Modulus (algebraic number theory)

Let K be a global field with ring of integers R. A modulus is a formal product{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§III.1}}

:$\backslash mathbf\{m\}\; =\; \backslash prod\_\{\backslash mathbf\{p\}\}\; \backslash mathbf\{p\}^\{\backslash nu(\backslash mathbf\{p\})\},\backslash ,\backslash ,\backslash nu(\backslash mathbf\{p\})\backslash geq0$

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,{{harvnb|Serre|1988|loc=§III.1}} and in the number field case, a modulus can be considered as special form of Arakelov divisor.{{harvnb|Neukirch|1999|loc=§III.1}}

The notion of congruence can be extended to the setting of moduli. If a and b are elements of K^×, the definition of a ≡^∗b (mod p^ν) depends on what type of prime p is:{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§III.1}}

• if it is finite, then

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

:where ord_p is the normalized valuation associated to p;

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

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

: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.

{{main|Ray class group}}

The ray modulo m is{{harvnb|Milne|2008|loc=§V.1}}{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§VI.6}}

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

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;{{harvnb|Janusz|1996|loc=§IV.1}}
• if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m.{{harvnb|Serre|1988|loc=§V.1}}

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).{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§VI.6}} 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.{{harvnb|Neukirch|1999|loc=§VII.6}}

When K is a number field, the following properties hold.{{harvnb|Janusz|1996|loc=§4.1}}

• When m = 1, the ray class group is just the ideal class group.
• The ray class group is finite. Its order is the ray class number.
• The ray class number is divisible by the class number of K.

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Category:Algebraic number theory