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In addition to the functor from curves C to one manifolds X, Riemann also considered two more functors, the symmetric products X^(d) and the Jacobian variety J(X), as well as a natural transformation between them X^(d)--->J(X), called the Abel map. The fibers of this map are the linear series |D|, ("Abel's theorem"), and the derivative of this map is the Roch ("B-N") matrix, (by the fundamental theorem of calculus). Hence the Riemann Roch theorem becomes the assertion that the fibers of the Abel map are non singular as schemes. I.e. the fiber dimension dim |D|, equals the dimension of the kernel of the derivative, d-g+h^0(K-D). This is the formulation of Mattuck and Mayer.

All this is actually proved essentially rigorously in his paper, appealing only to his extension theorem for holomorphic functions. What has been criticized as to rigor is the inverse correspondence that all compact connected complex one manifolds arise from plane curves. The method was to produce harmonic functions in plane regions by the Dirichlet principle, which method was justified by Hilbert and others later, as recorded in the books of Weyl and Siegel and Springer. More modern approaches occur in Gunning, and the article by Cornalba in his Trieste lectures.

This implies that your construction of a (non unique) plane curve from a function field, always yields birationally equivalent curves, hence isomorphic Riemann surfaces. Riemann himself considers the problem of birational equivalence in his paper and determines the lowest degree of a plane polynomial representation for a given Riemann surface in terms of the lowest degree of a map from that surface to the Riemann sphere.

All this is actually proved essentially rigorously in his paper, appealing only to his extension theorem for holomorphic functions. What has been criticized as to rigor is the inverse correspondence that all compact connected complex one manifolds arise from plane curves. The method was to produce harmonic functions in plane regions by the Dirichlet principle, which method was justified by Hilbert and others later, as recorded in the books of Weyl and Siegel and Springer. More modern approaches occur in Gunning, and the article by Cornalba in his Trieste lectures.

In that famous Abelsche Funktionen paper, Riemann goes on to deduce rigorously his famous inequality, by estimating the rank of a period matrix, assuming only the existence of sufficient meromorphic one forms of 1st and 2nd kinds, i.e. either holomorphic, or having zero residues at every pole. Although his proof of the existence of these forms in the manifold setting relies on his disputed use of the Dirichlet principle, he remarks in section 9 of the paper that one can simply write them down in the case of plane curves, and he actually does so for the holomorphic ones, using the "Poincare" residue principle. He says he could write down the others as well, but will not stop to do so. Such explicit expressions are given in the book on Plane Algebraic Curves of Brieskorn, by way of showing how to represent all cohomology classes on a curve by meromorphic forms of 1st and 2nd kinds. E.g. on the cubic curve y^2 = x(x-1)(x-t), the form x(x-1)dx/y^3 is an elementary form of 2nd kind with one double pole (at (t,0)) but zero residue.

If one grants that Riemann knew how to do this, as he said, then the foundation for his proof of the Riemann inequality is completely provided, and at least for plane curves, there is no need for Hilbert's analytic foundations to bolster Riemann's argument in the complex algebraic case. The 1965 paper of Roch, in which he completes Riemann's argument, rests solely on Green's theorem to compute Riemann's period matrix as a residue integral, hence is completely solid. 17 years later, Brill and Noether, using the same matrix computed by Roch, apparently showed that one can exploit the duality between divisors of form D and K-D to actually give the full proof using only the existence of the integrals of 1st kind. Since that paper was so influential, Roch's residue matrix (occurring in the middle of the second page of his paper) is now usually known as the Brill Noether matrix.

In that famous Abelsche Functionen paper, Riemann goes on to deduce rigorously his famous inequality, by estimating the rank of a period matrix, assuming only the existence of sufficient meromorphic one forms of 1st and 2nd kinds, i.e. either holomorphic, or having zero residues at every pole. Although his proof of the existence of these forms in the manifold setting relies on his disputed use of the Dirichlet principle, he remarks in section 9 of the paper that one can simply write them down in the case of plane curves, and he actually does so for the holomorphic ones, using the "Poincare" residue principle. He says he could write down the others as well, but will not stop to do so. Such explicit expressions are given in the book on Plane Algebraic Curves of Brieskorn, by way of showing how to represent all cohomology classes on a curve by meromorphic forms of 1st and 2nd kinds. E.g. on the cubic curve y^2 = x(x-1)(x-t), the form x(x-1)dx/y^3 is an elementary form of 2nd kind with one double pole (at (t,0)) but zero residue.

If one grants that Riemann knew how to do this, as he said, then the foundation for his proof of the Riemann inequality is completely provided, and at least for plane curves, there is no need for Hilbert's analytic foundations to bolster Riemann's argument in the complex algebraic case. The 1865 paper of Roch, in which he completes Riemann's argument, rests solely on Green's theorem to compute Riemann's period matrix as a residue integral, hence is completely solid. 17 years later, Brill and Noether, using the same matrix computed by Roch, apparently showed that one can exploit the duality between divisors of form D and K-D to actually give the full proof using only the existence of the integrals of 1st kind. Since that paper was so influential, Roch's residue matrix (occurring in the middle of the second page of his paper) is now usually known as the Brill Noether matrix.