What is recorded here?
Riemann organized Abelian integrals by their changes around cuts on a branched algebraic surface. Bilinear relations constrain their periods; a normalized period matrix has symmetry and a definite quadratic property that fits the convergence and transformation structure of multivariable theta series. Cuts allow an integral to be followed as a single-valued function while recording constant jumps when the cuts are crossed. Pairing the boundary sides yields bilinear restrictions on those jumps. The normalized periods then furnish the coefficients of a multivariable theta series with controlled convergence and quasi-periodic transformations. This is a new integrals/periods organization beyond the 1851 introduction of branched surfaces; Abel, Jacobi and Weierstrass remain predecessors and adjacent contributors.
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Scientific scope
This target concerns independent everywhere-finite Abelian integrals in the paper's compact algebraic-surface setting and their period/theta organization. Historical normalization uses a negative real quadratic form; modern normalizations change the convention. It is not the full Jacobi inversion proof, every moduli-dimension claim, the later Riemann–Roch equality or a certification of the historical Dirichlet-principle existence argument.
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People and systems
Bernhard Riemann
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English catalogue narrative
Riemann: periods and theta functions on algebraic surfaces
Riemann organized Abelian integrals by their changes around cuts on a branched algebraic surface. Bilinear relations constrain their periods; a normalized period matrix has symmetry and a definite quadratic property that fits the convergence and transformation structure of multivariable theta series.
Cuts allow an integral to be followed as a single-valued function while recording constant jumps when the cuts are crossed. Pairing the boundary sides yields bilinear restrictions on those jumps. The normalized periods then furnish the coefficients of a multivariable theta series with controlled convergence and quasi-periodic transformations. This is a new integrals/periods organization beyond the 1851 introduction of branched surfaces; Abel, Jacobi and Weierstrass remain predecessors and adjacent contributors.
Scientific scope
This target concerns independent everywhere-finite Abelian integrals in the paper's compact algebraic-surface setting and their period/theta organization. Historical normalization uses a negative real quadratic form; modern normalizations change the convention. It is not the full Jacobi inversion proof, every moduli-dimension claim, the later Riemann–Roch equality or a certification of the historical Dirichlet-principle existence argument.
People and systems
Bernhard Riemann