Modulus in algebraic number theory

2020 Mathematics Subject Classification: Primary: 11R [MSN][ZBL]

A formal product of places of an algebraic number field, also termed an extended ideal. It is used to encode ramification data for abelian extensions of a number field (cf Conductor of an Abelian extension).

Let $K$ be an algebraic number field with ring of integers $R$. A modulus is a formal product $$ \mathfrak{m} = \prod_{\mathfrak{p}} \mathfrak{p}^{\nu(\mathfrak{p})} $$ where $\mathfrak{p}$ runs over all places of $K$, finite or infinite, the exponents $\nu$ are zero except for finitely many $\mathfrak{p}$, for real places $\mathfrak{r}$ we have $\nu(\mathfrak{r})=0$ or $1$ and for complex places $\nu=0$.

We extend the notion of congruence to this setting. Let $x$ and $y$ be elements of K. For a finite place $\mathfrak{p}$, that is, a prime ideal of the ring of integers, we define $x$ and $y$ to be congruent modulo $\mathfrak{p}^n$ if x/y is in the valuation ring $R_{\mathfrak{p}}$ of ${\mathfrak{p}}$ and congruent to 1 modulo $\mathfrak{p}^n$ in $R_{\mathfrak{p}}$ in the usual sense of ring theory. For a real place $\mathfrak{r}$ we define $x$ and $y$ to be congruent modulo $\mathfrak{r}$ if $x/y$ is positive in the real embedding of $K$ associated to the place $\mathfrak{r}$. Finally, we define $x$ and $y$ to be congruent modulo $\mathfrak{m}$ if they are congruent modulo $\mathfrak{p}^{\nu(\mathfrak{p})}$ whenever $\nu(\mathfrak{p}) > 0$.

Ray class group

We split the modulus $\mathfrak{m}$ into $\mathfrak{m}_\text{fin}$ and $\mathfrak{m}_\text{inf}$, the product over the finite and infinite places respectively. Define $$ K_{\mathfrak{m}} = \left\lbrace a/b \in K \mid a,b \in R,~ ab ~\mbox{coprime to}~ \mathfrak{m}_\mbox{fin} \right\rbrace \,, $$ $$ K_{\mathfrak{m},1} = \left\lbrace x \in K_{\mathfrak{m}} \mid x \equiv 1 \pmod {\mathfrak{m}} \right\rbrace \ . $$

We call the group $K_{\mathfrak{m},1}$ the ray modulo $\mathfrak{m}$.

Further define the subgroup of the ideal group $I^{\mathfrak{m}}$ to be the subgroup generated by ideals coprime to $\mathfrak{m}_\text{fin}$. The ray class group modulo $\mathfrak{m}$ is the quotient $I^{\mathfrak{m}} / i(K_{\mathfrak{m},1})$, where $i$ is the map from $K$ to principal ideals in the ideal group. A coset of $i(K_{\mathfrak{m},1})$ is a ray class.

The ray class group is finite. Its order is the ray class number: this divides the class number of $K$. For the trivial modulus $\mathfrak{m} = 1$, the ray class group is just the ideal class group.

Hecke's original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus $\mathfrak{m}$.

References

• Harvey Cohn. "A classical invitation to algebraic numbers and class fields" (Springer-Verlag, 1978) ISBN 0-387-90345-3. pp.163-187
• Harvey Cohn. "Introduction to the construction of class fields". Cambridge studies in advanced mathematics 6 (Cambridge University Press, 1985) ISBN 0-521-24762-4.
• Gerald J. Janusz. "Algebraic Number Fields". Pure and Applied Mathematics 55 (Academic Press, 1973) ISBN 0-12-380250. pp.107-113.