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134 Chapter 5. Complex multiplication and elliptic curves5.1.2 Divisors of algebraic curvesThe group of divisors of a curve is the free abelian group generated by its geometric points. It isa powerful tool to study the curve itself.Definition 5.1.7 (Divisor [244, Section II.3], [128, Section II.6]). Let K be a perfect field and Can algebraic curve defined over K. A divisor D of C is a formal finite sum of geometric points ofC with integer coefficients:D =∑n P (P ) ,P ∈C(K)where only a finite number of coefficients n P ∈ Z are non-zero. The support of a divisor is theset of points with non-zero coefficients:supp(D) = { P ∈ C(K) | n P ≠ 0 } .The degree of a divisor is the sum of its coefficients:deg(D) =∑P ∈C(K)The Galois group of K naturally acts on divisors and we say that a divisor is rational over Kif it is invariant by this action.Definition 5.1.8 (Group of divisors). Let K be a perfect field and C an algebraic curve definedover K. The group of divisors on C is denoted by Div(C). The subgroup of divisors of degree 0is denoted by Div 0 (C).n P .If f ∈ K(C) ∗ is a non-zero function on C, then one can associate a divisor with it.Definition 5.1.9 (Divisor of a function). Let K be a perfect field and C an algebraic curvedefined over K. Let f ∈ K(C) ∗ be a non-zero function on C. The divisor of f, denoted by div(f),is defined asdiv(f) =∑ord P (f)(P ) ,where ord P (f) is the order of f at P .P ∈C(K)These divisors form the set of principal divisors, which is easily seen to be a subgroup ofDiv(C).Definition 5.1.10 (Principal divisors). Let K be a perfect field and C an algebraic curve definedover K. The group of divisors of the form div(f) for f ∈ K(C) ∗ is called the group of principaldivisors and denoted byPrin(C) = { div(f) ∈ Div(C) | f ∈ K(C) ∗} .We can then define an equivalence relation using them.Definition 5.1.11 (Linear equivalence). Let K be a perfect field and C an algebraic curvedefined over K. Let D and D ′ be two divisors on C. They are said to be linearly equivalent ifD − D ′ ∈ Prin(C). This is an equivalence relation that we denote byD ∼ D ′ .