Riemann: periods and theta functions on algebraic surfaces

What changed

Cut-system period increments obey bilinear constraints; normalized periods have symmetry and a definite quadratic form compatible with convergent multivariable theta series and their transformations.

Scientific scope

Cut-system period increments obey bilinear constraints; normalized periods have symmetry and a definite quadratic form compatible with convergent multivariable theta series and their transformations. 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.

Evidence

Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.

Contributors

• Bernhard Riemann · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.

• Bernhard Riemann, Theorie der Abel'schen Functionen, Journal für die reine und angewandte Mathematik 54 (1857), 115–155; Wilkins German transcription bundled with three preceding papers pp.101–114

Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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