Reflection theorem

In algebraic number theory, a reflection theorem or Spiegelungssatz (German for reflection theorem – see Spiegel and Satz) is one of a collection of theorems linking the sizes of different ideal class groups (or ray class groups), or the sizes of different isotypic components of a class group. The original example is due to Ernst Eduard Kummer, who showed that the class number of the cyclotomic field $\mathbb {Q} \left(\zeta _{p}\right)$ , with p a prime number, will be divisible by p if the class number of the maximal real subfield $\mathbb {Q} \left(\zeta _{p}\right)^{+}$ is. Another example is due to Scholz.[1] A simplified version of his theorem states that if 3 divides the class number of a real quadratic field $\mathbb {Q} \left({\sqrt {d}}\right)$ , then 3 also divides the class number of the imaginary quadratic field $\mathbb {Q} \left({\sqrt {-3d}}\right)$ .

For reflection principles in set theory, see reflection principle.

Leopoldt's Spiegelungssatz

Both of the above results are generalized by Leopoldt's "Spiegelungssatz", which relates the p-ranks of different isotypic components of the class group of a number field considered as a module over the Galois group of a Galois extension.

Let L/K be a finite Galois extension of number fields, with group G, degree prime to p and L containing the p-th roots of unity. Let A be the p-Sylow subgroup of the class group of L. Let φ run over the irreducible characters of the group ring Q_p[G] and let A_φ denote the corresponding direct summands of A. For any φ let q = p^φ(1) and let the G-rank e_φ be the exponent in the index

[A_{\phi }:A_{\phi }^{p}]=q^{e_{\phi }}.

Let ω be the character of G

\zeta ^{g}=\zeta ^{\omega (g)}{\text{ for }}\zeta \in \mu _{p}.

The reflection (Spiegelung) φ^* is defined by

\phi ^{*}(g)=\omega (g)\phi (g^{-1}).

Let E be the unit group of K. We say that ε is "primary" if $K({\sqrt[{p}]{\epsilon }})/K$ is unramified, and let E₀ denote the group of primary units modulo E^p. Let δ_φ denote the G-rank of the φ component of E₀.

The Spiegelungssatz states that

|e_{\phi ^{*}}-e_{\phi }|\leq \delta _{\phi }.

Extensions

Extensions of this Spiegelungssatz were given by Oriat and Oriat-Satge, where class groups were no longer associated with characters of the Galois group of K/k, but rather by ideals in a group ring over the Galois group of K/k. Leopoldt's Spiegelungssatz was generalized in a different direction by Kuroda, who extended it to a statement about ray class groups. This was further developed into the very general "T-S reflection theorem" of Georges Gras.[2] Kenkichi Iwasawa also provided an Iwasawa-theoretic reflection theorem.

References

A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, J. reine angew. Math., 166 (1932), 201-203.
Georges Gras, Class Field Theory: From Theory to Practice, Springer-Verlag, Berlin, 2004, pp. 157–158.

Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. pp. 147–149. ISBN 3-540-63003-1. Zbl 0819.11044.

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[1] A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, J. reine angew. Math., 166 (1932), 201-203.

[2] Georges Gras, Class Field Theory: From Theory to Practice, Springer-Verlag, Berlin, 2004, pp. 157–158.