Stickelberger ideal

Let $ m $ be a fixed natural number and $ K _ {m} $ the cyclotomic field generated by a primitive $ m $ th root of unity over the rational number field $ \mathbf Q $. Let $ G \simeq ( \mathbf Z/m \mathbf Z ) ^ \times $ be its Galois group and $ \mathbf Z [ G ] $ the group ring of $ G $ (cf. also Group algebra; Cross product) over the rational integers $ \mathbf Z $. They act on ideals and on the ideal class group $ C $ of $ K _ {m} $ (cf. Class field theory). The Stickelberger ideal $ S $ is an ideal in $ \mathbf Z [ G ] $ annihilating $ C $ and related with the relative class number $ h ^ {-} $ of $ K _ {m} $. It is defined as follows.

Let $ O $ be the ring of integers of $ K _ {m} $ and $ \mathfrak p $ a prime ideal of $ O $ that is prime to $ m $. Let $ p $ be a prime integer satisfying $ ( p ) = \mathfrak p \cap \mathbf Z $ and let $ \mathbf F _ {p} = \mathbf Z/p \mathbf Z $ be the prime field. Define a mapping $ {\psi _ {0} } : {\mathbf F _ {p} } \rightarrow {\mathbf C ^ \times = \mathbf C \setminus \{ 0 \} } $ by

$$ \psi _ {0} ( a ) = { \mathop{\rm exp} } \left ( { \frac{2 \pi i }{p} } a \right ) , $$

where $ \mathbf C $ is the complex number field. Let $ F = O/ \mathfrak p $ be the residue field and define $ {\psi _ {\mathfrak p} } : F \rightarrow {\mathbf C ^ \times } $ by composition of $ \psi _ {0} $ and the trace mapping $ T : F \rightarrow {\mathbf F _ {p} } $, i.e., $ \psi _ {\mathfrak p} = \psi _ {0} \circ T $. Let $ W _ {m} $ be the group of unities in $ K _ {m} $. Then there is an injection

$$ W _ {m} \rightarrow O \setminus \mathfrak p \rightarrow ( O/ \mathfrak p ) ^ \times = F ^ \times , $$

so that $ m $ divides $ q - 1 $ with $ q = N ( \mathfrak p ) = | F | $. This induces a bijection $ f : {W _ {m} } \rightarrow {( F ^ \times ) ^ {( q - 1 ) /m } } $. Define a character $ {\chi _ {\mathfrak p} } : {F ^ \times } \rightarrow {W _ {m} } $ of order $ m $ by

$$ \chi _ {\mathfrak p} ( x ) = f ^ {- 1 } ( x ^ {( q - 1 ) /m } ) $$

for $ x \in F ^ \times $. One can now make up the Gauss sum

$$ g _ {a} ( \mathfrak p ) = - \sum _ {x \in F ^ \times } \chi _ {\mathfrak p} ( x ) ^ {a} \cdot \psi _ {\mathfrak p} ( x ) $$

for $ a \in \mathbf Z $. For a real number $ x $, let $ \langle x \rangle $ be the number uniquely determined by $ x \equiv \langle x \rangle ( { \mathop{\rm mod} } \mathbf Z ) $ and $ 0 \leq \langle x \rangle < 1 $. For $ t ( { \mathop{\rm mod} } m ) $ with $ { \mathop{\rm gcd} } ( t,m ) = 1 $, let $ \sigma _ {t} $ be the element of $ G $ corresponding to $ t ( { \mathop{\rm mod} } m ) $ by $ G \simeq ( \mathbf Z/m \mathbf Z ) ^ \times $. For $ a \in \mathbf Z $, let $ \theta ( a ) \in \mathbf Q [ G ] $ be the element

$$ \theta ( a ) = \sum _ {t ( { \mathop{\rm mod} } m ) } \left \langle {- { \frac{at }{m} } } \right \rangle \sigma _ {t} ^ {- 1 } . $$

L. Stickelberger proved the following theorem: For $ r \geq 1 $, $ a _ {1} \dots a _ {r} ,b _ {1} \dots b _ {r} \in \mathbf Z $ one has $ \sum _ {i = 1 } ^ {r} a _ {i} \theta ( b _ {i} ) \in \mathbf Z [ G ] $ if and only if $ \sum a _ {i} b _ {i} \equiv 0 ( { \mathop{\rm mod} } m ) $. In this case, for any prime ideal $ \mathfrak p $ prime to $ m $ one has $ \prod _ {i = 1 } ^ {m} g _ {b _ {i} } ( \mathfrak p ) ^ {a _ {i} } \in K _ {m} $ and $ \mathfrak p ^ {\sum _ {i = 1 } ^ {r} a _ {i} \theta ( b _ {i} ) } $ is the principal ideal generated by $ \prod _ {i = 1 } ^ {m} g _ {b _ {i} } ( \mathfrak p ) ^ {a _ {i} } $.

The ideal $ S = \{ {\sum _ {i = 1 } ^ {r} a _ {i} \theta ( b _ {i} ) } : {\sum a _ {i} b _ {i} \equiv 0 ( { \mathop{\rm mod} } m ) } \} $ in $ \mathbf Z [ G ] $ is called the Stickelberger ideal for $ K _ {m} $, and an element of $ S $ is called a Stickelberger operator for $ K _ {m} $. Since any class of $ C $ contains a prime ideal $ \mathfrak p $ prime to $ m $, Stickelberger's theorem implies that $ S $ annihilates $ C $: $ S \cdot C = 0 $.

For a $ \mathbf Z [ G ] $-module $ A $, one defines $ A ^ {-} = \{ {a \in A } : {( 1 + J ) a = 0 } \} $, where $ J = \sigma _ {- 1 } $ corresponds to complex conjugation.

Let $ g $ be the number of primes that ramify in $ K _ {m} $. Now,

1) if $ g = 1 $, then $ [ \mathbf Z [ G ] ^ {-} :S ^ {-} ] = h ^ {-} $, where $ h ^ {-} $ is the relative class number of $ K _ {m} $ (Iwasawa's theorem);

2) if $ g \geq 2 $, then $ [ \mathbf Z [ G ] ^ {-} :S ^ {-} ] = 2 ^ {a} \cdot h ^ {-} $, where $ a = 2 ^ {g - 2 } - 1 $ (Sinnott's theorem).

These results have, to some extent, been generalized to absolute Abelian fields. Stickelberger ideals are also used to construct a $ p $-adic $ L $-function [a3].

References

[a1]	L. Stickelberger, "Über eine Verallgemeinerung der Kreistheilung" Math. Ann. , 37 (1890) pp. 321–367
[a2]	K. Iwasawa, "A class number formula for cyclotomic fields" Ann. of Math. (2) , 76 (1962) pp. 171–179
[a3]	K. Iwasawa, "Lectures on -adic -functions" , Ann. Math. Studies , 74 , Princeton Univ. Press (1972)
[a4]	W. Sinnott, "On the Stickelberger ideal and the circular units of a cyclotomic field" Ann. of Math. (2) , 108 (1978) pp. 107–134
[a5]	W. Sinnott, "On the Stickelberger ideal and the circular units of an abelian field" Invent. Math. , 62 (1980/1) pp. 181–234
[a6]	T. Kimura, K. Horie, "On the Stickelberger ideal and the relative class number" Trans. Amer. Math. Soc. , 302 (1987) pp. 727–739
[a7]	S. Lang, "Cyclotomic fields" , 1–2 , Springer (1990)
[a8]	L.C. Washington, "Introduction to cyclotomic fields" , Springer (1982)