Lemma 9 — explicit relative class-number bound

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Let $K/F$ be a CM extension with $d=[F:\mathbb Q]$, and put

$$\lambda=\operatorname{rd}_{K/F} =\left(\frac{|\Delta_K|}{|\Delta_F|}\right)^{1/d}.$$

Then

$$h^-(K)\leq 8\lambda^2 \left( \sqrt\lambda\log\lambda\frac{e}{4\pi} \right)^d. \tag{1}$$

Proof

The analytic input is Corollary 3 of Stéphane Louboutin, Explicit bounds for residues of Dedekind zeta functions, values of $L$-functions at $s=1$, and relative class numbers, Journal of Number Theory 85 (2000), 263--282, DOI 10.1006/jnth.2000.2545. In the exact form used here, it says

$$h^-(K)\leq 2Q_Kw_K\sqrt{\frac{|\Delta_K|}{|\Delta_F|}} \left( \frac{e}{4\pi d} \log\frac{|\Delta_K|}{|\Delta_F|} \right)^d, \tag{2}$$

where $Q_K$ is the Hasse unit index and $w_K$ is the number of roots of unity in $K$. Since $Q_K\in\{1,2\}$ and $|\Delta_K|/|\Delta_F|=\lambda^d$, (2) becomes

$$h^-(K)\leq 2Q_Kw_K \left(\sqrt\lambda\log\lambda\frac{e}{4\pi}\right)^d. \tag{3}$$

It remains to bound $w_K$. Root discriminants do not decrease under field extension. Because $K$ contains $\mathbb Q(\mu_{w_K})$,

$$\lambda\geq\operatorname{rd}_K \geq\operatorname{rd}_{\mathbb Q(\mu_{w_K})}. \tag{4}$$

The first inequality follows as well from the relative discriminant formula $|\Delta_K|\geq|\Delta_F|^2$. Proposition 2.7 of Lawrence C. Washington, Introduction to Cyclotomic Fields, second edition, Springer GTM 83 (1997), gives the cyclotomic discriminant formula, which in root-discriminant form is

$$\operatorname{rd}_{\mathbb Q(\mu_{w_K})} =\frac{w_K}{\prod_{p\mid w_K}p^{1/(p-1)}}. \tag{5}$$

The lower bound used by Sawin follows directly. If $w_K=\prod_p p^{a_p}$, then

$$\frac{\prod_{p\mid w_K}p^{2/(p-1)}}{w_K} =\prod_{p\mid w_K}p^{2/(p-1)-a_p}\leq2, \tag{6}$$

because the factor at $p=2$ is at most $2$, the factor at $p=3$ is at most $1$, and every factor for $p\geq5$ is at most $1$. Taking square roots in (6) and using (5) gives

$$\operatorname{rd}_{\mathbb Q(\mu_{w_K})}\geq \sqrt{\frac{w_K}{2}}. \tag{7}$$

Equations (4) and (7) imply $w_K\leq2\lambda^2$. Combining this with $Q_K\leq2$ in (3) proves (1).

External-input and source scope

This is Lemma 9 on physical p. 8 of the arXiv v1 manuscript. The substitutions from (2), the root-of-unity calculation, and the elementary inequality following Washington's formula are reproduced in full. Louboutin's Corollary 3, the Hasse unit-index fact, monotonicity of root discriminants, and Washington's cyclotomic discriminant formula are external results; their proofs were not recursively compiled here.

Used by. Proposition 10.

Bears on. Problem 90.