A POISSON SUMMATION FORMULA

FOR EXTENSIONS OF NUMBER FIELDS

Eduardo Friedman and Nils-Peter Skoruppa

Abstract

We establish a formula of Poisson summation type for sums involving relative norms in an extension of number fields.

1. Introduction

Sums of the form ∑ a_{n}f(n) can sometimes be handled by writing

∑ a_{n}f(n) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} L(s)F (s)ds,

where F (s) = ∫_{0}^{∞} x^{s}f(x)\frac{dx}{x} and L(s) = ∑ a_{n}n^{−s}. If L(s) has a meromorphic continuation to ℂ and a functional equation of the usual kind, relating L(s) to some L̃(1−s), one way of proceeding is to shift the contour to Re(s) = 1−c, and then apply the functional equation. The result is a formula

∑ a_{n}f(n) = A_{L,f} + ∑ b_{n}f̃(n),  (1.1)

where A_{L,f} is a term coming from possible residues of L(s)F (s), the b_{n} are the coefficients of L̃(s), and f̃ is an integral transform of the form f̃(x) = ∫_{0}^{∞} k(xy)f(y) dy, k being an inverse Mellin transform of a ratio of Γ-factors times an exponential.

When L(s) is the Riemann zeta function, (1.1) is just the Poisson summation formula [7, §2.8], which is well-known to be at the heart of the functional equation. When L(s) is the zeta function of an ideal class in some number field L, (1.1) relates

∑_{\begin{matrix}α∈𝔞/E_{L} \\ α≠0 \end{matrix}} f(| N_{L/ℚ} α|)  to  ∑_{\begin{matrix}β∈𝔞̂/E_{L} \\ β≠0 \end{matrix}} f̃(| N_{L/ℚ} β|).  (1.2)

Here the sums are taken modulo the natural action of the units E_{L}, N_{L/ℚ} denotes the norm from L to the rational field ℚ, 𝔞 is a fractional ideal and 𝔞̂ = 𝔞^{−1}𝔡_{L}^{−1}, where 𝔡_{L} denotes the (absolute) different ideal of L. Conversely, from (1.2) one can prove the

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 11R47, 11R41, 11R42.

Research supported by Fondecyt grant and 196-0867 and by European Union grant CI1*-CT93-0353.

functional equation of the ideal class zeta function by considering the Mellin transform of t ↦ ∑_{α} f(t · | N_{L/ℚ} α|). This is the essence of Hecke’s original proof.

We consider now (1.2) with ℚ replaced by a subfield K ⊂ L. The corresponding sum is

∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} f(||N_{L/K} α||),

where ||δ|| = (||δ||_{1}, ..., ||δ||_{m}) for δ ∈ K, the || ||_{j} run over the archimedean absolute values of K (normalized so that ∏_{j} ||δ||_{j} = | N_{K/ℚ} δ|), the sum over α is taken modulo the natural action of the relative units

E = E_{L/K} = {ε ∈ E_{L}| ||N_{L/K} ε||_{j} = 1 for 1 ≤ j ≤ m},  (1.3)

and f : ℝ_{+}^{m} → ℂ. By analogy with the classical case, to handle the sum one might consider the Dirichlet series in m complex variables s = (s_{1}, …, s_{m})

∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} ||N_{L/K} α||^{−s},  (1.4)

where ||δ||^{−s} := ∏_{j=1}^{m} ||δ||_{j}^{−s_{j}}. This series diverges for every s if m > 1. However, the formal inverse Mellin transform of (1.4) (times suitable Γ-factors) converges and satisfies a functional equation closely related to that of Hecke’s Θ-function [4, ch. XIII, §2]. We used it in [3] to prove a lower bound for the ratio of regulators Reg(L)/ Reg(K), a problem first raised by Bergé and Martinet [1] .

In this paper we prove

THEOREM. Let L/K be an extension of number fields and 𝔞 a fractional ideal of L. Then

∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} f(||N_{L/K} α||) = \frac{2^{r_{2}}}{\sqrt{d_{𝔞}}} ∑_{\begin{matrix}β∈𝔞̂/E \\ β≠0 \end{matrix}} f̃(||N_{L/K} β||) + \frac{2^{r_{1}}(2π)^{r_{2}} Reg(L/K)}{\sqrt{d_{𝔞}}w_{L}} ∫_{ℝ_{+}^{m}} f(y) dy.

Assumptions and notation are as follows. The function f : ℝ_{+}^{m} → ℂ is assumed C^{∞} and compactly supported (i. e. supp(f) ⊂ [a, b]^{m}, where 0 < a < b < ∞). This ensures that both sums above converge absolutely. In fact, the first sum is even finite. The transform f̃ is given on x = (x_{1}, …, x_{m}) ∈ ℝ_{+}^{m} by

f̃(x) = π^{[L:ℚ]/2} ∫_{ℝ_{+}^{m}} f(y) ∏_{j=1}^{m} k_{p_{j},q_{j}}(x_{j}y_{j}) dy_{1} ⋯ dy_{m},  (1.5)

where p_{j} (respectively, q_{j}) is the number of real (respectively, complex) places of L above || ||_{j}, and, for t > 0,

k_{p,q}(t) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} t^{−z}(\frac{π^{−z}Γ(\frac{z}{2})}{Γ(\frac{1−z}{2} )})^{p}(\frac{(2π)^{−2z}Γ(z)}{Γ(1 − z)})^{q} dz  (0 < c < \frac{1}{6}).  (1.6)

The term Reg(L/K) in the theorem is the relative regulator [1] [2]

Reg(L/K) = \frac{1}{[E_{K} : μ_{K} N_{L/K}(E_{L})]} \frac{Reg(L)}{Reg(K)}  (1.7),

where μ_{K} is the group of roots of unity in K. The remaining notation is as follows: r_{2} (respectively, r_{1}) is the number of complex (respectively, real) places of L,

w_{L} = |μ_{L}|, d_{𝔞} = |disc_{L}| N_{L/ℚ} 𝔞^{2},

where disc_{L} denotes the absolute discriminant of L. We remark that 𝔞̂ = 𝔞^{−1}𝔡_{L}^{−1} is still defined exactly as in the classical case, using the absolute different of L.

The expression k_{p,q} in (1.6) may look somewhat forbidding. In fact,

k_{1,0}(t) = cos(2πt), k_{0,1}(t) = J_{0}(4π\sqrt{t}), (J_{0} = Bessel function),

and, by the formalism of Mellin transforms, k_{p,q} = k_{1,0}^{∗p} ∗ k_{0,1}^{∗q}, i. e. k_{p,q} is just a multiplicative (Mellin) convolution of copies of cosine and J_{0}(4π\sqrt{t}). The function k_{2,0} is classical

k_{2,0}(t) = \frac{4}{π} K_{0}(4π\sqrt{t}) − 2Y_{0}(4π\sqrt{t}),

K_{0} and Y_{0} being the usual Bessel functions [7, p. 64]. In general, k_{p,q} behaves quite like k_{2,0}. As t ↦ 0^{+}, k_{p,q}(t) tends to +∞ like (− log t)^{p+q−1}. This can be checked by simply shifting the contour in (1.6) slightly to the left of the imaginary axis. As t ↦ +∞, k_{p,q}(t) oscillates in sign indefinitely and decays like t^{− \frac{1}{2} + \frac{1}{2(p+2q)}} [5, p. 191].

The proof of the theorem shows, essentially, that the theorem holds for f = κ_{E} that are averages over E = E ⊗_{ℤ} ℝ of a function κ on L_{∞} = L ⊗_{ℚ} ℝ satisfying the usual Poisson summation formula. The simplifying assumption we make in the theorem, that f be smooth and compactly supported, is certainly excessive. It would be interesting to obtain simple sufficient conditions for f to be of the form f = κ_{E}. We give, in a remark after Corollary 2.4, a necessary condition in terms of the Mellin transform of f.

2. Proof of the Theorem

We start by giving an outline of the proof, thereby introducing the notations used in the sequel.

The notation for the archimedean places that we used in §1, i. e. just numbering them, is convenient when dealing with one number field. It becomes unwieldy when one has to constantly pass from a field L to a subfield K. We therefore adopt for the proof a more intrinsic notation. We let A_{K} denote the set of archimedean places of K. We will consistently use v for elements of A_{L} and w for those of A_{K}. By v|w we mean, as usual, that v lies above w. We let

e_{v} = \{ \begin{matrix}1 & if v is real \\ 2 & if v is complex \end{matrix} .  (2.1)

By fixing a choice of embedding at each complex v, we shall identify L with its image in the sum of its archimedean completions

L_{∞} = ⊕L_{v} = ℝ^{r_{1}} × ℂ^{r_{2}}.

On the real vector space L_{∞} we define a symmetric, real, bilinear form <, > by

<y, z >= ∑_{v∈A_{L}} e_{v} Re(y_{v}z_{v}).  (2.2)

Note that <y, z > = Trace_{L/ℚ}(yz) for y and z in L.

Set

G = ℝ_{+}^{A_{L}}, H = ℝ_{+}^{A_{K}} .

Note that G ⊂ L_{∞} as ℝ_{+} ⊂ L_{v}. Let the homomorphism δ: G ↦ H and the subgroup E ⊂ G be defined by

(δ(g))_{w} := ∏_{v|w} g_{v}^{e_{v}},  E := ker(δ),  (2.3)

with e_{v} as in (2.1). Fix a section σ: H → G of δ by setting for v|w,

(σ(h))_{v} = hw^{\frac{1}{ew[L:K]}} .  (2.4)

Thus δ(σ(h)) = h, and the map

ρ: E × H → G,  (x, h) ↦ g = xσ(h).  (2.5)

is an isomorphism. Let μ_{E} be a Haar measure on E.

Choose a C^{∞} compactly supported function φ : E → ℝ such that

∫_{E} φ(x) dμ_{E}(x) = 1.

For y ∈ L_{∞}^{∗} (so y_{v} ≠ 0 for all v), set |y| := (|y_{v}|) ∈ G and

κ(y) := f(δ(|y|))φ(ν(|y|))  (2.6),

where, for any g ∈ G we use ν(g) for the unique element of E such that g = σ(δ(g)) ν(g).

As f (as in the theorem) and φ are C^{∞} and compactly supported, κ vanishes in a neighborhood of the coordinate axes. It therefore extends to a C^{∞}, compactly supported, coordinate-wise radial κ : L_{∞} → ℂ, i. e. κ((z_{v})) = κ((|z_{v}|)) for (z_{v}) ∈ L_{∞}. For γ : G → ℂ and h ∈ H, let

γ_{E}(h) = ∫_{E} γ(σ(h)x) dμ_{E}(x).  (2.7)

Thus,

κ_{E}(h) = ∫_{E} κ(σ(h)x)dμ_{E}(x) = ∫_{E} f(δ(σ(h)))φ(x)dμ_{E}(x) = f(h).

We note that even when L = K, so E is the trivial group, γ_{E} may not quite equal γ as δ is not the identity if K has complex places.

We now apply, for fixed x ∈ E, the usual Poisson summation formula to the function κ_{x}(z) := κ(xz). This yields

∑_{a∈𝔞} κ(xa) = \frac{2^{r_{2}}}{\sqrt{d_{𝔞}}} ∑_{b∈𝔞̂} κ̂(x^{−1}b).  (2.8)

Here, for any κ ∈ L^{1}(L_{∞}, dy), where dy is the standard Lebesgue measure, we define the transform κ̂ : L_{∞} → ℂ by

κ̂(z) = ∫_{L_{∞}} κ(y) e^{−2πi<y,z>} dy.

For the justification of formula (2.8) see Lemma 2.7 below. One needs also κ̂_{x}(b) = κ̂(x^{−1}b), which follows easily from the fact that the natural multiplicative action of E on L_{∞}, i. e. (z_{v}) ↦ (x_{v}z_{v}) for (x_{v}) ∈ E, is self-adjoint with respect to the bilinear form <, > defined in (2.2), and has determinant ∏_{v∈A_{L}} x_{v}^{e_{v}} = 1.

The relative units E = E_{L/K} map to E (see (1.3) and (2.3)) with kernel μ_{L} by ε ↦ |ε| = (|ε|_{v}), where | |_{v} is the absolute value corresponding to v ∈ A_{L} (the normalized absolute value is || ||_{v} = | |_{v}^{e_{v}}). Let Ẽ be the image of E in E and F ⊂ E a compact fundamental domain for the action of Ẽ on E.

We integrate (2.8) over a (compact) fundamental domain for the action of Ẽ on E. The left-hand side yields

∫_{x∈F} ∑_{a∈𝔞} κ(xa)dμ_{E}(x) = κ(0)μ_{E}(F) + ∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} ∑_{ε∈E} ∫_{x∈F} κ(x|ε|α)dμ_{E}(x)

= κ(0)μ_{E}(F) + w_{L} ∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} ∑_{|ε|∈Ẽ} ∫_{x∈|ε|F} κ(xα)dμ_{E}(x)

= κ(0)μ_{E}(F) + w_{L} ∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} ∫_{E} κ(xα)dμ_{E}(x)

= κ(0)μ_{E}(F) + w_{L} ∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} κ_{E}(||N_{L/K} α||).

In the last step we used the invariance of the Haar measure under translation by ν(α) ∈ E, together with α = σ(δ(α))ν(α) = σ(||N_{L/K} α||)ν(α) (see (2.3) to (2.7)). As F is compact, there is no problem in taking the integral over F across the sum, as we did above. The right-hand side of (2.8) is treated similarly. Since f is assumed compactly supported, κ(0) = 0. In short, as f = κ_{E},

∑_{\begin{matrix}α∈𝔞/E \\ α≠0 \end{matrix}} f(||N_{L/K} α||) = \frac{2^{r_{2}}}{\sqrt{d_{𝔞}}} (\frac{μ_{E}(F)}{w_{L}} κ̂(0) + ∑_{\begin{matrix}β∈𝔞̂/E \\ β≠0 \end{matrix}} (κ̂)_{E}(||N_{L/K} β||)),

where the sums converge absolutely.

In this formula μ_{E} could be any Haar measure. We now normalize it by

μ_{E}(F) = Reg(L/K),  (2.9)

where Reg(L/K) is given by (1.7). It remains to calculate κ̂(0) and (κ̂)_{E}. This will be done in a series of lemmas regarding Mellin transforms on the groups G and H (see Lemmas 2.1, 2.2, 2.3, 2.6 and Corollaries 2.4, 2.5). The result will be (see Corollary 2.5) that (κ̂)_{E} equals f̃ as given in equation (1.5). Moreover,

κ̂(0) = 2^{r_{1}}π^{r_{2}} ∫_{R_{+}^{A_{K}}} f(x) dx

(see Lemma 2.6). This then proves the theorem.

We now turn to Mellin transforms on G and H. For g = (g_{v}) ∈ G, let ||g||^{2} := ∑_{v∈A_{L}} |g_{v}|^{2}.

LEMMA 2.1. Let γ : G → ℂ be continuous and satisfy the estimate |γ(g)| = O(1 + ||g||^{2})^{−d} for some d > [L : ℚ]/2. Then the integral (2.7) defining γ_{E} converges absolutely. Furthermore, for h = (h_{w}) ∈ H, |γ_{E}(h)| < ∏_{w∈A_{K}} b_{w}(h_{w}), where b_{w} : ℝ_{+} → ℝ_{+} is continuous and satisfies

b_{w}(r) < \{ \begin{matrix}P_{w}(| log r|) & if r ≤ 1 \\ Cr^{−(1+Δ)} & if r ≥ 1 \end{matrix} ,

with constants C, Δ > 0, and P_{w} a real polynomial of degree strictly less than the number of places v lying above w.

Proof. For g ∈ G, we have for some constants c and ε > 0,

|γ(g)| < c ∏_{w∈A_{K}}(1 + ||g||^{2})^{−ε− \frac{1}{2} ∑_{v|w} e_{v}} ≤ c ∏_{w∈A_{K}} \frac{1}{(1 + ||g||_{w}^{2})^{ε+ \frac{1}{2} ∑_{v|w} e_{v}}} ,  (2.10)

where ||g||_{w}^{2} := ∑_{v|w} |g_{v}|^{2} and e_{v} is as in (2.1). We may assume L ≠ K, for otherwise the lemma follows easily from (2.10). The group E = ∏_{w} E_{w}, the Haar measure μ_{E} and the bound (2.10), all decompose naturally into a product of the corresponding objects for each w ∈ A_{K}. Thus we work at each w. We parametrize E_{w} by y = (y_{1}, y_{2}, …, y_{n}, Y), where the y_{i} are free positive variables for 1 ≤ i ≤ n, the number of v above w is n + 1, and

Y = ∏_{i=i}^{n} y_{i}^{−e_{i}/e_{n+1}},  e_{i} = 1 or 2,  ∑_{i=1}^{n+1} e_{i} = e_{w}[L : K].

The corresponding measure dμ_{w} is just \frac{dy_{1}}{y_{1}} ⋯ \frac{dy_{n}}{y_{n}}. In view of (2.7), (2.4) and (2.10), we

need to bound for t = r^{\frac{1}{ew[L:K]}} ,

b_{w}(r) = B_{n}(t) = ∫_{y_{n}=0}^{∞} ⋯ ∫_{y_{1}=0}^{∞} \frac{1}{(1 + t^{2}(y_{1}^{2} + y_{2}^{2} + ⋯ + y_{n}^{2} + Y^{2}))^{b}} \frac{dy_{1}}{y_{1}} ⋯ \frac{dy_{n}}{y_{n}} , (2.11)

where b = ε + \frac{1}{2} ∑_{j=1}^{n+1} e_{j} > \frac{n+1}{2}. After changing variables to x_{j} = ty_{j}, the above integral becomes

B_{n} = ∫_{x_{n}=0}^{∞} ⋯ ∫_{x_{1}=0}^{∞} \frac{1}{(1 + (r^{1/e}X)^{2} + ∑_{j=1}^{n} x_{j}^{2})^{b}} \frac{dx_{1}}{x_{1}} ⋯ \frac{dx_{n}}{x_{n}}

with X = ∏_{i=i}^{n} x_{i}^{−e_{i}/e_{n+1}} and e = e_{n+1}. Set Δ = ε/(e(2n + \frac{1}{2})). Then

(1 + (r^{1/e}X)^{2} + ∑_{j=1}^{n} x_{j}^{2})^{b} > r^{1+Δ}X^{e(1+Δ)}(1 + ∑_{j=1}^{n} x_{j}^{2})^{2neΔ+ \frac{1}{2} ∑_{i=1}^{n} e_{i}}

= r^{1+Δ} ∏_{j=1}^{n} x_{j}^{−(1+Δ)e_{j}}(1 + ∑_{j=1}^{n} x_{j}^{2})^{2neΔ+ \frac{1}{2} ∑_{i=1}^{n} e_{i}}

≥ \frac{r^{1+Δ}}{x_{1}x_{2} ⋯ x_{n}} ∏_{j=1}^{n} \frac{(1 + x_{j}^{2})^{2Δ+ \frac{1}{2} e_{j}}}{x_{j}^{e_{j}(1+Δ)−1}} .

Thus,

B_{n} < \frac{1}{r^{1+Δ}} ∏_{j=1}^{n} ∫_{x_{j}=0}^{∞} \frac{x_{j}^{e_{j}(1+Δ)−1}}{(1 + x_{j}^{2})^{2Δ+ \frac{1}{2} e_{j}}} dx_{j} = C/r^{1+Δ},

as e_{j} is 1 or 2. This proves the lemma for r ≥ 1.

For r < 1 we return to (2.11), but now make the substitution x_{j} = y_{j}^{2/f_{j}}, where f_{j} = 2e/e_{j} ≥ 1. Then

B_{n}(t) = ∫_{x_{n}=0}^{∞} ⋯ ∫_{x_{1}=0}^{∞} \frac{1}{(1 + t(X^{2} + ∑_{j=1}^{n} x_{j}^{f_{j}}))^{b}} \frac{dx_{1}}{x_{1}} ⋯ \frac{dx_{n}}{x_{n}} ,

which we must bound above, for t = r^{\frac{1}{ew[L:K]}} < 1, by a polynomial of degree n in | log t|. For n = 1 we have to estimate the integral

B_{1}(t) = ∫_{0}^{∞} \frac{1}{(1 + t(x^{f} + x^{−2}))^{b}} \frac{dx}{x},

where f ≥ 1 and b > 1. On dividing the region of integration into (0, 1) and (1, ∞), and transforming the former interval into the latter by x ↦ x^{−1}, we find that B_{1}(t) is bounded above by

2 ∫_{1}^{∞} \frac{1}{1 + tx} \frac{dx}{x} = 2 log(t + 1) − 2 log t ≤ log 4 + 2| log t|,

which is indeed a polynomial of degree 1 in | log t|. We compute B_{n+1}(t) by making the substitution x = x_{n+1}, z_{j} = x^{a/f_{j}}x_{j}, (1 ≤ j ≤ n), where

a = \frac{2}{1 + 2 ∑_{j=1}^{n} f_{j}^{−1}} .

After some algebra we obtain

B_{n+1}(t) = ∫_{0}^{∞} \frac{1}{(1 + tx^{f})^{b}} B_{n}(\frac{t}{(1 + tx^{f})x^{a}} ) \frac{dx}{x},

where f = f_{n+1}. For x > 0, the function g_{t}(x) := \frac{t}{(1+tx^{f})x^{a}} is monotone decreasing in x with range (0, ∞). Let α_{t} be such that g_{t}(α_{t}) = 1. Thus g_{t}(x) > 1 for x < α_{t} and g_{t}(x) < 1 for x > α_{t}. By induction on n we have then that B_{n+1}(t) is bounded above by,

C ∫_{0}^{α_{t}} \frac{1}{(1 + tx^{f})^{b}} \frac{(1 + tx^{f})x^{a}}{t} \frac{dx}{x} + ∫_{α_{t}}^{∞} \frac{1}{(1 + tx^{f})^{b}} P (| log(\frac{t}{(1 + tx^{f})x^{a}})|)\frac{dx}{x}

for some constant C and some real polynomial P of degree at most n (we have dropped the Δ from the upper bound B_{n}(g_{t}(x)) < Cg_{t}(x)^{−1−Δ}, as we may for g_{t}(x) > 1). The first integral is bounded above by

\frac{1}{t} ∫_{0}^{α_{t}} x^{a} \frac{dx}{x} = \frac{α_{t}^{a}}{a t} = \frac{1}{a(1 + tα_{t}^{f})} < \frac{1}{a}.

Set β_{t} := (t^{2}/2)^{1/a} and δ_{t} = t^{−1/f}. Then,

g(β_{t}) = \frac{2}{t + \frac{t^{2+(2f/a)}}{2^{f/a}}} > 1 (0 < t < 1).

Hence, β_{t} < α_{t}. To bound the second integral it suffices to take P to be a monomial of degree j ≤ n and to show, for 0 ≤ k ≤ j, that

(∫_{β_{t}}^{δ_{t}} + ∫_{δ_{t}}^{∞}) \{\frac{1}{(1 + tx^{f})^{b}} ( log(1 + tx^{f}))^{j−k}| log x|^{k} \frac{dx}{x}\} ,

is bounded above by a polynomial in | log t| of degree at most j + 1. But 1 ≤ 1 + tx^{f} ≤ 2 for β_{t} ≤ x ≤ δ_{t}. Hence the integral over the first interval is bounded above by

∫_{β_{t}}^{δ_{t}} | log x|^{k} \frac{dx}{x} ≤ \frac{1}{k + 1}(| log β_{t}|^{k+1} + | log δ_{t}|^{k+1}),

which is indeed a polynomial of degree at most j + 1 in | log t|. The integral over (δ_{t}, ∞) can be handled by the change of variable y = tx^{f}, which turns it into

\frac{1}{f^{k+1}} ∑_{l=0}^{k} \binom{k}{l} (− log t)^{l} ∫_{1}^{∞} \frac{( log y)^{k−l}( log(1 + y))^{j−k}}{(1 + y)^{b}} \frac{dy}{y} ,

which is again a polynomial of the kind required.

In Lemma 2.1, μ_{E} could be any Haar measure on E. We now make a choice of μ_{E} as in (2.9). For G = ℝ_{+}^{A_{L}} and H = ℝ_{+}^{A_{K}} we use the usual Haar measures

dμ_{G}(g) = ∏_{v∈A_{L}} \frac{dg_{v}}{g_{v}} , dμ_{H}(h) = ∏_{w∈A_{K}} \frac{dh_{w}}{h_{w}} .

The image μ_{G} ∘ ρ (see (2.5)) of the Haar measure μ_{G} under ρ is a Haar measure on E × H. Hence there exists a constant c such that

c μ_{G} ∘ ρ = μ_{E} × μ_{H}.  (2.12)

By using an expression for Reg(L/K) as a determinant of logarithms of units, one shows c = 2^{r_{2}} [3, proof of Prop. 3.1]. Thus, our choice of μ_{E} is a natural one.

Given f : H → ℂ, its Mellin transform M_{H}f is defined for s = (s_{w}) ∈ ℂ^{A_{K}} by

(M_{H}f)(s) = ∫_{H} f(h)h^{s} dμ_{H}(h), h^{s} := ∏_{w∈A_{K}} h_{w}^{s_{w}}.

Similarly we define for γ : G → ℂ a Mellin transform M_{G}γ with variable S = (S_{v}) ∈ ℂ^{A_{L}}.

LEMMA 2.2. For γ as in Lemma 2.1, (M_{H}γ_{E})(s) converges in the domain 0 < Re(s_{w}) < 1 + Δ, and (M_{G}γ)(S) converges for 0 < Re(S_{v}) < e_{v}(1 + Δ) for some Δ > 0 and e_{v} = 1 or 2 as in (2.1). For such s the two transforms are related by

(M_{H}γ_{E})(s) = 2^{r_{2}}(M_{G}γ)(es), where (es)_{v} := e_{v}s_{w} for v|w.

Proof. The convergence of M_{H}γ_{E} in the stated domain follows from Lemma 2.1. That of (M_{G}γ)(S) follows from

(1 + ||g||^{2})^{d} > (1 + ||g||^{2})^{\frac{(1+2Δ)[L:ℚ]}{2}} = ∏_{v}(1 + ||g||^{2})^{\frac{(1+2Δ)ev}{2}} ≥ ∏_{v}(1 + g_{v}^{2})^{\frac{ev}{2} +Δ},

where d > [L : ℚ]/2 satisfies |γ(g)| = O(1 + ||g||^{2})^{−d}), as in Lemma 2.1. Now,

(M_{H}γ_{E})(s) = ∫_{H} ∫_{E} γ(xσ(h))h^{s} dμ_{E}(x)dμ_{H}(h)  (see (2.7))

= ∫_{E×H} γ(xσ(h))h^{s} (dμ_{E} × dμ_{H})(x, h)

= c ∫_{E×H} γ(ρ(x, h))δ(ρ(x, h))^{s} d(μ_{G} ∘ ρ)(x, h) (see (2.12))

= c ∫_{G} γ(g) δ(g)^{s} dμ_{G}(g)

= c ∫_{G} γ(g) ∏_{w∈A_{K}} ∏_{v|w} g_{v}^{e_{v}s_{w}} dμ_{G}(g)

= c · (M_{G}γ)(es).

As c = 2^{r_{2}}, the lemma follows.

In the next lemma, κ is defined on L_{∞}. As G ⊂ L_{∞}, by M_{G}κ we mean the Mellin transform of the restriction of κ to G. A similar remark applies to κ_{E}.

LEMMA 2.3. Suppose κ : L_{∞} → ℂ satisfies |κ(y)| = O(1+||y||^{2})^{−d}, where d > [L : ℚ]/2, and that κ and all its partial derivatives of order ≤ [L : ℚ] + 2 are in L^{1}(L_{∞}, dy). Suppose also that κ is coordinate-wise radial, i. e. κ((z_{v})) = κ((|z_{v}|)) for (z_{v}) ∈ L_{∞}. Then, for any S = (S_{v}) with 0 < Re(S_{v}) < e_{v}, we have

(M_{G}κ̂)(S) = (M_{G}κ)(e − S) · π^{[L:ℚ]/2} ∏_{v∈A_{L}}(e_{v}π)^{−S_{v}} \frac{Γ(\frac{S_{v}}{2} )}{Γ(\frac{e_{v}−S_{v}}{2} )},

where (e − S)_{v} := e_{v} − S_{v}.

We shall give the proof after extracting two corollaries.

COROLLARY 2.4. Suppose f, g : H → ℂ are of the form f = κ_{E} and g = (κ̂)_{E} for some κ as in Lemma 2.3. Then for any s = (s_{w}) with 0 < Re(s_{w}) < 1,

(M_{H}g)(s) = (M_{H}f)(1 − s) · π^{[L:ℚ]/2} ∏_{w∈A_{K}}(\frac{π^{−s_{w}}Γ(\frac{s_{w}}{2} )}{Γ(\frac{1−s_{w}}{2} )})^{r_{1}(w)}(\frac{(2π)^{−2s_{w}}Γ(s_{w})}{Γ(1 − s_{w})})^{r_{2}(w)},

where (1−s)_{w} = 1−s_{w}, and r_{1}(w) (respectively, r_{2}(w)) is the number of real (respectively, complex) places of L above w.

Proof. We set c = 2^{r_{2}} and calculate, using Lemmas 2.2 and 2.3,

(M_{H}g)(s) = c · (M_{G}κ̂)(es) = c · (M_{G}κ)(e − es) · π^{\frac{[L:ℚ]}{2}} ∏_{w∈A_{K}} ∏_{v|w}(e_{v}π)^{−e_{v}s_{w}} \frac{Γ(\frac{e_{v}s_{w}}{2} )}{Γ(\frac{e_{v}−e_{v}s_{w}}{2} )}.

But

c · (M_{G}κ)(e − es) = (M_{H}κ_{E})(1 − s) = (M_{H}f)(1 − s).

We remark that Lemma 2.1 and the above corollary imply that if f = κ_{E}, then

(M_{H}f)(s) ∏_{w∈A_{K}} (\frac{π^{\frac{sw}{2}}}{Γ(\frac{s_{w}}{2} )})^{r_{1}(w)}(\frac{(2π)^{s_{w}}}{Γ(s_{w})} )^{r_{2}(w)}

can be continued analytically to −Δ < Re(s_{w}) < 1 + Δ for some Δ > 0 (and further if we place stronger decay and smoothness conditions on κ). This gives a necessary condition for f to be of the form f = κ_{E}.

Let k_{p,q} : ℝ_{+} → ℝ be given for p + 2q ≥ 3 by

k_{p,q}(t) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} t^{−z}(\frac{π^{−z}Γ(\frac{z}{2})}{Γ(\frac{1−z}{2} )})^{p}(\frac{(2π)^{−2z}Γ(z)}{Γ(1 − z)})^{q} dz (0 < c < \frac{1}{2} − \frac{1}{p + 2q} ). (2.13)

Stirling’s formula shows that the integral in (2.13) is absolutely convergent, and independent of c, for c and (p, q) as specified. For p + 2q < 3, set

k_{2,0}(t) = \frac{4}{π} K_{0}(4π\sqrt{t}) − 2Y_{0}(4π\sqrt{t}), k_{0,1}(t) = J_{0}(4π\sqrt{t}), k_{1,0}(t) = cos(2πt), (2.14)

K_{0}, Y_{0} and J_{0} being the usual Bessel functions. If p + 2q < 3, the integral (2.14) does equal the corresponding lim_{T ↦∞} ∫_{c−iT}^{c+iT} in (2.13) for any 0 < c < \frac{1}{2} [7, §7.9], but we prefer the explicit definition given.

COROLLARY 2.5. Suppose f = κ_{E} and g = (κ̂)_{E} for some κ as in Lemma 2.3. Then,

g(h) = π^{[L:ℚ]/2} ∫_{ℝ_{+}^{A_{K}}} f(x)k(xh) dx,  (h = (h_{w}) ∈ ℝ_{+}^{A_{K}} )  (2.15)

where dx is Lebesgue measure on ℝ^{A_{K}}, the integral (2.15) is (absolutely) convergent, xh = (x_{w}h_{w}) and

k(h) = ∏_{w∈A_{K}} k_{w}(h_{w}), k_{w} = k_{r_{1}(w),r_{2}(w)},

with k_{p,q} as in (2.13) and (2.14), and r_{1}(w), r_{2}(w) as in the previous corollary.

Proof of Corollary 2.5. Mellin inversion suggests

g(h) = \frac{1}{(2πi)^{m}} ∫_{I_{c}} h^{−s}(M_{H}g)(s) ds,  (2.16)

where m = |A_{K}| = r_{1}(K) + r_{2}(K), s ∈ ℂ^{m} and ∫_{I_{c}} ds denotes an m-fold integral over s = (s_{w}) satisfying Re(s_{w}) = c. Standard results [6, ch. 1, Cor. 1.21] show that (2.16) holds for almost all h if we are granted the (absolute) convergence for s ∈ I_{c} of (M_{H}g)(s) = ∫_{H} g(h)h^{s}dμ_{H}(h) and of the integral in (2.16). Lemma 2.1 shows that (M_{H}g)(s) does converges for s ∈ I_{c} if 0 < c < 1. After using Corollary 2.4, and Stirling’s formula to estimate |Γ(σ + iT)| for fixed real σ and large real T, one finds that (2.16) converges absolutely for any h if [L : K]e_{w} = r_{1}(w) + 2r_{2}(w) ≥ 3 and 0 < c < \frac{1}{2} − \frac{1}{p+2q} (we defer the case of conditional convergence, [L : K]e_{w} < 3, to the end of the proof). As g is continuous, (2.16) holds for all h.

Corollary 2.4, (2.16) and Fubini’s theorem (which applies because of absolute convergence) yield

\frac{(2πi)^{m}}{π^{[L:ℚ]/2}} g(h)

= ∫_{I_{c}} h^{−s} ∫_{ℝ_{+}^{m}} f(x)x^{−s}dx ∏_{w} (\frac{π^{−s_{w}}Γ(\frac{s_{w}}{2} )}{Γ(\frac{1−s_{w}}{2} )})^{r_{1}(w)}(\frac{(2π)^{−2s_{w}}Γ(s_{w})}{Γ(1 − s_{w})})^{r_{2}(w)} ds,

= ∫_{ℝ_{+}^{m}} ∫_{I_{c}} f(x)(xh)^{−s} ∏_{w} (\frac{π^{−s_{w}}Γ(\frac{s_{w}}{2} )}{Γ(\frac{1−s_{w}}{2} )})^{r_{1}(w)}(\frac{(2π)^{−2s_{w}}Γ(s_{w})}{Γ(1 − s_{w})})^{r_{2}(w)} dsdx

= (2πi)^{m} ∫_{ℝ_{+}^{m}} f(x)k(hx) dx.

This proves the corollary, except when [L : K]e_{w} < 3 for some w. The interesting case is [L : K] = 2 and e_{w} = 1 (when [L : K] = 1 the corollary and E are trivial). In this case we start by noting that the Mellin transform of k_{2,0} at z is absolutely convergent for 0 < Re(z) < \frac{1}{4}, its value being (π^{−z}Γ(\frac{z}{2})/Γ(\frac{1−z}{2} ))^{2} [7, p. 64]. A similar statement holds for k_{0,1}, which is the only other possibility when [L : K] = 2 and e_{w} = 1. Now apply M_{H} to the right-hand side of (2.15) for 0 < Re(s_{w}) < \frac{1}{4}. After reversing integrals, which we may by absolute convergence, and applying Corollary 2.4, we obtain (M_{H}g)(s). Therefore (2.15) holds in this case also.

Proof of Lemma 2.3. We compute κ̂(x) for x = (x_{v}) ∈ G, i. e. for x_{v} real and positive. Polar coordinates on each component L_{v} of L_{∞} yields L_{∞}^{∗} ≅ G × (S^{1})^{r_{2}} × {±1}^{r_{1}}, where S^{1} is the unit circle and L_{∞}^{∗} = {(z_{v}) ∈ L_{∞}| z_{v} ≠ 0 for all v ∈ A_{L}}. Now

κ̂(x) = ∫_{L_{∞}} e^{−2πi<x,z>}κ(z)dz

= ∫_{L_{∞}} e^{−2πi ∑_{v} e_{v} Re(x_{v}z_{v})}κ(z)dz (see (2.2))

= ∫_{G} κ(g)( ∏_{v real} 2 cos(2πx_{v}g_{v}))( ∏_{v complex} g_{v} ∫_{0}^{2π} e^{−4πi Re(x_{v}g_{v}e^{iθ})}dθ)dg

= 2^{r_{1}}(2π)^{r_{2}} ∫_{G} κ(g) ∏_{v real} cos(2πx_{v}g_{v}) ∏_{v complex} g_{v}J_{0}(4πx_{v}g_{v}) dg

= A ∫_{G} κ(g)g^{e} ∏_{v∈A_{L}} F_{v}(x_{v}g_{v}) dμ_{G}(g),

where J_{0} is the usual Bessel function and we have set

A = 2^{r_{1}}(2π)^{r_{2}}, F_{v}(r) = \{ \begin{matrix}cos(2πr) & if v is real \\ J_{0}(4πr) & if v is complex \end{matrix} .

For ε ∈ G, let Φ_{ε} : G → ℝ_{+} be given by Φ_{ε}((g_{v})) = ∏_{v∈A_{L}} e^{−ε_{v}g_{v}}. The smoothness assumption on κ allows us to apply Lemma 2.1 to κ̂. Thus (M_{G}κ̂)(S) is defined by an absolutely convergent integral for 0 < Re(S_{v}) < e_{v}, and

(M_{G}κ̂)(S) = lim_{ε↦0} (M_{G}(Φ_{ε}κ̂))(S).

Inserting Φ_{ε} will allow us to reverse integrals in the following calculation.

(M_{G}(Φ_{ε}κ̂))(S) = ∫_{G} Φ_{ε}(x)κ̂(x)x^{S} dμ_{G}(x)

= A ∫_{G} ∫_{G} κ(g)g^{e} ∏_{v} x_{v}^{S_{v}}e^{−ε_{v}x_{v}}F_{v}(x_{v}g_{v}) dμ_{G}(g)dμ_{G}(x)

= A ∫_{G} ∫_{G} κ(g)g^{e} ∏_{v} x_{v}^{S_{v}}e^{−ε_{v}x_{v}}F_{v}(x_{v}g_{v}) dμ_{G}(x)dμ_{G}(g)

= A ∫_{G} ∫_{G} g^{e−S}κ(g) ∏_{v} x_{v}^{S_{v}}e^{−ε_{v}x_{v}g_{v}^{−1}}F_{v}(x_{v}) dμ_{G}(x)dμ_{G}(g).

Recall that for any t ≥ 0, |J_{0}(t)| < C/(1 + \sqrt{t}) for some constant C. Hence for v complex and 0 < Re(S_{v}) < \frac{1}{2}, we can take the limit as ε_{v} ↦ 0 into the last integral above. On setting ε_{v} = δ > 0 for all real v, we find that \frac{1}{A}(M_{G}κ̂)(S) is given by

lim_{δ↦0} ∫_{G} ∫_{G} g^{e−S}κ(g) ∏_{v real} x_{v}^{S_{v}}e^{−δx_{v}g_{v}^{−1}}F_{v}(x_{v}) ∏_{v complex} x_{v}^{S_{v}}F_{v}(x_{v}) dμ_{G}(x)dμ_{G}(g). (2.17)

We now use

∫_{0}^{∞} t^{z}e^{−δt} cos(bt)\frac{dt}{t} = Γ(z)\frac{cos (z tan^{−1}(b/δ))}{(δ^{2} + b^{2})^{z/2}},  (b, Re(z), δ > 0)  (2.18)

and

∫_{0}^{∞} t^{z}J_{0}(tb)\frac{dt}{t} = 2^{z−1}b^{−z} \frac{Γ(\frac{z}{2})}{Γ(1 − \frac{z}{2})} (b > 0, 0 < Re(z) < \frac{1}{2}).

Note that for fixed z the absolute value of the right-hand side of (2.18) is bounded independently of δ. Hence we can explicitly evaluate the inner integrals in (2.17) and then, by dominated convergence, take the limit as δ ↦ 0 inside the integrand. Thus

(M_{G}κ̂)(S)

= A ∏_{v real}(2π)^{−S_{v}} Γ(S_{v}) cos (\frac{πS_{v}}{2} ) ∏_{v complex} \frac{(2π)^{−S_{v}}Γ(\frac{S_{v}}{2} )}{2Γ(1 − \frac{S_{v}}{2} )} ∫_{G} g^{e−S}κ(g)dμ_{G}

= A(M_{G}κ)(e − S) ∏_{v real} \frac{\sqrt{π} π^{−S_{v}} Γ(\frac{S_{v}}{2} )}{2 Γ(\frac{1−S_{v}}{2} )} ∏_{v complex} \frac{(2π)^{−S_{v}} Γ(\frac{S_{v}}{2} )}{2 Γ(1 − \frac{S_{v}}{2} )}

= π^{[L:ℚ]/2}(M_{G}κ)(e − S) ∏_{v∈A_{L}}(e_{v}π)^{−S_{v}} \frac{Γ(\frac{S_{v}}{2} )}{Γ(\frac{e_{v}−S_{v}}{2} )}.

This proves the lemma for 0 < Re(S_{v}) < \frac{1}{2}. By analytic continuation, it holds for 0 < Re(S_{v}) < e_{v}.

LEMMA 2.6. For κ as defined in (2.6), one has κ̂(0) = 2^{r_{1}}π^{r_{2}} ∫_{R_{+}^{A_{K}}} f(x)dx.

Proof. As in the proof of Lemma 2.3

κ̂(0) = 2^{r_{1}}(2π)^{r_{2}} ∫_{G} g^{e}κ(g)dμ_{G}(g) = 2^{r_{1}}π^{r_{2}}(M_{H}f)(1)

where, for the second identity, we used Lemma 2.2.

We finally give the proof of the Poisson summation formula as used in the proof of the theorem.

LEMMA 2.7. Suppose κ : L_{∞} → ℂ satisfies the estimate |κ(y)| = O(1 + ||y||^{2})^{−d}, where d > [L : ℚ]/2. Suppose also that κ and all its partial derivatives of order ≤ [L : ℚ] + 2 are in L^{1}(L_{∞}, dy). Then

∑_{a∈𝔞} κ(a) = \frac{2^{r_{2}}}{\sqrt{d_{𝔞}}} ∑_{b∈𝔞̂} κ̂(b),

where 𝔞 is a fractional ideal of L, 𝔞̂ = (𝔞𝔡_{L})^{−1}, and both sums above converge absolutely.

Proof. The bilinear form < y, z > defined in (2.2) is related to the standard real inner product and Euclidean norm

y · z = ∑_{v∈A_{L}} Re(y_{v}\overline{z}_{v}), ||y||^{2} = y · y = ∑_{v∈A_{L}} |y_{v}|^{2}

on L_{∞} by the formula

<y, z >= y · l(z) = l(y) · z,

where the linear transformation l : L_{∞} → L_{∞} is given by

(l(z))_{v} = e_{v} \overline{z}_{v}.  (2.19)

If in the definition following (2.8) of κ̂ we replace < y, z > by y · z, we obtain the usual Fourier transform of κ, which we denote temporarily by κ̃. Thus

κ̃(l(z)) = κ̂(z),

where l is given by (2.19). The smoothness assumption on κ implies the boundedness of (1 + ||y||^{2})^{\frac{[L:ℚ]+1}{2}} κ̃(y) for y ∈ L_{∞}. The Poisson summation formula then holds for κ and any lattice in L_{∞} [6, ch. VII, Cor. 2.6]. Thus,

∑_{a∈𝔞} κ(a) = \frac{1}{Vol(𝔞)} ∑_{c∈𝔞̃} κ̃(c),

where Vol(𝔞) denotes the volume of a fundamental domain for L_{∞}/𝔞 and

𝔞̃ = {z ∈ L_{∞} | z · a ∈ ℤ for all a ∈ 𝔞}.

As <, > gives the trace form on L, we have [4, ch. XIII, §2]

{z ∈ L_{∞} | <z, a> ∈ ℤ for all a ∈ 𝔞} = (𝔞𝔡_{L})^{−1} = 𝔞̂.

Thus, 𝔞̃ = l(𝔞̂). Hence

∑_{c∈𝔞̃} κ̃(c) = ∑_{b∈𝔞̂} κ̂(b).

As Vol(𝔞) = 2^{−r_{2}} \sqrt{d_{𝔞}} [4, ch. V, §2], the lemma follows.

References

[1] A.-M. Bergé and J. Martinet, ‘Notions relatives de régulateur et de hauteur’, Acta Arith. 54 (1989) 155–170.

[2] A. Costa and E. Friedman, ‘Ratios of regulators in totally real extensions of number fields’, J. Number Theory 37 (1991) 288–297.

[3] E. Friedman and N.-P. Skoruppa, ‘Relative regulators of number fields’, Invent. Math. (to appear).

[4] S. Lang, Algebraic number theory, (Addison–Wesley, Reading, 1970).

[5] Y. Luke, The special functions and their approximations, (Academic Press, New York, 1969).

[6] E. Stein and G. Weiss, Fourier Analyis on Euclidean Spaces, (Princeton Univ. Press, Prin ceton, 1975).

[7] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd edition, (Oxford Univ. Press, Oxford, 1948).

Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago 1, CHILE

E-mail address: friedman@abello.dic.uchile.cl

UFR de Mathématiques et Informatique, Université Bordeaux I, 351 cours de la Libération, 33405 Talence, FRANCE

E-mail address: skoruppa@math.u-bordeaux.fr