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 ${\displaystyle \mathbb {Q} \left(\zeta _{p}\right)}$, with p a prime number, will be divisible by p if the class number of the maximal real subfield ${\displaystyle \mathbb {Q} \left(\zeta _{p}\right)^{+}}$ 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 ${\displaystyle \mathbb {Q} \left({\sqrt {d}}\right)}$, then 3 also divides the class number of the imaginary quadratic field ${\displaystyle \mathbb {Q} \left({\sqrt {-3d}}\right)}$.

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

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

Let ω be the character of G

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

The reflection (Spiegelung) φ^* is defined by

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

Let E be the unit group of K. We say that ε is "primary" if ${\displaystyle K({\sqrt[{p}]{\epsilon }})/K}$ 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

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

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.