In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the field of algebraic number theory

Algebraic number theory

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field

Global field

In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...

 (i.e. an algebraic number field

Algebraic number field

In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...

 or a global function field). It is used to encode ramification

Ramification

In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...

 data for abelian extension

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....

s of a global field.

Definition

Let K be a global field with ring of integers R. A modulus is a formal product


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, and in the number field case, a modulus can be considered as an Arakelov divisor.

The notion of congruence

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

 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:

• if it is finite, then

where ord_p is the normalized valuation associated to p;

• if it is a real place (of a number field), then

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

The ray modulo m is

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;
• if K is a function field of an algebraic curve
  Algebraic curve
  In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
  
   over k, the group of divisors, rational over k, with support away from m.

In both case, there is a group homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

 i : K_m,1 → I^m obtained by sending a to the principal ideal

Principal ideal

In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...

 (resp. divisor) (a).

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

Erich Hecke

Erich Hecke

Erich Hecke was a German mathematician. He obtained his doctorate in Göttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students....

's original definition of Hecke character

Hecke character

In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to...

s may be interpreted in terms of character

Character (mathematics)

In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...

s of the ray class group with respect to some modulus m.

Properties

When K is a number field, the following properties hold.

• When m = 1, the ray class group is just the ideal class group
  Ideal class group
  In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an 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.