III How Jacobians and Theta Functions Arise

Among the uj s, the most important are those of 1st kind, i.e., without poles, and if we integrate all of them at once, we are led to the most important J of 

Note that since Jac is a group, translation — invariant 

He then showed that any such algebraic group-chunk prolonged automatically into an algebraic group J with S9C D JJ C J (some Zariski-open U ). 

An important point is that 0 is an integrated form of the canonical map C - Pfl 1 discussed at length above - 

The Jacobian has always been the corner-stone in the analysis of algebraic curves and compact Riemann surfaces. Its power lies in the fact that it abelianizes the curve and is a reification of Hi, e.g., 

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