Reflection theorem

Short description: One of several theorems linking the sizes of different ideal class groups

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 [math]\displaystyle{ \mathbb{Q} \left( \zeta_p \right) }[/math], with p a prime number, will be divisible by p if the class number of the maximal real subfield [math]\displaystyle{ \mathbb{Q} \left( \zeta_p \right)^{+} }[/math] is. Another example is due to Scholz. A simplified version of his theorem states that if 3 divides the class number of a real quadratic field [math]\displaystyle{ \mathbb{Q} \left( \sqrt{d} \right) }[/math], then 3 also divides the class number of the imaginary quadratic field [math]\displaystyle{ \mathbb{Q} \left( \sqrt{-3d} \right) }[/math].

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

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

Let ω be the character of G

[math]\displaystyle{ \zeta^g = \zeta^{\omega(g)} \text{ for } \zeta \in \mu_p . }[/math]

The reflection (Spiegelung) φ^* is defined by

[math]\displaystyle{ \phi^*(g) = \omega(g) \phi(g^{-1}) . }[/math]

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

[math]\displaystyle{ | e_{\phi^*} - e_\phi | \le \delta_\phi . }[/math]

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. Kenkichi Iwasawa also provided an Iwasawa-theoretic reflection theorem.

References

• 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.