In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality

$$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$

and compares with the bound due to Minkowski that

$$n-{1\over 12n}-\log\sqrt{2\pi n}-r_2\log\left({4\over\pi}\right)\le \log\sqrt{d}$$

where $n=[\mathbf{Z}:\Bbb Q]$ is the degree of a fixed, but arbitrary number field, $\mathbf{Z}$ (here $\mathbf{Z}$ is not the integers, at least not if I'm reading the German correctly), $d$ is the absolute discriminant of $\mathbf{Z}/\Bbb Q$, and $r_2$ is the number of pairs of complex embeddings of $\mathbf{Z}\to\Bbb C$.

My question: Does the inequality $(*)$ hold without any assumptions on the number field, $\mathbf{Z}$? If so, is there a reference to who proved this bound first/where? If not, what are the assumptions (I've noted some possibilities I've come up with below.)

The problem: My German is rusty, and it's difficult to process some of the terminology which is not always the same between literal words and mathematical usage. Like French, the Germans use their word for "bodies" in place of "fields," for example, so direct translation is harder to trust, and certainly I want to be sure I'm understanding the result as it is, and not as I'd like it to be.

Examples of complications I'm not sure of:

($1$) At the beginning he talks about the assumption that $w$, the number of roots of unity in $\mathbf{Z}$ being $2$ when the number of real embeddings, $r_1>0$.

It's hard to tell if he wants $r_1>0$ as a standing hypothesis and whether that affects the truth of the given inequality. It's not stated as an assumption in Satz 1, but it may be in Siegel's style, which I'm not intimate enough with to judge.

($2$) There is this constant

$$a=2^{-r_2}\pi^{-{n\over 2}}\sqrt{d}$$

which Siegel uses in ($3$) of his proof under an assumption

$$2^{r_1}h{R\over w}\le a^s\left(\Gamma\left({s\over 2}\right)\right)^{r_1}(\Gamma(s))^{r_2}s^{n+1}(s-1)^{1-n}$$

($h=|C_{\mathbf{Z}}|$ is the class number of $\mathbf{Z}$ and $R$ is its regulator)

and cites Landau as having taken $s=1+{1\over\log d}$ to get some other estimate.

This is part of the confusion, because it seems to indicate $w=2$ is not a standing assumption, but I could see it as being Siegel's style to keep the notation so that the reader can see where the factor of $2$ is coming from.

($3$) There is a stated assumption $\log a\le n-1$, which I cannot tell if the inequality ($*$) depends upon or not.

It's noted earlier in the proof, but not on the line where $(*)$ is presented, though it seems like it might be relevant, because it is stated in the line right before, but of course sometimes when we write papers, we write independent truths consecutively before putting them all together for the main theorem. Siegel even does this--as I mentioned earlier--in the inequality after $(*)$, where he quotes the Minkowski bound.

  • $\begingroup$ The inequality in (*) should give a lower bound for $d$ rather than an upper bound. It looks like you're interested in lower bounds for discriminants. Look up Odlyzko's survey article on this (available from his website). $\endgroup$ – Lucia Sep 19 '14 at 20:20
  • $\begingroup$ @Lucia sorry if it wasn't clear, I'm interested in the upper bound, actually, not the lower. The * inequality gives an upper bound after applying the exponential, at least at face value since $\exp$ is increasing, no? $\endgroup$ – Adam Hughes Sep 19 '14 at 20:21

I don't have access to Siegel's paper at the moment, but $(*)$ is clearly false in general. For example, for $n=2$ it would mean that there are only finitely many quadratic number fields. In fact $(*)$ is equivalent to the bound $\log a\leq n-1$ that you mention under (3).

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