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Let $h_m$ is the class number of $\mathbb{Q}[\sqrt m]$ and let $p>2$ a prime number. Is there a known connections between $h_p$ and $h_{-p}$? e.g. if $q^i$ divides $h_p$ then it also divides $h_{-p}$, or the other way around?

The only relevant result I've found is the following (ex' 10.6 in Washington's book "Intro to cyclotomic fields"): if $0 < d$ is an even, square-free integer, $r = $ 2-rank of $\mathbb{Q}[\sqrt d]$, $d = $ 2-rank of $\mathbb{Q}[\sqrt{-d}]$, then $r \leq s \leq r+1$.

Is there a similar result for any $d$?

Thanks

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There are many relations called Scholtz reflection principle. There are many articles and other MO questions on the topic. –  Dror Speiser Feb 5 '12 at 11:06

2 Answers 2

up vote 2 down vote accepted

Let $m$ be a squarefree number, and let $d$ run through the discriminants of quadratic number fields coprime to $m$. Then the $2$-rank of the class group of ${\mathbb Q}(\sqrt{dm})$ is, up to a small term depending on the residue class of $m$ modulo $4$ and the sign of the fundamental unit involved, essentially the number of prime factors of $md$ minus $1$. This follows from Dedekind's version of Gauss's genus theory in quadratic number fields. Thus the answer to your question is yes for any $d$ (I've just seen that I switched your $d$ and $m$).

You will get slightly cleaner formulas is you replace the usual class group by the class group in the strict sense, since the dependence on the fundamental unit will disappear.

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The short answer to your question is basically no, there's essentially no connection between the prime powers $q^i$ dividing $h_p$ and $h_{-p}$.

It's true that there's a general relationship between the 2-ranks of the class groups of $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{-m})$ as per Franz's answer and the result from Washington you cite (which, incidentally, can be pushed further to 4-ranks, 8-ranks, etc., getting pretty close to a full comparison of the 2-parts of the class numbers given knowledge of the fundamental unit of the real quadratic), but for your case this is almost contentless by genus theory. For other primes $q$, or the class number in whole, it's hard to give a conclusive justification for the "no relationship" claim, though two points bear mentioning:

  1. Larges tables of these class numbers are available or easily generated, a quick survey of which is pretty compellingly against any correlation; and
  2. There are all sorts of heuristics out there about how the two classes of class numbers should behave, some of which great imply a lack of relationship. For example, it's very unlikely that $q^i$ dividing $h_{-p}$ could tell you anything about $q^i$ dividing $h_p$ since $h_{-p}$ is non-trivial for all but nine examples, whereas (probably) $h_p=1$ infinitely often.
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