Picard group of an elliptic complex curve I've followed a course about complex tori and  complex elliptic curves. At the end of the course we introduced the $Pic_0$ group and showed it is isomorphic to the cubic seen as a group. I understood what we are doing formally, but i can't manage to grasp what we are really doing.
The fact that the picard zero group was isomorphic to the elliptic curve really surprised me,but I 'm still confused about what it should suggest to me.
What does it "measure" the $Pic_0$ of a curve? Why we introduce divisors?(Let me explain : for example for me De Rham cohomology measures the failure of a certain differential condition to better understand some geometrical aspects of manifold. Is there a similar way of thinking for the Picard group?)
 A: Cartier divisors on $X$ classify line bundles on $X$. They can give embedding in the projective space and are very useful tools. For example geometry of ruled surfaces heavily use line bundles.  
If $X$ is a smooth curve its Picard group, that is the group of line bundles up to isomorphism (or equivalenty the group of divisors up to linear equivalence) is $Pic(X) = Pic_0(X) \oplus \Bbb Z$. This is because $D \mapsto \deg(D)$ is a surjective morphism, and by picking a point $p \in X$ we can (non naturally) split this sequence by mapping $1 \to [p]$. 
In fact it turns out that $Pic_0(X) = Jac(X)$, the Jacobian of the curve which is a torus ! More precisely we have $Jac(X) \cong \Bbb C^g/\Lambda$ where $\Lambda$ is the lattice generated by periods, i.e the vectors $v_i = (\int_{c_i} \omega_1, \dots, \int_{c_i} \omega_g)$ where $c_1, \dots, c_{2g}$ is a basis of $H_1(X, \Bbb Z)$ and $\omega_1, \dots, \omega_g$ is a basis of $\Omega^1(X)$, the holomorphic 1-forms on $X$.
So to sum up, there is a "discrete part" (given by the degree) and a "continuous part" given by the Jacobian. This continuous part is when all the informations are hidden. For example $Pic(\Bbb P^1) = \Bbb Z$, which is very simple to understand. You saw that an elliptic curve is isomorphic to its Jacobian, in fact it is a deep theorem (the Torelli theorem) that any smooth curve $X$ is uniquely determined by its Jacobian. 
I suggest reading Rick Miranda's book on Riemann surfaces which contains a very nice exposition of the Abel-Jacobi theorem, and a more abstract interpretation of the Jacobian, using divisors and line bundles. 
