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The Scholz reflection principle says, among other things, that if $D < 0$ is a negative fundamental discriminant, not $-3$, then the 3-ranks of the class group of $\mathbb{Q}(\sqrt{D})$ is either equal to that of $\mathbb{Q}(\sqrt{-3D})$, or one larger.

Does anyone know of (and recommend) any introductory reading on this fact? Why it is true, what context to view it in, etc.? Googling reveals some highbrow perspectives on it, some interesting applications, and citations to Scholz's 1932 article (which I'm having difficulty accessing for the moment). All of this is interesting, but there doesn't seem to be any obvious place to begin.

Thank you!

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Interesting. How did you get to know about this principle? –  Joël Sep 9 '11 at 18:37
    
I'm interested in cubic fields and 3-torsion in class groups from the zeta function perspective; see e.g. my work with Taniguchi arxiv.org/abs/1102.2914 -- and I'm trying to learn other techniques that have been used to study related problems. –  Frank Thorne Sep 9 '11 at 19:44
    
This work of Cohen and Morra perso.univ-rennes1.fr/anna.morra/these.pdf (Chapter 1 of Morra's thesis) is, IMHO, particularly interesting. –  Frank Thorne Sep 9 '11 at 19:45
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You might have a look at section 6 of Guillermo Mantilla-Soler's paper arxiv.org/abs/1104.4598, which expresses the Scholz reflection very concretely in terms of binary quadratic forms (which are in bijection with class groups by Gauss>) –  JSE Sep 10 '11 at 3:15

4 Answers 4

up vote 9 down vote accepted

This is simple class field theory plus Galois theory. Consider a quadratic number field $K$ with class number divisible by $3$. For constructing an unramified cyclic cubic extension $L/K$, adjoin the cube root of unity, and denote the resulting field by $K'$. The Kummer generator of the Kummer extension $L' = K'(\sqrt[3]{\mu})$ must be an ideal cube for the extension to be unramified: $(\mu) = {\mathfrak m}^3$. Since $L'/K$ is abelian, Galois theory shows that the ideal class of ${\mathfrak m}$ must come from the quadratic subfield $F$ different from $K$ and ${\mathbb Q}' = {\mathbb Q}(\sqrt{-3})$. Thus the unramified cubic extensions of $K$ correspond roughly to the $3$-class group of $F$; any differences come from the fact that $\mu$ might be a unit.

The reflection theorem was found independently by Reichardt and then generalized by Leopoldt. For a dvi file of Scholz's article, see here.

Edit.Here's an English translation.

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Excellent, thank you! If you have an English translation convenient I'd be grateful to see it; if not, I'll do my best with the German. Thanks! –  Frank Thorne Sep 9 '11 at 19:49
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Apparently I have not yet translated this one. I'll put it here when I'm done. Until then, Chapter 2 in rzuser.uni-heidelberg.de/~hb3/publ/pcft.pdf contains an introduction to Leopoldt's Spiegelungssatz which I'd like to rewrite eventually since a lot of it can be done in a cleaner way. –  Franz Lemmermeyer Sep 11 '11 at 18:09

This is covered in Ralph Greenberg's book-in-progress "Topics in Iwasawa theory" http://www.math.washington.edu/~greenber/book.pdf It also contains lots of other interesting stuff on class groups.

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Gras' book Class field theory: from theory to practice has an entire section devoted to the reflection principle you mention and its generalizations. It also shows up in Manjul Bhargava's work, but given what you say you're interested in, you likely already know about his work.

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Hi Frank,

There are two places that I remember reading, and enjoying, when learning about the reflection principle:

i) Washington's book on cyclotomic fields section 10.2.(This book is just so great, so in case you don't own a copy this might be a good excuse to buy it.)

ii) Reflection principles and bounds for class group torsion By Ellenberg and Venkatesh--This is a VERY cool paper, but in case you are short on time you just need to read Lemmas 4 and 5.

Also some of the answers to this question Explicit map for Scholz reflection principle might help a bit.

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