Modulus (algebraic number theory)

For the operation that gives a number's remainder, see Modulo operation.

In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle,^[1] or extended ideal^[2]) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field.

Definition

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Let K be a global field with ring of integers R. A modulus is a formal product^[3]^[4]

\mathbf {m} =\prod _{\mathbf {p} }\mathbf {p} ^{\nu (\mathbf {p} )},,円,円\nu (\mathbf {p} )\geq 0

{\displaystyle \mathbf {m} =\prod _{\mathbf {p} }\mathbf {p} ^{\nu (\mathbf {p} )},,円,円\nu (\mathbf {p} )\geq 0}

where p runs over all places of K, finite or infinite, the exponents ν(p) are zero except for finitely many p. If K is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places. If K is a function field, ν(p) = 0 for all infinite places.

In the function field case, a modulus is the same thing as an effective divisor,^[5] and in the number field case, a modulus can be considered as special form of Arakelov divisor.^[6]

The notion of congruence can be extended to the setting of moduli. If a and b are elements of K^×ばつ, the definition of a ≡^∗b (mod p^ν) depends on what type of prime p is:^[7]^[8]

if it is finite, then

a\equiv ^{\ast }\!b,円(\mathrm {mod} ,円\mathbf {p} ^{\nu })\Leftrightarrow \mathrm {ord} _{\mathbf {p} }\left({\frac {a}{b}}-1\right)\geq \nu

{\displaystyle a\equiv ^{\ast }\!b,円(\mathrm {mod} ,円\mathbf {p} ^{\nu })\Leftrightarrow \mathrm {ord} _{\mathbf {p} }\left({\frac {a}{b}}-1\right)\geq \nu }

where ord_p is the normalized valuation associated to p;

if it is a real place (of a number field) and ν = 1, then

a\equiv ^{\ast }\!b,円(\mathrm {mod} ,円\mathbf {p} )\Leftrightarrow {\frac {a}{b}}>0

{\displaystyle a\equiv ^{\ast }\!b,円(\mathrm {mod} ,円\mathbf {p} )\Leftrightarrow {\frac {a}{b}}>0}

under the real embedding associated to p.

if it is any other infinite place, there is no condition.

Then, given a modulus m, a ≡^∗b (mod m) if a ≡^∗b (mod p^ν(p)) for all p such that ν(p) > 0.

Ray class group

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Main article: Ray class group

The ray modulo m is^[9]^[10]^[11]

K_{\mathbf {m} ,1}=\left\{a\in K^{\times }:a\equiv ^{\ast }\!1,円(\mathrm {mod} ,円\mathbf {m} )\right\}.

{\displaystyle K_{\mathbf {m} ,1}=\left\{a\in K^{\times }:a\equiv ^{\ast }\!1,円(\mathrm {mod} ,円\mathbf {m} )\right\}.}

A modulus m can be split into two parts, m_f and m_∞, the product over the finite and infinite places, respectively. Let I^m to be one of the following:

if K is a number field, the subgroup of the group of fractional ideals generated by ideals coprime to m_f;^[12]
if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m.^[13]

In both case, there is a group homomorphism i : K_m,1 → I^m obtained by sending a to the principal ideal (resp. divisor) (a).

The ray class group modulo m is the quotient C_m = I^m / i(K_m,1).^[14]^[15] A coset of i(K_m,1) is called a ray class modulo m.

Erich Hecke's original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus m.^[16]