Relative regulators of number fields

By Eduardo Friedman and Nils-Peter Skoruppa*

Introduction

Some years ago Zimmert [Zi] devised a new analytic method to prove that the regulator of a number field L satisfies a lower bound that increases exponentially with the degree [L : ℚ]. He found numerical constants c_{0} > 0 and c_{1} > 1 such that the regulator Reg(L) satisfies

(0.1)  Reg(L) ≥ c_{0} c_{1}^{[L:ℚ]}

for any number field L.

In 1987, Bergé and Martinet [BM1-3] defined a relative regulator associated to an extension L/K of number fields. Essentially, the relative regulator turned out to be the ratio Reg(L)/ Reg(K). After studying these relative regulators by geometric methods, Bergé and Martinet asked how Zimmert’s result (0.1) might extend to them ([BM2], p. 16).

In this paper we prove:

Main Theorem. There exist absolute constants d_{0} > 0 and d_{1} > 1 such that for any extension L/K of number fields, the ratio of their regulators satisfies

(0.2)  \frac{Reg(L)}{Reg(K)} ≥ (d_{0} d_{1}^{[L:K]})^{[K:ℚ]}.

In §4 (Theorem 4.1 and Corollaries) we give precise values for d_{0} and d_{1}, as well as some improvements depending on the distribution of real and complex places of L.

For large relative degrees [L : K], inequality (0.2) implies

(0.3)  \frac{Reg(L)}{Reg(K)} ≥ d_{1}^{(1−ε)[L:ℚ]}

Typeset by AMS-TEX

*Research supported by Chilean Fondecyt grants 1068-92, 194-0353, 196-0867, by Fundación Andes, by DFG-Forschungsprojekt ‘Automorphe Formen’ Heidelberg/Mannheim AZ Pi 214/3–1, and by European Commission Grant CI1*-CT93-0353.

for a suitable small ε > 0.

By considering infinite class-field towers L_{0} ⊂ L_{1} ⊂ … one finds that Reg(L)/ Reg(K) need not grow any faster than exponentially in [L : ℚ]. Indeed, the crude estimate Reg(L) ≤ C^{[L:ℚ]}|D_{L}|, where D_{L} is the discriminant and C an absolute constant [La], p. 261, and the identity for discriminants in unramified towers |D_{L_{i}}| = |D_{L_{0}}|^{[L_{i}:L_{0}]}, imply that Reg(L_{i}) ≤ C_{1}^{[L_{i}:ℚ]} with a C_{1} depending only on L_{0}. Since by Zimmert’s bound, 1/ Reg(K) ≤ 1/.056 for any number field K [Zi], we find that

\frac{Reg(L_{i})}{Reg(K_{i})} ≤ 20 C_{1}^{[L_{i}:ℚ]},

for any subfield K_{i} ⊂ L_{i}. Thus, if we leave aside the difficult problem of finding the optimal value of d_{1}, the Main Theorem gives the best possible bound for large [L : K]. In this sense, for large relative degrees, one has a complete answer to Bergé and Martinet’s question.

The same cannot be said for small relative degrees [L : K], as our d_{0} works out to be very small. For fixed small [L : K], inequality (0.2) gives a bound for Reg(L)/ Reg(K) which is exponentially decreasing in [L : ℚ]. This is most likely far off the truth. Indeed, when L is totally real, it is known [CF], p. 290, that Reg(L)/ Reg(K) increases exponentially with [L : ℚ] as soon as [L : K] ≥ 2.

Zimmert proved his bound (0.1) by establishing a lower bound for the residue of any positive Dirichlet series having a functional equation of the kind one usually meets in number theory. Inequality (0.1) then followed by applying this lower bound to the partial zeta function ζ(s, 𝔞) of an ideal class of L. It is not known whether Zimmert’s method can be extended to the case of a relative extension L/K. The main obstacle is that there is no known natural analogue of the Dirichlet series ζ(s, 𝔞) which can be associated to a relative extension.

Instead of such a Dirichlet series we use a relative analogue Θ_{L/K} of the inverse Mellin transform of a partial zeta function ζ(s, 𝔞) completed by its Γ-factors. More precisely,

Θ_{L/K}(t; 𝔞) = \frac{Reg(L/K)}{w_{L}} + ∑_{\begin{matrix}a∈𝔞/∼ \\ a≠0 \end{matrix}} g(t, N_{L/K}(a)),

where t is a positive real variable, 𝔞 is a fractional ideal of L, N_{L/K} is the relative norm, the sum is over the nonzero elements of 𝔞 modulo the relative units (defined below), w_{L} is the number of roots of unity of L, and g is a function depending only on the splitting pattern in L/K of the archimedean places. Finally, Reg(L/K) is the relative regulator as defined in §1 below, which enters here as a volume term. Like Bergé and Martinet’s regulator, it is basically Reg(L)/ Reg(K). Although the Mellin transform of Θ_{L/K} does not in general result in a Dirichlet series, it still satisfies the usual kind of transformation rule

(0.4)  Θ_{L/K}(t^{−1}; 𝔞) = t^{[L:ℚ]/2}Θ_{L/K}(t; (𝔞𝔡)^{−1}),

where 𝔡 is the different of L. The left-hand side turns out for trivial reasons to be increasing. Thus the derivative of the right hand side is positive. On expanding its derivative, we obtain a complicated inequality for Reg(L/K). On taking for 𝔞 the ring of integers of L, a convexity argument allows us to drop all terms but the one with a = 1. This leads to the simple inequality (0.2).

We note in passing that our convexity argument hinges on an asymptotic study of integrals of the form

\frac{1}{2πi} ∫_{c−i∞}^{c+i∞} t^{−s} Γ(s)^{p+q} Γ(s + \frac{1}{2})^{q} ds (p, q, t, c > 0, p + 2q ≫ 0).

This method, though simple, seems to be quite strong. In fact, a variation of it was used in [Sk] to give a short and easy proof of Zimmert’s inequality (0.1). It should also apply to more general situations involving sums of thetaseries type, e.g. for obtaining upper bounds for lattice kissing numbers. We hope to come back to this in the future.

As indicated above, the Main Theorem gives a weak bound for small [L : K]. For such degrees one cannot hope in general for a bound for relative regulators which is exponentially increasing in [L : ℚ]. Indeed, the first accumulation point of Salem numbers is associated to an infinite set of quadratic extensions L/K with [L : ℚ] unbounded and Reg(L/K) < 0.28 [Sa] [CF]. Hence the inequality in the Main Theorem must be rephrased to apply better to small degrees. It seems quite likely to us that

(0.5)  Reg(L/K) ≥ f_{0} f_{1}^{r_{L/K}}

holds for any extension L/K. Here r_{L/K} is the difference of the unit ranks of L and K, while f_{0} > 0 and f_{1} > 1 should be absolute constants. In support of (0.5) one can say that that it does holds if L is totally real [CF], p. 290, and that, for [L : K] large, it is equivalent to the bound of the Main Theorem.

Inequality (0.5) would imply [CF], p. 291, that the smallest positive Salem number is strictly greater than one. In fact, as r_{L/K} = 1 for the family of extensions L/K associated to Salem numbers, (0.5) would yield |γ| ≥ exp(f_{0}f_{1}) for any Salem number γ. A proof of this would be an important step toward Lehmer’s conjecture [Le] giving a sharp lower bound for the height of any algebraic number.

We are grateful to the referee for suggesting significant improvements on the asymptotic estimates in §5.

1. Notation

We summarize the notations which will be used throughout. We use K ⊂ L for an extension of number fields, and we set n = [L : ℚ], d = [L : K]. We let A_{L} and A_{K} be the set of archimedean places of L and K. As usual, we let r_{1}(L) and r_{2}(L) denote, respectively, the number of real and complex places of L. For v ∈ A_{L} and a ∈ L, we let |a|_{v} denote the absolute value of a at v, and we set

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

For a place w ∈ A_{K} we use p_{w} and q_{w}, respectively, for the number of real and complex places of L extending w. The norm from L to K is denoted by N_{L/K}. Thus

(1.1)  | N_{L/K}(a)|_{w}^{e_{w}} = ∏_{\begin{matrix}v∈A_{L} \\ v|w \end{matrix}} |a|_{v}^{e_{v}},  ∑_{\begin{matrix}v∈A_{L} \\ v|w \end{matrix}} e_{v} = e_{w}[L : K].

We use O_{L}^{∗} for the group of units of L, and we let E_{L/K} be the relative units of L/K, i.e. the subgroup of elements ε of O_{L}^{∗} such that N_{L/K}(ε) is a root of unity. The rank of this group is r_{L/K} = |A_{L}| − |A_{K}|. By Reg(L/K) we denote the regulator of L/K. This can be described as follows: Choose fundamental relative units ε_{1}, ε_{2}, …, ε_{r} (r = r_{L/K}), i.e. free generators of E_{L/K} modulo torsion. Fix for each place w ∈ A_{K} an extension w̃ ∈ A_{L} lying above w. Let A′_{L} = {v_{1}, …, v_{r}} denote the set of archimedean places of L after deletion of the w̃ for all w ∈ A_{K}. The relative regulator is then given by

(1.2)  Reg(L/K) = | det(e_{v} log |ε_{j}|_{v})\begin{matrix}v∈A′_{L} \\ 1≤j≤r \end{matrix}|.

In [CF], p. 289, it is shown that the relative regulator satisfies

(1.3)  Reg(L/K) = \frac{1}{[O_{K}^{∗} : μ_{K} N_{L/K}(O_{L}^{∗})]} \frac{Reg(L)}{Reg(K)},

where μ_{K} is the group of roots of unity in K.

2. Theta-series associated to subgroups of the units

Let L be a number field, let E ⊂ O_{L}^{∗} be a subgroup of its group of units, and let E_{ℝ} = E ⊗_{ℤ} ℝ. Embed E_{ℝ} into ℝ_{+}^{A_{L}} by

x = ∑_{j} ε_{j} ⊗ ξ_{j} ↦ (x_{v})_{v∈A_{L}}, x_{v} = ∏_{j} |ε_{j}|_{v}^{ξ_{j}},

where A_{L} is the set of archimedean places of L, ε_{j} ∈ E, ξ_{j} ∈ ℝ, and | |_{v} denotes the absolute value at v. Note that x ↦ (log |x|_{v})_{v∈A_{L}} is the usual logarithmic embedding (up to conventional factors of e_{v}). Since E ⊂ O_{L}^{∗}, we have

(2.1)  ∏_{v∈A_{L}} x_{v}^{e_{v}} = 1,

with e_{v} as in Notation. For a ∈ L and x ∈ E_{ℝ} set

||ax||^{2} := ∑_{v∈A_{L}} e_{v} |a|_{v}^{2} x_{v}^{2}.

It is easily checked that for ε ∈ E, one has

(2.2)  ||(aε)x||^{2} = ||a((ε ⊗ 1)x)||^{2}.

Fix a Haar measure μ on E_{ℝ}. For a fractional ideal 𝔞 of L and real t > 0 we set

(2.3)  Θ_{E}(t; 𝔞) := \frac{μ (E_{ℝ}/E)}{|E_{tor}|} + ∑_{\begin{matrix}a∈𝔞/E \\ a≠0 \end{matrix}} ∫_{E_{ℝ}} exp(−c_{𝔞} t ||ax||^{2}) dμ(x).

Here μ(E_{ℝ}/E) is the volume of any fundamental domain for the natural action of E on E_{ℝ}, E_{tor} is the torsion subgroup of E, the sum in (2.3) is over a complete set of representatives a of the non-zero E-orbits in 𝔞,

c_{𝔞} := π(\sqrt{|D_{L}|} N_{L/ℚ}(𝔞))^{−2/n},

and D_{L} is the discriminant of L. Note that by (2.2), the integral in (2.3) indeed depends only on the E-orbit of a.

Proposition 2.1. The integral in (2.3 ) is finite, the infinite sum defining Θ_{E}(t; 𝔞) is absolutely convergent for all t > 0, and Θ_{E} satisfies the identity

(2.4)  Θ_{E}(t^{−1}; (𝔞𝔡)^{−1}) = t^{n/2}Θ_{E}(t; 𝔞),

where 𝔡 is the different of L.

Proof. For positive real numbers y_{v} (where v ∈ A_{L}) set

ϑ({y_{v}}; 𝔞) := ∑_{a∈𝔞} exp(− c_{𝔞} ∑_{v∈A_{L}} e_{v}|a|_{v}^{2} y_{v}).

This function satisfies [La], pp. 252–253,

(2.5)  ϑ({y_{v}^{−1}}; (𝔞𝔡)^{−1}) = ϑ({y_{v}}; 𝔞) ∏_{v∈A_{L}} y_{v}^{e_{v}/2}.

By “unfolding the integral” one verifies

∫_{E_{ℝ}/E} ϑ({tx_{v}^{2}}; 𝔞) dμ(x) = |E_{tor}|Θ_{E}(t; 𝔞).

From this the proposition is now obtained by replacing t by t^{−1}, 𝔞 by (𝔞𝔡)^{−1}, changing variables x ↦ x^{−1} (which leaves invariant the Haar measure), and then applying (2.5). The product resulting from the right hand side of (2.5) equals t^{n/2}, as one sees by using (2.1) and n = ∑_{v} e_{v}.

Our main point in this paper is that one should view equation (2.4) not as a transformation law, but rather as a formula for the co-volume μ(E_{ℝ}/E) from which inequalities for it can be derived. In fact, we shall only use the following

Corollary. For all t > 0 one has

\frac{2}{n}t^{1− \frac{n}{2}} \frac{d}{dt}(t^{n/2}Θ_{E}(t; 𝔞)) = Θ_{E}(t; 𝔞) + \frac{2}{n} t Θ′_{E}(t; 𝔞) ≥ 0 .

Proof. The equality is immediate. To prove the inequality note that Θ_{E}(t^{−1}; (𝔞𝔡)^{−1}) is increasing in t, as is immediate from its definition. By the transformation law (2.4), t^{n/2}Θ_{E}(t; 𝔞) is then increasing too, whence has derivative ≥ 0.

An easy consequence of the corollary is

Proposition 2.2. For any real t > 0 we have

\frac{μ (E_{ℝ}/E)}{|Etor|} ≥ (2t − 1) ∫_{E_{ℝ}} exp(− t||x||^{2})dμ(x).

Proof. If we write out the inequality in the corollary we obtain

(2.6)  \frac{μ (E_{ℝ}/E)}{|E_{tor}|} ≥ ∑_{\begin{matrix}a∈𝔞/E \\ a≠0 \end{matrix}} ∫_{E_{ℝ}} (\frac{2}{n}tc_{𝔞}||ax||^{2} − 1) exp(−tc_{𝔞}||ax||^{2}) dμ(x).

Using “arithmetic mean ≥ geometric mean” and n = ∑_{v} e_{v} we find

\frac{||ax||^{2}}{n} ≥ ∏_{v∈A_{L}}(|a|_{v}x_{v})^{2e_{v}/n} = | N_{L/ℚ}(a)|^{2/n} (a ∈ 𝔞, x ∈ E_{ℝ}),

where for the last identity we used (2.1). Take 𝔞 to be the unit ideal of L. Then for a ∈ 𝔞, a ≠ 0, we have ||ax||^{2}/n ≥ | N_{L/ℚ}(a)| ≥ 1. Replacing tc_{𝔞} by t, we find that the integral in (2.6) can be bounded below by

(2t − 1) ∫_{E_{ℝ}} exp(−t||ax||^{2}) dμ(x).

Thus for t > \frac{1}{2} all terms in the sum in (2.6) are positive, and the proposition follows on dropping all terms except the one corresponding to a = 1. For t ≤ \frac{1}{2} the proposition is trivially true.

3. Relative Theta-series

Recall that the group of relative units E_{L/K}, associated to an extension L/K of number fields, consists of those units of L whose norm to K is a root of unity. By considering the norm map from O_{L}^{∗} to O_{K}^{∗} one sees that

r_{L/K} := rank_{ℤ}(E_{L/K}) = rank_{ℤ}(O_{L}^{∗}) − rank_{ℤ}(O_{K}^{∗}).

If 𝔞 is a fractional ideal of L and t > 0 we define the relative Θ-series by taking E = E_{L/K} in (2.3)

Θ_{L/K}(t; 𝔞) := Θ_{E_{L/K}}(t; 𝔞),

where the Haar measure μ on E_{ℝ} (implicit in the definition of Θ_{E}) is chosen so that

μ(E_{ℝ}/E) = Reg(L/K),

Reg(L/K) being the relative regulator defined by (1.2).

Proposition 3.1. Let μ be the Haar measure on E_{ℝ} = E_{L/K} ⊗_{ℤ} ℝ normalized as above. Then, for t > 0 and a ∈ L^{∗},

∫_{E_{ℝ}} exp(−t ||ax||^{2}) dμ(x) = A ∏_{w∈A_{K}} f_{w}(a_{w} + log t),

where

f_{w}(y) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} e^{−se_{w}dy} Γ(s)^{p_{w}+q_{w}} Γ(s + \frac{1}{2})^{q_{w}} ds (y ∈ ℝ, any c > 0),

a_{w} = \frac{2}{d} log | N_{L/K}(a)|_{w}, A = 2^{−r_{L/K}}π^{−r_{2}(L)/2},

and where p_{w} and q_{w} denote, respectively, the number of real and complex places of L extending the place w ∈ A_{K}. Finally, A_{K}, e_{w}, r_{2}(L), r_{L/K} and d = [L : K] are as in Notation.

Remark 3.2. As a consequence we have the following formula for Θ_{L/K}

(3.1)  Θ_{L/K}(t; 𝔞) = \frac{Reg(L/K)}{w_{L}} + A ∑_{\begin{matrix}a∈𝔞/E_{L/K} \\ a≠0 \end{matrix}} ∏_{w∈A_{K}} f_{w}(log(tc_{𝔞}) + a_{w}),

where w_{L} is the number of roots of unity in L.

Before proving the proposition we take up its main application.

Proposition 3.3. For any y ∈ ℝ,

\frac{Reg(L/K)}{w_{L}} ≥

A ∑_{\begin{matrix}a∈𝔞/E_{L/K} \\ a≠0 \end{matrix}} \{ − 1 − \frac{2}{[L : ℚ]} ∑_{w∈A_{K}} \frac{f′_{w}}{f_{w}} (a_{w} + y)\} ∏_{w∈A_{K}} f_{w}(a_{w} + y),

where w_{L} is the number of roots of unity in L, and the notation is as in Proposition 3.1.

Proof. Write out the inequality of the corollary to Proposition 2.1, using formula (3.1) for Θ_{L/K}, and then choose t so that y = log(tc_{𝔞}).

Proof of Proposition 3.1. Set G = ℝ_{+}^{A_{L}}, H = ℝ_{+}^{A_{K}}, E = E_{ℝ} = E_{L/K} ⊗ ℝ. We regard the elements g of G as maps

A_{L} ∋ v ↦ g_{v} ∈ ℝ_{+},

and similarly for H. We identify E with its image in G under the embedding x ↦ (x_{v}) described in the preceding section. For g ∈ G and w ∈ A_{K}, let δ: G ↦ H be the group homomorphism defined by δ(g) = h, where h is given by

h_{w} = ∏_{v|w} g_{v}^{e_{v}}.

Thus we have an exact sequence

1 → E → G →^{δ} H → 1.

Fix a section σ: H → G of δ (this is possible since G and H may be viewed as real vector spaces). Thus δ(σ(h)) = h, and the map

α: E × H → G, (x, h) ↦ xσ(h)

defines an isomorphism.

For g ∈ G set

Ψ(g) = ∫_{E} exp(−||gx||^{2}) dμ(x), ||g||^{2} = ∑_{v∈A_{L}} e_{v}g_{v}^{2}.

We are interested in Ψ(g) for g such that g_{v} = \sqrt{t} |a|_{v} with a ∈ L, a ≠ 0. We observe that Ψ(g) only depends on g modulo E. Thus it suffices to compute ψ = Ψ ∘ σ. For g with g_{v} = \sqrt{t} |a|_{v} as above, we then find Ψ(g) = ψ(h) with h = δ(g), i.e.

h_{w} = ∏_{v|w}(\sqrt{t}|a|_{v})^{e_{v}} = t^{e_{w}d/2} | N_{L/K}(a)|_{w}^{e_{w}} = exp(e_{w}d(a_{w} + log t)/2)

(cf. (1.1)). Thus the formula in Proposition 3.1 is equivalent to

ψ(h) = A ∏_{w∈A_{K}} \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} h_{w}^{−2s_{w}} Γ(s_{w})^{p_{w}+q_{w}}Γ (s_{w} + \frac{1}{2})^{q_{w}} ds_{w} (c > 0).

To calculate ψ we compute its (multi-dimensional) Mellin transform:

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

Here s = (s_{w}) ∈ ℂ^{A_{K}} with Re(s_{w}) > 0 for all w ∈ A_{K}, and μ_{H} is a Haar measure on H.

Now let μ_{G} and μ_{H} denote the natural Haar measures on G and H, i.e.

dμ_{H}(h) = ∏_{w∈A_{K}} \frac{dh_{w}}{h_{w}} ,

and similarly for μ_{G}. Then the image μ_{G} ∘ α of the Haar measure μ_{G} under α is a Haar measure on E × H. Hence there exists a constant c such that

c μ_{G} ∘ α = μ × μ_{H},

where μ is the measure on E = E_{ℝ} as in Proposition 3.1. We shall determine c below.

For the Mellin transform of ψ(h) we find

(Mψ)(s) = ∫_{E×H} exp(−||xσ(h)||^{2}) h^{s} (dμ × dμ_{H})(x, h)

= c ∫_{E×H} exp(−||α(x, h)||^{2}) δ(α(x, h))^{s} d(μ_{G} ∘ α)(x, h)

= c ∫_{G} exp(−||g||^{2}) δ(g)^{s} dμ_{G}(g)

= c ∏_{w∈A_{K}} ∏_{\begin{matrix}v∈A_{L} \\ v|w \end{matrix}} ∫_{ℝ_{+}} exp(−e_{v}t^{2}) t^{e_{v}s_{w}} \frac{dt}{t}

= c · 2^{−|A_{L}|} ∏_{w∈A_{K}} Γ (\frac{s_{w}}{2} )^{p_{w}} (2^{−s_{w}}Γ(s_{w}))^{q_{w}}

= c · 2^{−|A_{L}|} · (2\sqrt{π})^{−r_{2}(L)} ∏_{w∈A_{K}} Γ(\frac{s_{w}}{2} )^{p_{w}+q_{w}}Γ(\frac{s_{w} + 1}{2})^{q_{w}},

where, for the last identity we used 2^{−s}Γ(s) = (2\sqrt{π})^{−1}Γ (\frac{s}{2}) Γ (\frac{s+1}{2}) . On taking the inverse Mellin transform (and substituting in the inverse Mellin integrals 2s_{w} for s_{w}) we obtain the formula to be proven with

A = c · 2^{−|A_{L}|} · (2\sqrt{π})^{−r_{2}(L)} · 2^{|A_{K}|},

the last factor coming in by the substitution of 2s_{w} for s_{w}. Thus it remains to show c = 2^{r_{2}(L)}.

For computing c we view H, G and E as real vector spaces. Let ε_{j} (1 ≤ j ≤ r) be a ℤ-basis of E_{L/K} modulo torsion. Thus, the x_{j} := ε_{j} ⊗ 1 form an ℝ-basis of E. Extend this to a basis x_{j}, g_{k} of G by adjoning vectors g_{k} (1 ≤ k ≤ q := |A_{K}|). Then the δ(g_{k}) form a basis of H, and δ(g_{k}) ↦ g_{k} defines a section σ of δ. Computing the volume of the convex hull in G of this basis x_{j}, g_{k} and the volume of its inverse image under α (which equals the cartesian product of convex hull in E of the x_{j} and the convex hull in H of the δ(g_{k})) we find

c μ_{G}([x_{1}, …, x_{r}, g_{1}, …, g_{q},]) = μ([x_{1}, …, x_{r}]) · μ_{H}([(δ(g_{1}), …, δ(g_{q})]),

where […] means convex hull. The first factor on the right is Reg(L/K) by the very normalisation of μ. For computing the others, let w_{1}, …, w_{q} be the infinite places of K and choose, for each w_{k}, a v_{k} ∈ A_{L} extending it. We take g_{k} such that

(g_{k})_{v} = \{ \begin{matrix}exp(1/e_{v}) & if v = v_{k} \\ 1 & otherwise \end{matrix}

(recall that G is written multiplicatively). The convex hull of the δ(g_{k}) is the “unit cube”, hence has volume 1. For the volume on the left side of the above formula for c we find by a simple calculation that it equals 2^{−r_{2}(L)} · Reg(L/K), with Reg(L/K) given as the determinant (1.2). This proves the Proposition.

We can now give a crude lower bound for Reg(L/K), which we shall later need for d = [L : K] < 40.

Proposition 3.4. Reg(L/K) > 0.1^{[L:ℚ]} for any L/K.

Proof. We apply Proposition 2.2 to E = E_{L/K} with t = 1. Then

\frac{Reg(L/K)}{w_{L}} ≥ A ∏_{w∈A_{K}} f_{w}(0),

where we replaced the integral in Proposition 2.2 by the formula in Proposition 3.1 with a = t = 1. To estimate f_{w}(0) from below we write

f_{w}(0) = (g_{1} ∗ ⋯ ∗ g_{m})(1),

where the asterisk denotes Mellin (multiplicative) convolution, where g_{j}(x) = exp(−x) for 1 ≤ j ≤ p_{w} + q_{w} and g_{j}(x) = \sqrt{x} exp(−x) for p_{w} + q_{w} < j ≤ m = p_{w} + 2q_{w}. Recall that the Mellin convolution of m functions g_{j} is given by

(g_{1} ∗ ⋯ ∗ g_{m})(t) = ∫_{0}^{∞} ⋯ ∫_{0}^{∞} g_{1}(t/x_{1}) g_{2}(x_{2}) ⋯ g_{m}(x_{m}) \frac{dx_{2}}{x_{2}} ⋯ \frac{dx_{m}}{x_{m}} ,

where x_{1} = x_{2} ⋯ x_{m}. By shrinking the integration region to x_{j} ≥ 1 for 2 ≤ j ≤ m, we obtain

f_{w}(0) ≥ exp(−1) Γ(0, 1)^{p_{w}+q_{w}−1} Γ(1/2, 1)^{q_{w}} ≥ Γ(0, 1)^{m},

where Γ(s, a) = ∫_{a}^{∞} x^{s} exp(−x) \frac{dx}{x} and we used exp(−1) > Γ(0, 1). Using A ≥ 1/2^{[L:ℚ]} and Γ(0, 1) > 0.2, we obtain the inequality in the Proposition.

4. Proof of the Main Theorem

We begin by outlining the proof of the Main Theorem given in the Introduction. Our starting point is the inequality in Proposition 3.3 with 𝔞 = O_{L}, the ring of integers of L. The sum over a then contains the term corresponding to the orbit of a = 1. We would like to drop all the other terms. To do this, we need to insure that each term in the sum over a is non-negative. As f_{w} > 0, this means that we must choose the free variable y ∈ ℝ so that for all nonzero a ∈ 𝔞 we have

(4.1)  −\frac{2}{[L : ℚ]} ∑_{w∈A_{K}} \frac{f′_{w}}{f_{w}} (y + a_{w}) ≥ 1.

The problem we meet is that we know very little about the a_{w}. We can only say that

∑_{w∈A_{K}} e_{w}a_{w} = \frac{2}{d} log | N_{L/ℚ}(a)| ≥ 0,

since a is a nonzero algebraic integer. This together with the shape of (4.1) suggests bounding the functions −\frac{f′_{w}}{f_{w}} from below by a suitable convex function. Indeed we shall prove, for large relative degrees d = [L : K],

(4.2)  −\frac{f′_{w}}{f_{w}} (x) ≥ e_{w}d ρ(x)

for a certain convex and increasing function ρ. We thus obtain that the left hand side of (4.1) is bounded below by

\frac{2}{[K : ℚ]} ∑_{w∈A_{K}} e_{w}ρ(y + a_{w}) ≥ 2 ρ(y + \frac{2}{d} log | N_{L/ℚ}(a)|) ≥ 2ρ(y).

Thus the terms corresponding to a ≠ 1 will be positive if 2ρ(y) ≥ 1. Hence we obtain

(4.3) \frac{Reg(L/K)}{w_{L}} ≥ A (2ρ(y) − 1) ∏_{w∈A_{K}} f_{w}(y)  (A = 2^{−r_{L/K}}π^{−r_{2}(L)/2}).

Estimating the f_{w} by explicitly calculable functions will then give numerical lower bounds for the relative regulator.

To describe these lower bounds we need to introduce some functions. Define for Re(s) > 0 and 0 ≤ k ≤ 1

α_{k}(s) = k log Γ(s) + (1 − k) log Γ(s + \frac{1}{2}).

Here the logarithms are taken real for s real. Then

f_{w}(y) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} exp(m_{w}α_{κ_{w}}(s) − m_{w}ys) ds,

where c > 0 is arbitrary and

(4.4)  κ_{w} := \frac{p_{w} + q_{w}}{m_{w}} ,  m_{w} = e_{w}d = p_{w} + 2q_{w},

with notation as in Proposition 3.1. Note that, for s positive,

α′_{k}(s) = kΨ(s) + (1 − k)Ψ(s + \frac{1}{2}), (Ψ(s) = \frac{Γ′(s)}{Γ(s)})

is concave and strictly increasing with image (−∞, ∞), since Ψ(s) has these properties. Thus its inverse function is convex and increasing. We denote it by

(4.5)  σ_{k}(y) = (α′_{k})^{−1}(y).

Using these functions our main result can be stated as follows

Theorem 4.1. Suppose d = [L : K] ≥ 40 and let \frac{1}{2} ≤ k ≤ 1 − \frac{q_{w}}{e_{w}d} for all w ∈ A_{K}. Then for all real y

\frac{Reg(L/K)}{w_{L}} ≥ [\frac{(1 − \frac{6.4}{d})}{\sqrt{πdΨ′(s)}}]^{|A_{K}|} ·

· (cs − 1)(\frac{Γ(u)}{2})^{r_{1}}Γ(v)^{r_{2}} exp(− uΨ(u)r_{1} − vΨ(v)r_{2}),

where r_{1} = r_{1}(L) and r_{2} = r_{2}(L) are, respectively, the number of real and complex places of L, c = (2 − \frac{4.84}{d}), and where s = σ_{k}(y), u = σ_{1}(y), v = 2σ_{1/2}(y), with σ_{k} as in (4.5 ).

Before giving the proof we draw some numerical results. It is worth noting that Reg(L)/ Reg(K) ≥ Reg(L/K), by (1.3). Hence any lower bound for Reg(L/K) is also valid for Reg(L)/ Reg(K).

Corollary 1. If [L : K] ≥ 40 , then

\frac{Reg(L/K)}{w_{L}} > 0.014 · 1.15^{[L:ℚ]+r_{1}(L)}.

If, furthermore, L is totally real, then Reg(L/K) > 0.028 · 1.97^{[L:ℚ]}.

Corollary 2. If [L : K] ≫ 0, then Reg(L/K) > 1.33^{[L:ℚ]} 1.18^{r_{1}(L)}. If, furthermore, L is totally real, then Reg(L/K) > 2.36^{[L:ℚ]}.

The 1.33 and 2.36 are, respectively, decimal approximations from below of exp(γ/2) and of \sqrt{π} exp(γ/2), where γ is Euler’s constant. The 1.18 approximates \frac{1}{2}Γ(u) exp(u(γ + log 2) − \frac{1}{2}γ), where u = 0.682 ⋯ is the unique positive solution of Ψ(u) = −γ − log 2 = \frac{1}{2}(Ψ(\frac{1}{2}) + Ψ(1)).

Corollary 3. (Main Theorem in Introduction)

\frac{Reg(L)}{Reg(K)} ≥ (d_{0} d_{1}^{[L:K]})^{[K:ℚ]},

where d_{0} = (0.1/1.15)^{39} and d_{1} = 1.15.

Proof of Corollary 1. We write the inequality of the theorem in the form

\frac{Reg(L/K)}{w_{L}} ≥ [\frac{a}{\sqrt{d}}]^{|A_{K}|} · B · C^{r_{1}} · D^{r_{2}},

where

a = \frac{1 − \frac{6.4}{d}}{\sqrt{πΨ′(s)}}, B = cs − 1, C = \frac{Γ(u)}{2} exp(−uΨ(u)), D = Γ(v) exp(−vΨ(v)).

We shall choose y below so that \frac{1}{2} < s ≤ 0.54. Hence 0 < a, B < 1. Thus,

\frac{Reg(L/K)}{w_{L}} ≥ B · (\frac{a^{1/d}}{d^{1/2d}} \sqrt{D})^{[L:ℚ]} · (\frac{C}{\sqrt{D}} )^{r_{1}} ≥ B(\frac{a^{1/40}}{40^{1/80}} \sqrt{D})^{[L:ℚ]} · (\frac{C}{\sqrt{D}} )^{r_{1}},

where we also used |A_{K}| ≤ [K : ℚ] and d = [L : K] ≥ 40. To prove the first inequality in the corollary, set k = \frac{1}{2} and choose y so that

s = σ_{\frac{1}{2}} (y) = \frac{v}{2} = 0.54.

Thus, by (4.5),

y = α_{\frac{1}{2}} (0.54) = \frac{1}{2}(Ψ(0.54) + Ψ(0.54 + \frac{1}{2})) = −1.145946 ⋯

and

u = σ_{1}(y) = 0.72693 ⋯ < 0.72694.

As Γ(u) exp(−uΨ(u)) is decreasing for u > 0 (direct derivative check), in calculating C we may take u = 0.72694. We thus obtain

\sqrt{D} > 1.251, \frac{C}{\sqrt{D}} > 1.15, a ≥ \frac{1 − \frac{6.4}{40}}{\sqrt{πΨ′(0.54)}} > 0.2276, B > 0.014.

Substituting these numerical results in the above inequality for \frac{Reg(L/K)}{w_{L}}, we obtain the first inequality in Corollary 1. The second inequality is obtained similarly on letting k = 1 and y = Ψ(0.54) in the theorem, and noting that w_{L} = 2 if L is real.

Proof of Corollary 2. We proceed just as above, taking k = \frac{1}{2} (or k = 1 if L is totally real), but we choose y so that s = σ_{k}(y) = \frac{1}{2} + \frac{2}{d}.

Proof of Corollary 3. The claim follows from (1.3) and Proposition 3.4 when d < 40. For d ≥ 40 it follows from Corollary 1.

To prove Theorem 4.1 we shall use the following estimates, whose proof using the saddle-point method will be given in §5 (See Lemmas 5.5 and 5.8, noting that f_{w}(y) = F_{κ_{w}}(y, m_{w}) in these lemmas, with κ_{w} and m_{w} as in (4.4)).

Proposition 4.2. Let y ∈ ℝ. Then, for m_{w} ≥ 40,

f_{w}(y) = \begin{matrix}e & m_{w}(α(σ)−yσ) \\ √ & \overline{2πm_{w}α′′(σ)} \end{matrix} (1 + \frac{γ}{m_{w}} ), |γ| = |γ(y, p_{w}, q_{w})| < 6.4,

and

−\frac{f′_{w}}{f_{w}} (y) = m_{w} σ · (1 + \frac{β}{m_{w}} ), |β| = |β(y, p_{w}, q_{w})| < 2.42,

where σ = σ_{κ_{w}}(y) and α = α_{κ_{w}}.

We shall also need the following properties of σ_{κ} and α_{κ}.

Lemma 4.3. For 1 ≥ κ ≥ k ≥ 0 and any y ∈ ℝ, one has σ_{κ}(y) ≥ σ_{k}(y).

Proof. It suffices to check the reverse inequality for the inverse functions α′_{κ} and α′_{k}, as these are strictly increasing functions. Since Ψ(s) < Ψ(s + \frac{1}{2}) (s > 0), we find

α′_{κ}(s) − α′_{k}(s) = (κ − k)(Ψ(s) − Ψ(s + \frac{1}{2})) ≤ 0.

Lemma 4.4. For 0 ≤ k ≤ 1 and fixed real y, let g(k) = α_{k}(σ_{k}(y)) − yσ_{k}(y). Then g(k) is a concave function.

Proof. Fix y and write σ_{k} for σ_{k}(y). From the definition of α_{k}

\frac{dg}{dk} = log Γ(σ_{k}) − log Γ(σ_{k} + \frac{1}{2}) + α′_{k}(σ_{k}) · \frac{dσ_{k}}{dk} − y \frac{dσ_{k}}{dk} .

The last two terms cancel since, by the very definition (4.5) of σ_{k}, α′_{k}(σ_{k}) = y. Hence

\frac{d^{2}g}{dk^{2}} = (Ψ(σ_{k}) − Ψ(σ_{k} + \frac{1}{2}))\frac{dσ_{k}}{dk} ≤ 0,

since \frac{dσ_{k}}{dk} ≥ 0 (Lemma 4.3) and Ψ′ > 0.

Proof of Theorem 4.1. Note that κ_{w} ≥ k for w ∈ A_{K} since, by hypothesis, k ≤ 1 − \frac{q_{w}}{e_{w}d}. By the second estimate of Proposition 4.2, and since σ_{κ_{w}}(y) ≥ σ_{k}(y) (Lemma 4.3), we obtain inequality (4.2) with

ρ(y) = (1 − \frac{2.42}{d})σ_{k}(y) = cs/2.

Here c and s = σ_{k}(y) are as in the theorem. Hence we have proved inequality (4.3) for the relative regulator with the above ρ.

To estimate the f_{w}(y) occurring in (4.3), we apply the first inequality of Proposition 4.2 to obtain

f_{w}(y) ≥ \frac{(1 − \frac{6.4}{m_{w}} )e^{m_{w}g(κ_{w})}}{\sqrt{2πm_{w}α′′_{κ_{w}}(σ_{κ_{w}}(y))}},

with

g(k) = α_{k}(σ_{k}(y)) − yσ_{k}(y)

as in Lemma 4.4.

For estimating the denominator we use

0 < α′′_{κ_{w}}(σ_{κ_{w}}(y)) ≤ α′′_{κ_{w}}(σ_{k}(y)) ≤ Ψ′(σ_{k}(y)),

where we used σ_{κ_{w}}(y) ≥ σ_{k}(y) (Lemma 4.3) and the fact that Ψ′(s) is positive and decreasing in s for s > 0. Thus,

∏_{w} f_{w}(y) ≥ (\frac{(1 − \frac{6.4}{d})}{\sqrt{4πdΨ′(σ_{k}(y))}})^{|A_{K}|} exp( ∑_{w∈A_{K}} m_{w}g(κ_{w})),

where we also used m_{w} = e_{w}d ≤ 2d. Since g is concave (Lemma 4.4) and 1 ≥ κ_{w} ≥ \frac{1}{2} we have

g(κ_{w}) ≥ 2(g(1) − g(\frac{1}{2}))(κ_{w} − \frac{1}{2}) + g(\frac{1}{2}).

Thus, by a simple computation, using m_{w} = p_{w} + 2q_{w}, ∑_{w} p_{w} = r_{1}(L) and ∑_{w} q_{w} = r_{2}(L), we obtain

∑_{w∈A_{K}} m_{w}g(κ_{w}) ≥ ∑_{w∈A_{K}} m_{w}(2(g(1) − g(\frac{1}{2}))(κ_{w} − \frac{1}{2}) + g(\frac{1}{2}))
= g(1)r_{1} + 2g(\frac{1}{2})r_{2}.

But g(1) = α_{1}(u) − α′_{1}(u)u with u = σ_{1}(y) as in Theorem 4.1. Hence

exp(g(1)) = Γ(u) exp(− uΨ(u)).

Similarly, using also 2^{−2s}Γ(2s) = (2\sqrt{π})^{−1}Γ(s)Γ(s + \frac{1}{2}), we obtain

exp(2g(\frac{1}{2})) = Γ(\frac{v}{2})Γ(\frac{v}{2} + \frac{1}{2}) exp(− \frac{v}{2}[Ψ(\frac{v}{2}) + Ψ(\frac{v}{2} + \frac{1}{2})])
= 2\sqrt{π} Γ(v) exp(− vΨ(v)),

with v = 2σ_{\frac{1}{2}} (y). Collecting all the factors, using A = 2^{|A_{K}|}2^{−r_{1}}(2\sqrt{π})^{−r_{2}}, and (4.3), we obtain the theorem.

5. Asymptotic Estimates

In this section we find an asymptotic expression for integrals of the form

G(t) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} t^{−s} Γ(s)^{p+q} Γ(s + \frac{1}{2})^{q} ds =: \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} t^{−s} g(s) ds (t, c > 0),

where p and q are non-negative and g = g_{p,q}. We prove in Lemmas 5.5 and 5.8 below the estimates

(1 − \frac{6.4}{p+2q} )G^{0}(t) ≤ G(t) ≤ (1 + \frac{6.4}{p+2q} )G^{0}(t) (p + 2q ≥ 40),
(1 − \frac{2.42}{p+2q} )τ ≤ −t\frac{G′(t)}{G(t)} ≤ (1 + \frac{2.42}{p+2q} )τ, (p + 2q ≥ 40),

where τ = τ_{p,q}(t) is the unique positive critical point of s ↦ t^{−s}g(s), and

G^{0}(t) = G_{p,q}^{0}(t) = \frac{1}{\sqrt{2π(log g)′′(τ)}}t^{−τ} g(τ)

is the stationary phase approximation to G(t).

Inverse Mellin transforms such as G can be readily approximated by the saddle-point method [deB], which yields G(t) = G^{0}(t)(1 + O(\frac{1}{p+2q})), with the O constant depending on t. However, for our applications in §4, it is essential that the estimate of t\frac{G′(t)}{G(t)} be independent of t. To prove this uniformity we will now painstakingly proceed through the necessary estimates. While the succession of lemmas below makes sense only in the above context, we actually assume no knowledge of the saddle-point method.

Recall from the previous section that we set, for 0 ≤ κ ≤ 1,

(5.1)  α_{κ}(s) = κ log Γ(s) + (1 − κ) log Γ(s + \frac{1}{2}), σ_{κ}(y) = (α′_{κ})^{−1}(y).

For m > 0 and y ∈ ℝ, let

(5.2)  F_{κ}(y, m) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} exp(m(α_{κ}(s) − ys)) ds  (c > 0).

Thus, for κ = \frac{p+q}{p+2q} and m = p + 2q,

G(t) = F_{κ}(\frac{1}{m} log t, m), τ_{p,q}(t) = σ_{κ}(\frac{log t}{m}).

We shall need [GR], §8.363, §8.326,

(5.3)  \begin{matrix}\frac{Ψ^{(n)}(s)}{n!} = & (−1)^{n+1} ∑_{k=0}^{∞} \frac{1}{(s + k)^{n+1}} ,  (Ψ(s) = \frac{Γ′}{Γ} (s)) \\ |\frac{Γ(x + iy)}{Γ(x)}|^{2} = & ∏_{k=0}^{∞} (1 + \frac{y^{2}}{(x + k)^{2}} )^{−1}. \end{matrix}

The first formula is valid for n ≥ 1 and Re(s) > 0 [WW], §12.6. The second one is true for real x, y with x > 0 (as follows from the Weierstrass product expansion of Γ(z)^{−1} ).

Lemma 5.1. For y ∈ ℝ and 0 ≤ κ ≤ 1, let α = α_{κ} and σ = σ_{κ}(y) be as in (5.1 ), and set α(σ +it)−iyt = α(σ)− \frac{α′′(σ)}{2} t^{2} +ρ(y, t). Then, for 0 ≤ t ≤ σ,

0 ≤ Im(ρ(y, t)) ≤ −t^{3} \frac{α^{(3)}(σ)}{3!}, 0 ≤ Re(ρ(y, t)) ≤ t^{4} \frac{α^{(4)}(σ)}{4!}.

Proof. Recall that α′(σ) = y, by definition of σ. Expand α(s) about s = σ to get

α(σ + it) = α(σ) + iyt − \frac{α′′(σ)}{2}t^{2}+

+ i ∑_{j=1}^{∞}(−1)^{j} \frac{α^{(2j+1)}(σ)}{(2j + 1)!}t^{2j+1} + ∑_{j=2}^{∞}(−1)^{j} \frac{α^{(2j)}(σ)}{(2j)!}t^{2j}.

By (5.3) and (5.1), each of the two sums above is alternating and

|\frac{α^{(l)}(σ)}{l!}| = \frac{κ}{l} ∑_{k=0}^{∞} \frac{1}{(σ + k)^{l}} + \frac{1 − κ}{l} ∑_{k=0}^{∞} \frac{1}{(σ + \frac{1}{2} + k)^{l}} (l ≥ 2).

Comparing term by term, we find

(5.4)  |\frac{α^{(l)}(σ)}{l!} t^{l}| ≥ \frac{l + 1}{l} |\frac{α^{(l+1)}(σ)}{(l + 1)!} t^{l+1}| ≥ |\frac{α^{(l+1)}(σ)}{(l + 1)!} t^{l+1}| .

Hence

\frac{−α^{(3)}(σ)}{3!} t^{3} ≥ ∑_{j=1}^{∞}(−1)^{j} \frac{α^{(2j+1)}(σ)}{(2j + 1)!} t^{2j+1} = Im(ρ(y, t)) ≥ 0

and

\frac{t^{4}}{4!}α^{(4)}(σ) ≥ ∑_{j=2}^{∞}(−1)^{j} \frac{α^{(2j)}(σ)}{(2j)!} t^{2j} = Re(ρ(y, t)) ≥ 0.

Lemma 5.2. For y ∈ ℝ and 0 < κ ≤ 1, let α = α_{κ} and σ = σ_{κ}(y) be as in (5.1 ). Then, for any integer n ≥ 2,

\frac{|α^{(n)}(σ)|}{(α′′(σ))^{n/2}} ≤ \frac{(n − 1)!}{κ^{\frac{n}{2} −1}}.

Proof. Set

t = \frac{1 − κ}{κ} ≥ 0, A = Ψ′(σ), B = Ψ′(σ + \frac{1}{2}),

C = (−1)^{n} \frac{Ψ^{n−1}(σ)}{(n − 1)!}, D = (−1)^{n} \frac{Ψ^{n−1}(σ + \frac{1}{2})}{(n − 1)!}.

The lemma is equivalent to the positivity of

f(t) := (A + tB)^{\frac{n}{2}} − C − tD.

We shall prove that f(t) ≥ 0 by showing f(0) ≥ 0, f′(0) ≥ 0 and f′′(t) ≥ 0 for all t ≥ 0.

Equation (5.3) shows that f(0) ≥ 0 is equivalent to

(5.5)  (∑_{k=0}^{∞} \frac{1}{(σ + k)^{2}} )^{n} ≥ (∑_{k=0}^{∞} \frac{1}{(σ + k)^{n}} )^{2} .

On letting x_{k} = 1/(σ + k)^{n}, (5.5) follows from the concavity argument

∑_{k=0}^{N−1} xk^{\frac{2}{n}} ≥ N(\frac{∑_{k=0}^{N−1} x_{k}}{N} )^{\frac{2}{n}} ≥ ( ∑_{k=0}^{N−1} x_{k})^{\frac{2}{n}} (x_{k} ≥ 0, n ≥ 2).

Now,

f′(0) = \frac{n}{2}A^{\frac{n}{2} −1}B − D ≥ A^{\frac{n}{2} −1}B − D > B^{\frac{n}{2}} − D ≥ 0,

where we used A > B (Proof: Ψ′ is decreasing), and B^{n} > D^{2} (Proof: Take (5.5) with σ replaced with σ + \frac{1}{2}). Finally,

f′′(t) = B^{2} \frac{n}{2}(\frac{n}{2} − 1)(A + tB)^{\frac{n}{2} −2} ≥ 0.

Lemma 5.3. For 0 < κ ≤ 1, let α = α_{κ} be as in (5.1 ), let ε = 0 or ε = 1, and suppose u, c > 0 and mκ > 2. Then,

∫_{uc}^{∞} |t^{ε}e^{mα(c+it)}| dt ≤ \frac{e^{mα(c)}c^{1+ε}u^{ε−1}(1 + u^{2})}{(mκ − 2)(1 + \frac{u^{2}}{4} )^{mκ[c]/2}(1 + u^{2})^{mκ/2}} ,

where [c] denotes the largest integer smaller than or equal to c.

Proof. Using (5.3) we have

∫_{uc}^{∞} |t^{ε}e^{m(α(c+it)−α(c))}| dt

= ∫_{uc}^{∞} t^{ε} ∏_{k=0}^{∞} (1 + \frac{t^{2}}{(c + k)^{2}} )^{−mκ/2}(1 + \frac{t^{2}}{(\frac{1}{2} + c + k)^{2}} )^{−m(1−κ)/2} dt

≤ ∫_{uc}^{∞} t^{ε} ∏_{k=0}^{[c]} (1 + \frac{t^{2}}{(c + k)^{2}} )^{− \frac{mκ}{2}} dt ≤ ∫_{uc}^{∞} t^{ε}(1 + \frac{t^{2}}{c^{2}} )^{− \frac{mκ}{2}} (1 + \frac{t^{2}}{4c^{2}} )^{− \frac{mκ[c]}{2}} dt

= (uc)^{1+ε} ∫_{1}^{∞} x^{ε}(1 + u^{2}x^{2})^{−mκ/2} (1 + \frac{u^{2}x^{2}}{4} )^{−mκ[c]/2} dx

≤ (1 + \frac{u^{2}}{4} )^{− \frac{mκ[c]}{2}} (uc)^{1+ε} ∫_{1}^{∞} x(1 + u^{2}x^{2})^{− \frac{mκ}{2}} dx = \frac{c^{1+ε}u^{ε−1}(1 + u^{2})^{1− \frac{mκ}{2}}}{(mκ − 2)(1 + \frac{u^{2}}{4} )^{mκ[c]/2}} .

Lemma 5.4. For y ∈ ℝ and 0 < κ ≤ 1, let α = α_{κ} and σ = σ_{κ}(y) be as in (5.1 ). Then, for any positive D and m subject to D < m^{1/3}\sqrt{κ} and mκ > 2, we have

∫_{−∞}^{+∞} e^{m(α(σ+it)−iyt)} dt = \begin{matrix}\sqrt{2π} & e & mα(σ) \\ √ & \overline{mα′′(σ)} \end{matrix} (1 + φ),

where φ = φ(y, m, κ) satisfies

|φ| ≤ \frac{2^{3/2}σ\sqrt{mα′′(σ)}}{\sqrt{π}(mκ − 2)1.25^{mκ[σ]/2}2^{mκ/2}} + \frac{2^{3/2} exp(−m^{1/3}D^{2}/4)}{\sqrt{π}m^{1/6}D}

+\frac{\sqrt{2} exp(−m^{1/3}D^{2}/2)}{\sqrt{π}m^{1/6}D} + \frac{5}{6mκ} + \frac{e^{M} − 1}{M} \frac{3}{4mκ}, (M = \frac{D^{4}}{4m^{\frac{1}{3}} κ}).

Proof. Set

(5.6)  δ = \frac{D}{m^{1/3}\sqrt{α′′_{κ}(σ)}}.

Note that

(5.7)  α′′_{κ}(σ) = ∑_{k=0}^{∞} \frac{κ}{(σ + k)^{2}} + ∑_{k=0}^{∞} \frac{1 − κ}{(\frac{1}{2} + σ + k)^{2}} > \frac{κ}{σ^{2}} .

Therefore, for all σ > 0,

(5.8)  \frac{δ}{σ} = \frac{D}{m^{1/3}\sqrt{σ^{2}α′′_{κ}(σ)}} < \frac{D}{m^{1/3}\sqrt{κ}} < 1,

20 EDUARDO FRIEDMAN AND NILS-PETER SKORUPPA

by the assumption on D in the lemma. Write

e^{−mα(σ)} ∫_{−∞}^{∞} e^{m(α(σ+it)−iyt)} dt

= (∫_{σ}^{∞} + ∫_{δ}^{σ} + ∫_{−δ}^{δ} + ∫_{−σ}^{−δ} + ∫_{−∞}^{−σ}) e^{m(α(σ+it)−α(σ)−iyt)} dt
=: I_{1} + I_{2} + I_{3} + \overline{I_{2}} + \overline{I_{1}}.

Lemma 5.3, with u = 1, ε = 0 and c = σ, gives

|I_{1}| ≤ \frac{2σ}{(mκ − 2)1.25^{mκ[σ]/2}2^{mκ/2}} .

By Lemma 5.1, for 0 ≤ t ≤ σ,

(5.9)  |e^{m(α(σ+it)−α(σ)−iyt)}| ≤ e^{m(− \frac{α′′(σ)}{2} t^{2}+ \frac{α^{(4)}(σ)}{4!} t^{4})} ≤ e^{−mα′′(σ)t^{2}/4},

where in the last step we used (5.4) twice. Thus, from (5.6), (5.8) and (5.9),

(5.10) |I_{2}| ≤ ∫_{δ}^{σ} e^{−mα′′(σ)t^{2}/4}dt < ∫_{δ}^{∞} e^{−mα′′(σ)t^{2}/4}dt ≤ \frac{2e^{−m^{1/3}D^{2}/4}}{m^{2/3}D\sqrt{α′′(σ)}},

where we used ∫_{x}^{∞} e^{−t^{2}} dt < e^{−x^{2}}/2x (Proof: ∫_{x}^{∞} e^{−t^{2}} dt = \frac{1}{x} ∫_{x}^{∞} xe^{−t^{2}} dt < \frac{1}{x} ∫_{x}^{∞} te^{−t^{2}} dt = e^{−x^{2}}/2x).

To handle I_{3}, which gives the main term, write

I_{3} = ∫_{−δ}^{δ} e^{m(α(σ+it)−α(σ)−iyt)} dt

= ∫_{−δ}^{δ} e^{−mt^{2}α′′(σ)/2}e^{mρ(y,t)} dt =: I_{4} + I_{5},

where ρ(y, t) is as in Lemma 5.1 and

I_{4} = ∫_{−δ}^{δ} e^{−mt^{2}α′′(σ)/2}(e^{mρ(y,t)} − 1) dt, I_{5} = ∫_{−δ}^{δ} e^{−mt^{2}α′′(σ)/2} dt. We compute

I_{5} = ∫_{−δ}^{δ} e^{−mt^{2}α′′(σ)/2} dt = ∫_{−∞}^{∞} e^{−mt^{2}α′′(σ)/2} dt − 2 ∫_{δ}^{∞} e^{−mt^{2}α′′(σ)/2} dt
=\begin{matrix}√ & \overline{2π} \\ √ & \overline{mα′′(σ)} \end{matrix} − 2 ∫_{δ}^{∞} e^{−mt^{2}α′′(σ)/2} dt.

As in (5.10),

∫_{δ}^{∞} e^{−mt^{2}α′′(σ)/2} dt ≤ \frac{e^{−m^{1/3}D^{2}/2}}{m^{2/3}D\sqrt{α′′(σ)}}.

RELATIVE REGULATORS OF NUMBER FIELDS 21

Now

I_{4} = 2 ∫_{0}^{δ} e^{−mt^{2}α′′(σ)/2}Re(e^{mρ(y,t)} − 1) dt.

Quite generally, if 0 ≤ u ≤ R, we have

(5.11)  |Re(e^{u+iv} − 1)| ≤ \frac{v^{2}}{2} + u\frac{e^{R} − 1}{R}.

Indeed,

|Re(e^{u+iv} − 1)| = |e^{u} cos v − 1| = |(e^{u} − 1) cos v − (1 − cos v)|

≤ (e^{u} − 1) + (1 − cos v) ≤ u\frac{e^{R} − 1}{R} + \frac{v^{2}}{2} .

From Lemma 5.1 and δ ≤ σ (see (5.8)), we have for 0 ≤ t ≤ δ,

0 ≤ mRe(ρ(y, t)) ≤ mt^{4} \frac{α^{(4)}(σ)}{4!}, 0 ≤ mIm(ρ(y, t)) ≤ mt^{3} \frac{|α^{(3)}(σ)|}{3!}.

Lemma 5.2 implies

(5.12)  mRe(ρ(y, t)) ≤ mδ^{4} \frac{α^{(4)}(σ)}{4!} ≤ \frac{D^{4}}{4m^{\frac{1}{3}} κ} = M

Hence, by (5.11),

|I_{4}| ≤ 2 ∫_{0}^{δ} e^{−mt^{2}α′′(σ)/2} (\frac{e^{M} − 1}{M}m\frac{t^{4}}{4!}α^{(4)}(σ) + \frac{(m\frac{t^{3}}{3!} α^{(3)}(σ))^{2}}{2} ) dt

≤ \frac{2(e^{M} − 1)mα^{(4)}(σ)}{M4!} ∫_{0}^{∞} e^{−mt^{2}α′′(σ)/2}t^{4} dt

+\frac{(mα^{(3)}(σ))^{2}}{(3!)^{2}} ∫_{0}^{∞} e^{−mt^{2}α′′(σ)/2}t^{6} dt

= \frac{\sqrt{2π}}{m\sqrt{mα′′(σ)}} (\frac{e^{M} − 1}{8M} \frac{α^{(4)}(σ)}{(α′′(σ))^{2}} + \frac{5(α^{(3)}(σ))^{2}}{24(α′′(σ))^{3}} )

≤ \frac{\sqrt{2π}}{m\sqrt{mα′′(σ)}} (\frac{e^{M} − 1}{M} \frac{3}{4κ} + \frac{5}{6κ}) ,

where in the last step we used Lemma 5.2.

Putting together the above estimates for I_{1} to I_{5} proves the lemma. Lemma 5.5. Let y ∈ ℝ, let \frac{1}{2} ≤ κ ≤ 1 and let F_{κ} be as defined in (5.2 ).

If m ≥ 40, then

F_{κ}(y, m) = \begin{matrix}e & mα(σ)−myσ \\ √ & \overline{2πmα′′(σ)} \end{matrix} (1 + \frac{γ}{m}), |γ| < 6.4,

where σ = σ_{κ}(y) and α(σ) = α_{κ}(σ_{κ}(y)), in the notation of (5.1 ).

Proof. Take, as we may, c = σ = σ_{κ}(y) in the definition of F_{κ}. Then the present lemma is seen to be equivalent to showing that φ in the previous lemma satisfies |φ| < \frac{6.4}{m} for all y ∈ ℝ. Now,

\frac{2^{\frac{3}{2}} σ\sqrt{mα′′(σ)}}{\sqrt{π}(mκ − 2)1.25^{mκ[σ]/2}2^{mκ/2}} = \frac{4}{mκ^{3/2}\sqrt{π}} \frac{\sqrt{mκ/2}}{2^{mκ/2}} \frac{mκ}{mκ − 2} \frac{\sqrt{σ^{2}α′′(σ)}}{1.25^{mκ[σ]/2}}

≤ \frac{5\sqrt{10}}{9 · 2^{5}\sqrt{π}m} < \frac{0.031}{m}.

We have used mκ ≥ 20 and κ ≥ \frac{1}{2} to bound the terms which do not involve σ. For the term involving σ, note that \sqrt{σ^{2}α′′(σ)} < \sqrt{1 + σ}, as follows from (5.3) on estimating the sum by an integral. Hence, \sqrt{σ^{2}α′′(σ)}/1.25^{mκ[σ]/2} < \sqrt{2}.

Let D = 1.73 < 40^{1/3}\sqrt{\frac{1}{2}}, which is allowed in Lemma 5.4, and set

q(m) = \frac{5}{3} + \frac{3}{2} \frac{(e^{N_{m}} − 1)}{N_{m}} + \frac{\sqrt{2}m^{5/6}}{D\sqrt{π}} ( exp(−m^{1/3}D^{2}/2) + 2 exp(−m^{1/3}D^{2}/4)),

where N_{m} := \frac{D^{4}}{2m^{1/3}} . Since κ ≥ \frac{1}{2}, Lemma 5.4 shows

|φ| < \frac{0.031 + q(m)}{m}.

As max_{m≥40} q(m) < 6.369, we have |φ| < \frac{6.4}{m}.

Lemma 5.6. For y ∈ ℝ and 0 < κ ≤ 1, let α = α_{κ} and σ = σ_{κ}(y) be as in (5.1 ). Then, if mκ > 2,

|∫_{−∞}^{+∞} te^{m(α(σ+it)−iyt)} dt| ≤ \frac{4e^{mα(σ)}σ^{2}}{(mκ − 2)1.25^{mκ[σ]/2}2^{mκ/2}} + \frac{e^{mα(σ)}\sqrt{2π}}{\sqrt{κ}m^{\frac{3}{2}} α′′(σ)}.

Proof.

e^{−mα(σ)} |∫_{−∞}^{+∞} te^{m(α(σ+it)−iyt)} dt|

≤ |∫_{−σ}^{σ} te^{−mα′′(σ)t^{2}/2+mρ}dt| + 2 ∫_{σ}^{∞} |te^{m(α(σ+it)−α(σ)−iyt)}| dt =: J_{1} + 2J_{2},

where ρ = ρ(y, t) is as in Lemma 5.1. By the estimate in Lemma 5.3, with ε = 1 = u and c = σ,

J_{2} ≤ \frac{2σ^{2}}{(mκ − 2)1.25^{mκ[σ]/2}2^{mκ/2}} .

RELATIVE REGULATORS OF NUMBER FIELDS 23

As for J_{1},

\frac{J_{1}}{2} = |∫_{0}^{σ} te^{−mα′′(σ)t^{2}/2} sin(mIm(ρ))dt| ≤ ∫_{0}^{σ} te^{−mα′′(σ)t^{2}/2}|mIm(ρ)|dt

≤ m\frac{|α^{(3)}(σ)|}{3!} ∫_{0}^{∞} t^{4}e^{−mα′′(σ)t^{2}/2}dt = \frac{\sqrt{2π}}{4m^{\frac{3}{2}} α′′(σ)} \frac{|α^{(3)}(σ)|}{(α′′(σ))^{3/2}} ≤ \frac{\sqrt{2π}}{2\sqrt{κ}m^{\frac{3}{2}} α′′(σ)}

where we used Lemmas 5.1 and 5.2.

Lemma 5.7. Let y ∈ ℝ, \frac{1}{2} ≤ κ ≤ 1, m ≥ 40 and let α = α_{κ} and σ = σ_{κ}(y) be as in (5.1 ). Then

|∫_{−∞}^{+∞} te^{m(α(σ+it)−iyt)} dt| ≤ \frac{2.031σ}{m} \begin{matrix}\sqrt{2π} & e & mα(σ) \\ √ & \overline{mα′′(σ)} \end{matrix} .

Proof. By the previous lemma,

e^{−m(α(σ))} |∫_{−∞}^{+∞} te^{m(α(σ+it)−iyt)} dt|

< \frac{σ\sqrt{2π}e^{mα(σ)}}{\sqrt{mα′′(σ)}} (\frac{2^{\frac{3}{2}} σ\sqrt{mα′′(σ)}}{\sqrt{π}(mκ − 2)1.25^{mκ[σ]/2}2^{mκ/2}} + \frac{1}{m\sqrt{κσ^{2}α′′(σ)}}).

We showed in the proof of Lemma 5.5 that

\frac{2^{\frac{3}{2}} σ\sqrt{mα′′(σ)}}{\sqrt{π}(mκ − 2)1.25^{mκ[σ]/2}2^{mκ/2}} ≤ \frac{0.031}{m}.

Finally, by (5.7),

\frac{1}{m\sqrt{κσ^{2}α′′(σ)}} < \frac{1}{mκ} ≤ \frac{2}{m}.

Lemma 5.8. For y ∈ ℝ, \frac{1}{2} ≤ κ ≤ 1 and m ≥ 40, let F_{κ} be as defined in (5.2 ). Then

−\frac{1}{m} \frac{\frac{∂F_{κ}}{∂y}}{F_{κ}} (y, m) = σ_{κ}(y) (1 + \frac{β}{m}), |β| = |β(y, m, κ)| < 2.42 ,

where σ = σ_{κ}(y) and α(σ) = α_{κ}(σ_{κ}(y)), in the notation of (5.1 ).

Proof. From (5.2), for any c > 0,

F_{κ}(y, m) = \frac{e^{−myc}}{2π} ∫_{−∞}^{+∞} e^{m(α(c+it)−iyt)} dt ,

where α = α_{κ}. Therefore,

−\frac{1}{m} \frac{∂F_{κ}}{∂y} = cF_{κ}(y, m) + \frac{ie^{−myc}}{2π} ∫_{−∞}^{+∞} te^{m(α(c+it)−iyt)} dt .

Now take c = σ := σ_{κ}(y). Then

−\frac{1}{m} \frac{\frac{∂F_{κ}}{∂y}}{F_{κ}} (y, m) = σ + \frac{i ∫_{−∞}^{+∞} te^{m(α(σ+it)−iyt)} dt}{2πe^{myσ}F_{κ}(y, m)}.

By Lemmas 5.5 and 5.7,

|\frac{i ∫_{−∞}^{+∞} te^{m(α(σ+it)−iyt)} dt}{2πe^{myσ}F_{κ}(y, m)}| < σ \frac{2.031}{m} \frac{1}{1 − \frac{6.4}{m}} < σ \frac{2.42}{m},

since m ≥ 40.

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

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

References

[AS] M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions, Dover Pubs., New York.

[BM1] A.-M. Berg^{}é and J. Martinet, Sur les minorations géométriques des régulateurs, in Séminaire de Théorie des Nombres de Paris 1987-1988, C. Goldstein, ed., Birkhäuser Verlag, Boston, Mass., 1990, 23–50.

[BM2] , Minorations de hauteurs et petits régulateurs relatifs, Sémin. Th. de Nombres de Bordeaux 1987-1988.

[BM3] , Notions relatives de régulateur et de hauteur, Acta Arith. 54 (1989), 155– 170.

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

[deB] N. G. de Bruijn, Asymptotic Methods in Analysis, North Holland, Amsterdam, 1957.

[GR] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, Orlando, 1980.

[La] S. Lang, Algebraic number theory, Addison–Wesley, Reading, Mass., 1970.

[Le] D. H. Lehmer, Factorization of certain cyclotomic functions. Annals of Math. (2) 34 (1933), 461–479.

[Sa] R. Salem, Power series with integral coefficients, Duke Math. J. 12 (1945), 153–172.

[Sk] N-P. Skoruppa, Quick lower bounds for regulators of number fields, Enseign. Math. t. 39 (1993), 137–141.

[WW] E. T. Whittaker and J. N. Watson, A course of Modern Analysis, 4th ed., Cambridge Univ. Press, Cambridge, 1927.

[Zi] R. Zimmert, Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. Math. 62 (1981), 367–380.