4
$\begingroup$

Related question: Totally real number fields with bounded regulators

Given a number field $K$ with degree $n$ and determinant $D$, what is the "best" upper bound for its regulator $R$, if any? I know that there are many studies about lower bounds (for example "Analytic Formula for the Regulator of a Number Field"), but I have not found any reference on upper bounds.

I am also interested (rather then bounding the regulator of any number fields), on the existence of number fields of small determinant, for general $n$ and $K$ (maybe assuming a totally complex/totally real extension).

$\endgroup$
2
  • 4
    $\begingroup$ Franz Lemmermeyer's answer to the related question you link to actually provides an upper bound for the regulator (R < c(n)*D^(1/2)*log^(n-1)(D)) due to Landau and cites a subsequent improvement due to Remak. Of course both of these results are over 80 years old... $\endgroup$
    – user1073
    Nov 9, 2015 at 16:32
  • 1
    $\begingroup$ This answers the firs part of the question - but since my German is, let's say, null, I couldn't see any details of the proof in the paper. Also, is it possible to describe how is the constant c(n) increasing with n? $\endgroup$
    – Campello
    Nov 10, 2015 at 9:18

1 Answer 1

7
$\begingroup$

This is an extended version of the comment I made above and a response to the OP's follow-up question.

Franz Lemmermeyer's response to this question provides an upper bound for the regulator of a number field due to Landau.

  • E. Landau, Verallgemeinerung eines Polyaschen Satzes auf algebraische Zahlkörper Gött. Nachr. 1918, 478--488

Landau's bound is $$R<c(n)D^{1/2}\log^{n-1}(D),$$

and was later improved upon by Remak.

  • R. Remak, Elementare Abschätzungen von Fundamentaleinheiten und des Regulators eines algebraischen Zahlkörpers, J. Reine Angew. Math. 167 (1932), 360-378.

Note that Siegel also has related results.

  • C. L. Siegel, Abschätzung von Einheiten, Nachr. Göttingen 9 (1969) 71-86.

All of these results are in german. The best english reference I could find is due to Sands.

  • J. Sands, Generalization of a theorem of Siegel. Acta Arithmetica 58, 47–56 (1991).

To be more precise, Sands considered the problem of obtaining an (effective) upper bound for the regulator $R_\mathcal{O}$ of an order $\mathcal{O}$ contained in the ring of integers $\mathcal{O}_K$ of $K$. Let $h_\mathcal{O}$ be the class number of $\mathcal{O}$, $w_\mathcal{O}$ the order of the torsion subgroup of $\mathcal{O}$ and $r_1$ the number of real places of $K$.

Sand's main theorem is:

Theorem: We have the inequality $$2^{r_1}R_\mathcal{O}h_\mathcal{O}/w_\mathcal{O} < 4\left(\frac{4}{n-1}\right)^{n-1}|d_\mathcal{O}|^{1/2}(\log|d_\mathcal{O}|)^{n-1}(\log\log|d_\mathcal{O}|)^{n/2}.$$

When $\mathcal{O}=\mathcal{O}_K$ (the case you are interested in) Sands notes that one can remove the $(\log\log|d_\mathcal{O}|)^{n/2}$ term.

While I do not have an answer to your second question I merely want to mention the discriminant bounds of Odlyzko (see this paper and this one).

Theorem: Let $\gamma=0.57721\dots$ be the Euler–Mascheroni constant and $K$ be a number field of signature $(r_1,r_2)$ (hence degree $n=r_1+2r_2$) and absolute value of discriminant $D$. Then $$D^{1/n} > (4\pi e^{1+\gamma})^{r_1/n}\left(4\pi e^\gamma\right)^{2r_2/n}-O(n^{-2/3}).$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.