarnak obtain a subconvexity estimate of automorphic L-functions L(s, ϕ) for a Maass cusp form ϕ of SL(2, Z[i]). In the proof they consider the twists with Grossencharacters and take an avarage in the form

∑ ∫ |L(\frac{1}{2} + it, ϕ ⊗ λ^{m})|^{2}dt,

where the summation and the integration are taken over certain range of (m, t). Consequently they succeed in obtaining the subconvexities in the both m and t aspects.

2. Approximate Functional Equation. We require the functional equation for the Hecke L-function given by Mordell [Mo] (5.14):

L(s, χλ) = W (χλ)A^{1−2s}G(1 − s, χ̂λ̄)L(1 − s, χ̄λ̄),  (2.1)

where |W (χλ)| = 1 and A = (\frac{d_{K}N(𝔣)}{π^{n}2^{2r_{2}}})^{1/2} with d_{K} the discriminant of K, χ̂ is the real character induced by χ on the group of narrow ideal classes lying over the broad class of principal ideals, and G(s, χ̂λ) is defined by

G(s, χ̂λ) = ∏_{v=1}^{r_{1}} \frac{Γ(\frac{1}{2}(s + a_{v} + ib_{v}))}{Γ(\frac{1}{2}(1 − s + a_{v} − ib_{v}))} ∏_{v=r_{1}+1}^{r_{1}+r_{2}} \frac{Γ(s + \frac{1}{2}|a_{v}| + ib_{v})}{Γ(1 − s + \frac{1}{2}|a_{v}| − ib_{v})}.

We denote each factor by G_{v}, namely, G(s, χ̂λ) = ∏_{v=1}^{r_{1}+r_{2}} G_{v}(s, χ̂λ).

Lemma 2.1. The function G(s, χ̂λ) is holomorphic except at the following simple poles:

s = −2k − a_{v} − ib_{v} (1 ≤ v ≤ r_{1})

with residue \frac{(−1)^{k}2^{2(k+a_{v})+1}(k + a_{v})!}{\sqrt{π}k!(2(k + a_{v}))!} ∏_{v′≠v} G_{v′}(−2k − a_{v} − ib_{v})

s = −k − \frac{|a_{v}|}{2} − ib_{v} (r_{1} < v ≤ r_{1} + r_{2})

with residue \frac{(−1)^{k}}{k!(k + |a_{v}|)!} ∏_{v′≠v} G_{v′}(−k − \frac{|a_{v}|}{2} − ib_{v})

with k ≥ 0, k ∈ Z, and the sum ∑_{ρ} \frac{Res_{s=ρ}G(s,χ̂λ)}{ρ} over all poles of G(s, χ̂λ) absolutely converges. In particular, G(s, χ̂λ) is holomorphic in Re(s) > 0.

Proof. We first treat the case 1 ≤ v ≤ r_{1}. The numerator of

G_{v}(s, χ̂λ) = \frac{Γ(\frac{1}{2}(s + a_{v} + ib_{v}))}{Γ(\frac{1}{2}(1 − s + a_{v} − ib_{v}))}

has poles at s = −2k − a_{v} − ib_{v}, none of which agrees to any zero coming from a pole of the denominator, since k ≥ 0. By the fact that the residue of Γ(z) at z = −k is \frac{(−1)^{k}}{k!} , the residue of G_{v}(s, χ̂λ) at s = −2k − a_{v} − ib_{v} is computed as

\frac{2(−1)^{k}}{k!Γ (\frac{1+2k+2a_{v}}{2} )} = \frac{(−1)^{k}2^{2(k+a_{v})+1}(k + a_{v})!}{\sqrt{π}k!(2k + 2a_{v})!},

where we used the identity

Γ (N + \frac{1}{2}) = \frac{\sqrt{π}(2N)!}{2^{2N}N!}

for any nonnegative integer N. The case r_{1} + 1 ≤ v ≤ r_{1} + r_{2} can be treated similarly.

Huxley’s expression of G_{v}(s, χ̂λ) (See (2.7) below) and Stirling’s formula show that G_{v′}(−2k − a_{v} − ib_{v}) and G_{v′}(−k − \frac{|a_{v}|}{2} − ib_{v}) decay exponentially as k → ∞. It gives the absolute convergence of the sum. □

In what follows we define for z ∈ C \ R^{−}

z^{z} := exp(z log z) = exp(z(log |z| + i arg(z)))

with | arg(z)| < π.

Lemma 2.2. Put z = s + iy ∈ C with s = σ + it, σ, t ∈ R being fixed. Then we have

z^{z} = (s + iy)^{s+iy} = |y|^{s+iy} exp (s + sgn(y)\frac{πiz}{2} ) (1 + O (\frac{1}{y} ))

as |y| → ∞.

Proof. Putting |z| = r and arg(z) = θ give

z^{z} = r^{σ} exp(−θ(t + y) + i((t + y) log r + θσ)).

By putting φ = \frac{π}{2} − |θ| = \frac{π}{2} − sgn(t + y)θ, we have sin φ = φ + O(φ^{3}). On the other hand we have sin φ = \frac{σ}{\sqrt{σ^{2}+y^{2}}} = \frac{σ}{|y|} + O(\frac{1}{y^{2}} ). Thus θ(t + y) ≥ 0, and

φ = \frac{σ}{|y|} + O (\frac{1}{y^{2}} ) ,  (2.3)

which leads to

exp(−θ(t + y)) = exp (−θt + sgn(t + y) (−\frac{π}{2} + φ) y)

= exp (−θt − \frac{π}{2} |y| + σ + O (\frac{1}{y^{2}} ))

= exp (−θt − \frac{π}{2} |y| + σ) (1 + O (\frac{1}{y^{2}} )) .  (2.4)

Next we easily see that

r^{σ} = (σ^{2} + (t + y)^{2})^{σ/2} = |y|^{σ} (1 + O (\frac{1}{y} )) .  (2.5)

By the facts that

log r = log |y| + \frac{1}{2} log ((1 + \frac{t}{y} )^{2} + \frac{σ^{2}}{y^{2}} ) = log |y| + \frac{t}{y} + O (\frac{1}{y^{2}} )

and that |θ| = \frac{π}{2} − φ = \frac{π}{2} + O (\frac{1}{y} ) which is deduced from (2.3), we compute that

(t + y) log r + θσ = (t + y) log |y| + sgn(y)\frac{π}{2} σ + t + O(1).  (2.6)

Combining (2.4), (2.5) and (2.6) gives that

z^{z} = r^{σ}e^{−θ(t+y)+i((t+y) log r+θσ)}

= |y|^{s+iy}e^{s+sgn(y) \frac{πiz}{2}} (1 + O (\frac{1}{y} )) .

The proof is complete. □

Lemma 2.3. As |y| → ∞, it holds that

Γ(s + iy) = |y|^{z− \frac{1}{2}} exp (i (sgn(y)\frac{π}{2} z − (2 − sgn(y))\frac{π}{4} − y)) \sqrt{2π} (1 + O (\frac{1}{y} ))

with s ∈ C being fixed and z = s + iy. Proof. Stirling’s formula and the previous lemma show that

Γ(z) = e^{−z}z^{z− \frac{1}{2}} \sqrt{2π} (1 + O (\frac{1}{|z|}))

= |y|^{z}e^{i(sgn(y) \frac{πz}{2} −y)}z^{−1/2}\sqrt{2π} (1 + O (\frac{1}{|z|})) .

By y = e^{\frac{1−sgn(y)}{2} πi}|y| and thus

(s + iy)^{−1/2} = |y|^{−1/2}e^{−(2−sgn(y))πi/4}(1 + O(y^{−1})).

We have the conclusion. □ Lemma 2.4. As |m| → ∞,

G(s, χ̂λ^{m}) = 2^{−r_{1}s}|m|^{n(s− \frac{1}{2} )} ∏_{v=1}^{n} c̃_{v}(m)|b_{v}|^{s− \frac{1}{2}} (1 + O (\frac{1}{m})) ,

where c̃_{v}(m) is given by

c̃_{v}(m) = (−1)^{\frac{av}{2}} (1 − sgn(mb_{v})i) (\frac{|mb_{v}|}{2e})^{imb_{v}}

for 1 ≤ v ≤ r_{1}, and by

c̃_{v}(m)^{2} = −sgn(mb_{v})ic_{v}(m)

for r_{1} + 1 ≤ v ≤ r_{1} + r_{2} with

c_{v}(m) = \{ \begin{matrix}1 − (−1)^{\frac{mav}{2}} (2s − 1) log \frac{2ib_{v}}{a_{v}+2ib_{v}} & (ma_{v} ∈ 2Z) \\ 1 − (−1)^{\frac{mav −1}{2}} (2s − 1) log \frac{2ib_{v}}{a_{v}+2ib_{v}} & (ma_{v} ∈ 1 + 2Z). \end{matrix}

Proof. Huxley [H] p.115 computes that

G_{v}(s, χ̂λ)

= \{ \begin{matrix}2^{1−s−ib_{v}}π^{−1/2} cos \frac{(s+ib_{v})π}{2} Γ(s + ib_{v}) & (1 ≤ v ≤ r_{1}, a_{v} = 0) \\ 2^{1−s−ib_{v}}π^{−1/2} sin \frac{(s+ib_{v})π}{2} Γ(s + ib_{v}) & (1 ≤ v ≤ r_{1}, a_{v} = 1) \\ \frac{sin(s+ \frac{1}{2} |a_{v}|+ib_{v})π}{π} Γ(s + \frac{|a_{v}|}{2} + ib_{v})Γ(s − \frac{|a_{v}|}{2} − ib_{v}). & (r_{1} + 1 ≤ v ≤ r_{1} + r_{2}) \end{matrix}

(2.7)

When 1 ≤ v ≤ r_{1}, the cos or sin part in (2.7) is asymptotically equal to

\frac{(−1)^{\frac{av +1}{2}}}{2} e^{sgn(b_{v}) \frac{π}{2} (b_{v}−i(s+1))}

up to an exponentially small error term. Hence from Lemma 2.3 we compute that

G_{v}(s, χ̂λ^{m}) = \frac{(−1)^{\frac{av}{2}}}{2^{s+imb_{v}− \frac{1}{2}}} |mb_{v}|^{s+imb_{v}− \frac{1}{2}} e^{−(sgn(mb_{v}) \frac{π}{4} +mb_{v})i} (1 + O (\frac{1}{m})) .

Since e^{−sgn(mb_{v}) \frac{π}{4} i} = 2^{−1/2}(1 − sgn(mb_{v})i), we have

G_{v}(s, χ̂λ^{m}) = 2^{−s}c̃_{v}(m)|mb_{v}|^{s− \frac{1}{2}} (1 + O (\frac{1}{m})) .  (2.8)

When r_{1} + 1 ≤ v ≤ r_{1} + r_{2}, (2.7) can further be calculated as follows. We iterate the formula of the Gamma function Γ(z + 1) = zΓ(z) to get

G_{v}(s, χ̂λ)

= \{ \begin{matrix}^{\frac{(−1)^{\frac{|av |}{2}} sin(s+ib_{v})π}{π}} Γ(s + ib_{v})Γ(s − ib_{v}) ∏_{k=1}^{\frac{|av |}{2}} \frac{s+ib_{v}+k−1}{s−ib_{v}−k} (a_{v} ∈ 2Z) \\ ^{\frac{(−1)^{\frac{|av |−1}{2}} cos(s+ib_{v})π}{π}} Γ(s + \frac{1}{2} + ib_{v})Γ(s − \frac{1}{2} − ib_{v}) ∏_{k=1}^{\frac{|av |−1}{2}} \frac{s+ib_{v}+k− \frac{1}{2}}{s−ib_{v}−k− \frac{1}{2}} \end{matrix}

(a_{v} ∈ 1 + 2Z)

(2.9)

We estimate the product over k in (2.9) for G_{v}(s, χ̂λ^{m}) as |m| → ∞. When ma_{v} ∈ Z, it is

∏_{k=1}^{\frac{|mav |}{2}} \frac{s + imb_{v} + k − 1}{s − imb_{v} − k} = ∏_{k=1}^{\frac{|mav |}{2}} (−1 + \frac{2s − 1}{s − imb_{v} − k} )

= (−1)^{\frac{|mav |}{2}} − (2s − 1) ∑_{k=1}^{\frac{|mav |}{2}} \frac{1}{s − imb_{v} − k} + O (\frac{1}{m})

= (−1)^{\frac{|mav |}{2}} − (2s − 1) log \frac{2ib_{v}}{a_{v} + 2ib_{v}} + O (\frac{1}{m}) .

When ma_{v} ∈ 1 + 2Z, we have

∏_{k=1}^{\frac{|av |−1}{2}} \frac{s + ib_{v} + k − \frac{1}{2}}{s − ib_{v} − k − \frac{1}{2}} = (−1)^{\frac{|mav |−1}{2}} − (2s − 1) log \frac{2ib_{v}}{a_{v} + 2ib_{v}} + O (\frac{1}{m}) .

Next by applying Lemma 2.3 we estimate the product of the gamma functions in (2.9) as |m| → ∞ as follows:

Γ(s + imb_{v})Γ(s − imb_{v}) = −2π|mb_{v}|^{2s−1}e^{−π|mb_{v}|} (1 + O (\frac{1}{m}))

and

Γ(s+imb_{v} + \frac{1}{2})Γ(s−imb_{v} − \frac{1}{2}) = −2πisgn(mb_{v})|mb_{v}|^{2s−1}e^{−π|mb_{v}|} (1 + O (\frac{1}{m})) .

Then (2.9) is written as

G_{v}(s, χ̂λ^{m}) = c̃_{v}(m)^{2}e^{−sgn(mb_{v})isπ}|mb_{v}|^{2s−1} (1 + O (\frac{1}{m})) .  (2.10)

Multiplying (2.8) and (2.10) for all v leads to the lemma. □ Corollary 2.5. Put G̃(s, χ̂λ̄^{m}) = |m|^{−n(s− \frac{1}{2} )}G(s, χ̂λ̄^{m}), and we have

G̃(s, χ̂λ̄^{m}) = O(1)

as |m| → ∞. Proof. By the definition of c̃_{v}(m) in Lemma 2.4, we have c̃_{v}(m) = O(1) as |m| → ∞. □

We follow the method of Luo-Sarnak [LS] for proving the approximate functional equation. Put for σ > 1/2

F (X) = \frac{1}{2πi} ∫_{(σ)} Γ(s + l)X^{−s} \frac{ds}{s},

F (w, X) = \frac{1}{2πi} ∫_{(σ)} Γ(−s + l)G(w̄ + s, χ̂λ̄^{m})X^{−s} \frac{ds}{s}

where w ∈ C is fixed with Re(w) = 1/2. Lemma 2.6. Both F (X) and F (w, X) rapidly decrease as X → ∞, and we have

F (X) = O(1) for X ≪ 1

F (w, X) = O(1) for X ≪ |m|^{n}

Proof. By integrating by parts

F (X) = ∫_{X}^{∞} e^{−ξ}ξ^{l−1}dξ = e^{−X}X^{l−1} + (l − 1) ∫_{X}^{∞} e^{−ξ}ξ^{l−2}dξ.

It is O(1) as X → 0, and is estimated as

F (X) = e^{−X}X^{l−1} (1 + O (\frac{1}{X} ))

as X → ∞.

Next we change the order of integrals to get

F (w, X) = ∫_{0}^{∞} e^{−ξ}ξ^{l−1} \frac{1}{2πi} ∫_{(σ)} (\frac{|m|^{n}ξ}{X} )^{s+it_{0}} (\frac{ξ}{X} )^{−it_{0}} G̃(w̄ + s, χ̂λ̄^{m})\frac{ds}{s}dξ,

where we put w = \frac{1}{2} + it_{0}. We first compute the inner integral on s. When X > |m|^{n}ξ, we shift the contour from (σ) to (+∞). By Lemma 2.1 there are no poles in the region. and we deduce that the integral goes to 0. When X < |m|^{n}ξ, by shifting the contour to (−∞), we have

\frac{1}{2πi} ∫_{(σ)} (\frac{|m|^{n}ξ}{X} )^{s+it_{0}} (\frac{ξ}{X} )^{−it_{0}} G(w̄ + s, χ̂λ̄^{m})\frac{ds}{s}

= G(w, χ̂λ̄^{m}) + ∑_{ρ} (\frac{|m|^{n}ξ}{X} )^{ρ+it_{0}} \frac{Res_{s=ρ}G̃(w̄ + s, χ̂λ̄^{m})}{ρ} (\frac{ξ}{X} )^{−it_{0}} (2.11)

where the sum is taken over the poles ρ of G(w̄+s, χ̂λ̄^{m}). By Lemma 2.1 we see that the sum absolutely converges. We also have G(w, χ̂λ̄^{m}) = O(|m|^{n(w− \frac{1}{2} )}) = O(1) since Re(w) = \frac{1}{2}. Therefore we have

F (w, X) ≪ ∫ ∞_{\frac{X}{|m|^{n}}} e^{−ξ}ξ^{l−1}dξ,

which can be estimated again by integrating by parts. □

Proposition 2.7. Let a_{m}(k) be the k-th coefficient in the Dirichlet series expansion of the Hecke L-function L(w, χλ^{m}), that is,

L(w, χλ^{m}) = ∑_{k=1}^{∞} \frac{a_{m}(k)}{k^{w}}.

Then with Re(w) = 1/2 and w, χ, λ being fixed, it holds that

L(w, χλ^{m})Γ(l) = ∑_{k=1}^{∞} \frac{a_{m}(k)}{k^{w}}F (kx) − W (χλ^{m})A^{1−2w} ∑_{k=1}^{∞} \frac{\overline{a_{m}(k)}}{kw̄}F (w, k/xA^{2}),

where x > 0 and l ∈ Z, l > 0.

Proof. Consider the integral

I = \frac{1}{2πi} ∫_{(σ)} L(w + s, χλ^{m})Γ(s + l)x^{−s} \frac{ds}{s},  (2.12)

where σ > 1/2. We compute

I = ∑_{k=1}^{∞} \frac{a_{m}(k)}{k^{w}}F (kx).

On the other hand, since the integrand in (2.12) has a pole at s = 0, Cauchy’s theorem shows

I = L(w, χλ^{m})Γ(l) + \frac{1}{2πi} ∫_{(−σ)} L(w + s, χλ^{m})Γ(s + l)x^{−s} \frac{ds}{s}.

By changing the variable s ↦ −s, the second term I − L(w, χλ^{m})Γ(l) is equal to

−\frac{1}{2πi} ∫_{(σ)} L(w − s, χλ^{m})Γ(−s + l)x^{s} \frac{ds}{s}.

We deduce by the functional equation (2.1) that

I−L(w, χλ^{m})Γ(l)

= −\frac{1}{2πi} ∫_{(σ)} L(w̄ + s, χ̄λ̄^{m})Γ(−s + l)W (χλ^{m})A^{1−2(w−s)}G(w̄ + s, χ̂λ̄^{m})x^{s} \frac{ds}{s}

= −\frac{W (χλ^{m})A^{1−2w}}{2πi} ∑_{k=1}^{∞} \frac{\overline{a_{m}(k)}}{kw̄} ∫_{(σ)} Γ(−s + l)G(w̄ + s, χ̂λ̄^{m}) (\frac{k}{A^{2}x})^{−s} \frac{ds}{s}.

This gives the conclusion. □

Corollary 2.8. For fixed w ∈ C with Re(w) = 1/2, we have

L(w, χλ^{m}) = O ( ∑_{{𝔟:ideal \atop N(𝔟)<|m|^{n/2}}} \frac{χλ^{m}(𝔟)}{N(𝔟)^{w}} )

Proof. Choose x = |m|^{−n/2}, and the corollary follows from |W (χλ^{m})| = 1. □

3. Reduction to Cubes. Let A, B ∈ R satisfy 0 < A < B ≤ 2A. Let L(𝔞) be the set of principal ideals divisible by 𝔞. In this section we estimate the exponential sum

S_{𝔞}(A, B) = ∑_{{𝔟⊂L(𝔞) \atop A≤N(𝔟)<B}} \frac{χλ^{m}(𝔟)}{N(𝔟)^{it}}  (3.1)

as |m| → ∞.

Lemma 3.1. Let ε_{1}, ..., ε_{r} (r = r_{1} + r_{2} − 1) be the fundamental units of K. Let (α_{ik}) be the inverse of the square matrix (log |ε_{k}^{(i)}|)_{1≤i,k≤r}. Let 𝔐 be the set of μ ∈ 𝔞∩O_{K} such that A ≤ |N(μ)| < B and that 0 ≤ t_{k}(μ) < 1 where for k = 1, ..., r,

t_{k}(μ) = ∑_{i=1}^{r} α_{ik} (log |μ^{(i)}| − \frac{1}{n} log |N(μ)|) .

Then

S_{𝔞}(A, B) = \frac{1}{w} ∑_{μ∈𝔐} \frac{χλ^{m}(μ)}{N(μ)^{it}},

where w is the number of roots of unity in K.

Proof. We follow the method of Mitsui [M] Lemma 1. Put a_{k} = [t_{k}(μ)] (k = 1, ..., r). We have μ_{1} := μ ∏_{k=1}^{r} ε_{k}^{−a_{k}} ∈ 𝔐. The product μ_{1}ε (ε ∈ O_{K}^{×}) belongs to 𝔐 if and only if ε is a root of unity. Therefore the number of generators of the ideal (μ) which belongs to 𝔐 is equal to w. □

Let γ_{1}, ...γ_{n} be the basis of 𝔞. Let J be a subset of {1, ..., r_{1}}. We define a subset X_{J} of R^{n} as

X_{J} := \{(x_{1}, ..., x_{n}) | \begin{matrix}∑_{i=1}^{n} x_{i}γ_{i}^{(q)} > 0 & for q ∈ J \\ ∑_{i=1}^{n} x_{i}γ_{i}^{(q)} < 0 & for q ∉ J, 1 ≤ q ≤ r_{1} \\ ∑_{i=1}^{n} x_{i}γ_{i}^{(q)} ≠ 0 & for r_{1} + 1 ≤ q ≤ n \end{matrix} \} .

A map f_{J} : X_{J} → R^{n} is defined as follows:

f_{J}(x_{1}, ..., x_{n}) = (y_{0}, y_{1}, ..., y_{r}, θ_{1}, ..., θ_{r_{2}})

where

y_{0} = ∑_{q=1}^{n} log |∑_{i=1}^{n} x_{i}γ_{i}^{(q)}|

y_{k} = ∑_{q=1}^{r} α_{qk} (log |∑_{i=1}^{n} x_{i}γ_{i}^{(q)}| − \frac{1}{n}y_{0}) (k = 1, ..., r)

θ_{p} = arg ∑_{i=1}^{n} x_{i}γ_{i}^{(p+r_{1})} (p = 1, ..., r_{2}).

We see that f_{J} is injective. Put

V = \{(y_{0}, y_{1}, ..., y_{r}, θ_{1}, ..., θ_{r_{2}}) | \begin{matrix}log A ≤ y_{0} < log B \\ 0 ≤ y_{k} < 1 (k = 1, ..., r) \\ 0 ≤ θ_{p} < 2π (p = 1, ..., r_{2}) \end{matrix} \} .

Then μ = ∑_{i=1}^{n} m_{i}γ_{i} ∈ 𝔞 ∩ O_{K} belongs to 𝔐 if and only if (m_{1}, ..., m_{n}) ∈ f_{J}^{−1}(V) for some J. Putting for μ ∈ K^{×}

F (μ) := exp(−it log |N(μ)| + log χ(μ) + m log λ(μ)),

we have

S_{𝔞}(A, B) = \frac{1}{w} ∑_{J⊂\{1,2,...,r_{1}\}} ∑_{{(m_{1},...,mn)∈f{−1 \atop J}(V ) \atop m_{j} ∈Z}} F (∑_{i=1}^{n} m_{i}γ_{i}) .  (3.2)

Lemma 3.2. For k_{1}, ..., k_{n} ∈ Z, we define a cube Q ⊂ R^{n} as

Q = Q(k_{1}, ..., k_{n}; M)
= {(x_{1}, ..., x_{n}) | k_{q}M ≤ x_{q} < (k_{q} + 1)M (q = 1, ..., n)},  (3.3)

where M = [A^{8/9n}]. Then

∑_{{(m_{1},...,mn)∈f{−1 \atop J}(V ) \atop m_{j} ∈Z}} F (∑_{i=1}^{n} m_{i}γ_{i}) = ∑_{Q⊂f_{J}^{−1}(V)} ∑_{(m_{1},...,m_{n})∈Q} F (∑_{i=1}^{n} m_{i}γ_{i})+O(A^{1− \frac{1}{9n}}).

Proof. This is a corollary of the proof of Mitsui’s lemma ([M] Lemma 2). His lemma is stated in the case when M = [A^{6/7n}] with remainder O(A^{1− \frac{1}{7n}}), but the proof does not rely on the exponent of A. Actually in his proof he shows in page 237 that the remainder term is bouded by δA ≪ MA^{1− \frac{1}{n}} with δ the maximum of the diameters of the f_{J}(Q) having points with V in common. □

4. Main Theorem.

Lemma 4.1 ([T] Lemma 5.11). Let f(x) be real and have continuous derivatives up to the third order, and let λ_{3} ≤ f′′′(x) ≤ hλ_{3}, or λ_{3} ≤ −f′′′(x) ≤ hλ_{3}, and b − a ≥ 1. Then

∑_{a<n≤b} e^{2πif(n)} = O(h^{1/2}(b − a)λ_{3}^{1/6}) + O((b − a)^{1/2}λ_{3}^{−1/6}).

Lemma 4.2. For fixed k_{1}, ..., k_{n} ≥ 0 and the cube Q = Q(k_{1}, ..., k_{n}) defined in Lemma 3.2, we have

∑_{(m_{1},...,m_{n})∈Q} F (∑_{i=1}^{n} m_{i}γ_{i}) ≪ M^{n− \frac{1}{2}} |m|^{\frac{1}{6}} + M^{n}|m|^{− \frac{1}{6}} .

Proof. The left hand side of the lemma is

∑_{m_{1}} ⋯ ∑_{m_{n−1}} ∑_{m_{n}}

exp (−it log |N(μ)| + log χ(μ) + m log ( ∏_{v=1}^{r_{1}+r_{2}} |μ^{(v)}|^{ib_{v}} ∏_{v=r_{1}+1}^{r_{1}+r_{2}} (\frac{μ^{(v)}}{|μ^{(v)}|})^{a_{v}})) ,

where m_{q} runs through the integers in k_{q}M ≤ m_{q} < (k_{q} + 1)M (q = 1, ..., n) with μ = m_{1}γ_{1} + ⋯ + m_{n}γ_{n}. We will trivially estimate the sum over m_{1}, ..., m_{n−1}, and will obtain a nontrivial estimate for the sum over m_{n} by van der Corput’s technique. In what follows we write μ^{(v)} = x_{1}γ_{1}^{(v)} + ⋯ + x_{n}γ_{n}^{(v)} with real variables x_{1}, ..., x_{n}. Fix x_{1}, ..., x_{n−1} and put

f(x_{n}) = −\frac{1}{2π}

× (t ∑_{v=1}^{n} log |μ^{(v)}| − arg χ(μ) − m (∑_{v=1}^{n} b_{v} log |μ^{(v)}| − i ∑_{v=r_{1}+1}^{r_{1}+r_{2}} a_{v} log \frac{μ^{(v)}}{|μ^{(v)}|})) .

Let

A^{(v)} = ∑_{j=1}^{n−1} x_{j}Re(γ_{j}^{(v)}) and B^{(v)} = ∑_{j=1}^{n−1} x_{j}Im(γ_{j}^{(v)}).

Then we have

Re(μ^{(v)}) = ∑_{j=1}^{n} x_{j}Re(γ_{j}^{(v)}) = x_{n}Re(γ_{n}^{(v)}) + A^{(v)}

Im(μ^{(v)}) = ∑_{j=1}^{n} x_{j}Im(γ_{j}^{(v)}) = x_{n}Im(γ_{n}^{(v)}) + B^{(v)}.

We compute

−i\frac{∂}{∂x_{n}} log \frac{μ^{(v)}}{|μ^{(v)}|} = \frac{∂}{∂x_{n}} (tan^{−1} \frac{Im(μ^{(v)})}{Re(μ^{(v)})})

= \frac{A^{(v)}Im(γ_{n}^{(v)}) − B^{(v)}Re(γ_{n}^{(v)})}{(x_{n}Re(γ_{n}^{(v)}) + A^{(v)})^{2} + (x_{n}Im(γ_{n}^{(v)}) + B^{(v)})^{2}} .

Since A^{(v)} and B^{(v)} are linear in x_{1}, ..., x_{n−1}, the condition (x_{1}, ..., x_{n}) ∈ Q implies that

\frac{∂}{∂x_{n}} log \frac{μ^{(v)}}{|μ^{(v)}|} ≈ M^{−1},

where ≈ means ≪ and ≫. Therefore

f′(x_{n}) = −\frac{1}{2π} (t ∑_{v=1}^{n} \frac{γ_{n}^{(v)}}{|μ^{(v)}|} − m (∑_{v=1}^{n} \frac{b_{v}γ_{n}^{(v)}}{μ^{(v)}} − ia_{v} ∑_{v=r_{1}+1}^{r_{1}+r_{2}} \frac{∂}{∂x_{n}} log \frac{μ^{(v)}}{|μ^{(v)}|}))
≈ mM^{−1}

Similarly we compute

\frac{∂^{2}}{∂x_{n}^{2}} log \frac{μ^{(v)}}{|μ^{(v)}|}

= \frac{−2(A^{(v)}Im(γ_{n}^{(v)}) − B^{(v)}Re(γ_{n}^{(v)}))(|γ_{n}^{(v)}|^{2}x_{n} + A^{(v)}Re(γ_{n}^{(v)}) + B^{(v)}Im(γ_{n}^{(v)}))}{((x_{n}Re(γ_{n}^{(v)}) + A^{(v)})^{2} + (x_{n}Im(γ_{n}^{(v)}) + B^{(v)})^{2})^{2}}

≈ M^{−2}

and

\frac{∂^{3}}{∂x_{n}^{3}} log \frac{μ^{(v)}}{|μ^{(v)}|}

= \frac{−2(A^{(v)}Im(γ_{n}^{(v)}) − B^{(v)}Re(γ_{n}^{(v)}))}{((x_{n}Re(γ_{n}^{(v)}) + A^{(v)})^{2} + (x_{n}Im(γ_{n}^{(v)}) + B^{(v)})^{2})^{3}}

× (|γ_{n}^{(v)}|^{2}|μ^{(v)}|^{2} − 4(|γ_{n}^{(v)}|^{2}x_{n} + A^{(v)}Re(γ_{n}^{(v)}) + B^{(v)}Im(γ_{n}^{(v)}))^{2})

≈ M^{−3}.

These give

f′′(x_{n}) ≈ \frac{m}{M^{2}} and f′′′(x_{n}) ≈ \frac{m}{M^{3}} .

By Lemma 4.1 we have

∑_{k_{n}M≤m_{n}<(k_{n}+1)M} e^{2πif(m_{n})} ≪ M (\frac{m}{M^{3}} )^{1/6} + M^{1/2} (\frac{m}{M^{3}} )^{−1/6}
= M^{1/2}m^{1/6} + Mm^{−1/6}.

By estimating the sums over m_{q} (1 ≤ q ≤ n − 1) trivially, we have

∑_{(m_{1},...,m_{n})∈Q} F (∑_{i=1}^{n} m_{i}γ_{i}) ≪ M^{n−1}(M^{1/2}|m|^{1/6} + M|m|^{−1/6})
= M^{n− \frac{1}{2}} |m|^{\frac{1}{6}} + M^{n}|m|^{− \frac{1}{6}} .

This gives the conclusion. □

Corollary 4.3. Put S_{𝔞}(A) := S_{𝔞}(A, 2A) which is defined by (3.1). Then

S_{𝔞}(A) ≪ A^{1− \frac{4}{9n}} |m|^{\frac{1}{6}} + A|m|^{− \frac{1}{6}} .

Proof. The number of cubes Q such that Q ⊂ f_{J}^{−1}(V) in (3.3) is O(AM^{−n}) = O(A^{1/9}). The number of subsets J in (3.2) is a constant 2^{r_{1}}. Then from (3.2), Lemma 3.2 and Lemma 4.2 with B = 2A, we reach the conclusion. □

Theorem 4.4. The Hecke L-functions for arbitrary number field K of degree n satisfies the following estimate for fixed vector a and as |m| → ∞:

L(\frac{1}{2} + it, λ^{ma}) ≪_{t,ε} |m|^{\frac{n}{4} − \frac{1}{18} +ε}

Proof. Let 𝔞_{1}, ..., 𝔞_{h} be the representatives of ideal classes of K. By Corollary 2.8 it suffices to estimate the sum

∑_{{𝔟:ideal \atop 0<N(𝔟)≤|m|^{n/2}}} \frac{χλ^{m}(𝔟)}{N(𝔟)^{\frac{1}{2} +it}} = ∑_{j=1}^{h} \frac{χλ^{m}(𝔞_{j})}{N(𝔞_{j})^{\frac{1}{2} +it}} ∑_{{𝔟∈L(𝔞_{j}) \atop 0<N(𝔟)≤|m|^{n/2}}} \frac{χλ^{m}(𝔟)}{N(𝔟)^{\frac{1}{2} +it}} .

The inner sum over principal ideals 𝔟 = (μ) with μ ∈ O_{K} is estimated by the standard diadic decomposition

∑_{k=1}^{ν} \frac{S_{𝔞}(2^{k})}{2^{k/2}} ≪ |m|^{\frac{n}{4} − \frac{1}{18} +ε}

with ν = \frac{n}{2} log_{2} |m|. □

References

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Department of Mathematics, Keio University, 3 Hiyoshi, Kohoku, Yokohama 223-8522, Japan