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    "summary": "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.\n\nCuts 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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    "year": 1857,
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            "citation": "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",
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      "scope": {
        "description": "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.",
        "domain": "Abelian integrals and complex curves"
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      "sources": [
        {
          "citation": "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",
          "url": "https://www.maths.tcd.ie/pub/HistMath/People/Riemann/AbelFn/AbelFn.pdf"
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      "statement": "# Riemann: periods and theta functions on algebraic surfaces\n\nRiemann 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.\n\nCuts 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.\n\n## Scientific scope\n\nThis 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.\n\n## People and systems\n\nBernhard Riemann\n\n\n\n## Original sources\n\n- [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](https://www.maths.tcd.ie/pub/HistMath/People/Riemann/AbelFn/AbelFn.pdf)\n",
      "summary": "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.\n\nCuts 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.",
      "title": "Riemann: periods and theta functions on algebraic surfaces"
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    "title": "Riemann: periods and theta functions on algebraic surfaces",
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    "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.",
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    "references": [
      {
        "title": "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",
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