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5.1. Further background on elliptic curves 135And so we can define the quotient of the group of divisors by the subgroup of principal divisors:the Picard group.Definition 5.1.12 (Picard group). The quotient of Div(C) by Prin(C), denoted by Pic(C), iscalled the divisor class group, or the Picard group, of C.For a principal divisor, the degree is always zero.Proposition 5.1.13 ([244, Proposition II.3.1], [128, Corollary II.6.10]). Let K be a perfect fieldand C a smooth curve defined over K. Let f ∈ K(C) ∗ be a non-zero function on C. Then1. div(f) = 0 if and only if f ∈ K;2. deg(div(f)) = 0.Hence, for a smooth curve, the group of principal divisors is in fact a subgroup of the groupof divisors of degree zero. Therefore, we can also define the quotient of the zero degree part ofthe groups of divisors by the subgroup of principal divisors.Definition 5.1.14. Let K be a perfect field and C a smooth curve defined over K. We denoteby Pic 0 (C) the quotient of Div 0 (C) by Prin(C).It is a general fact that this quotient group can be given the structure of an algebraic variety:the Jacobian variety. In the case of elliptic curve, its description as a variety is surprisinglysimple.Proposition 5.1.15 ([244, Proposition III.3.4], [128, Example II.6.10.2]). Let K be a perfectfield and E an elliptic curve defined over K. Then the following map is an isomorphism:And the following corollary is very useful.E(K) → Pic 0 (E) ,P ↦→ [(P ) − (O E )] .Corollary 5.1.16 ([244, Corollary III.3.5]). Let K be a perfect field and E an elliptic curvedefined over K. Let D = ∑ P ∈E(K) n P (P ), n P ∈ Z almost all zero, be a divisor on E. Then Dis principal if and only if1. deg D = 0,2. ∑ P ∈K(E) [n P ]P = O E .To conclude this subsection, we should define a few more tools relating morphisms and divisorswhich are useful to define pairings on elliptic curves.A function can be evaluated at a divisor if their support are disjoint as follows.Definition 5.1.17 (Evaluation of a function at a divisor). Let K be a perfect field and Can algebraic curve defined on K. If f ∈ K(C) and D = ∑ P ∈C(K) n P (P ) with supp(D) ∩supp(div(f)) = ∅, then we define the value of f at D asf(D) =∏P ∈C(K)f(P ) n P.This operation is well-behaved with regard to the group law on divisors.