An explicit formula of the normalized Mumford form
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We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented by an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. By this formula, we have a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves.
12 pages. arXiv admin note: text overlap with arXiv:1411.3058
- Saga University Japan
Local ground fields in algebraic geometry, 14H10, 14H15, 14C40, 81T30, moduli space of algebraic curves, Ramanujan delta function, FOS: Physical sciences, Arithmetic ground fields for curves, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Mathematical Physics (math-ph), Families, moduli of curves (analytic), Mathematics - Algebraic Geometry, normalized Mumford form, FOS: Mathematics, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Theta functions and curves; Schottky problem, Families, moduli of curves (algebraic), Arithmetic varieties and schemes; Arakelov theory; heights, Polyakov string measure, Algebraic Geometry (math.AG), Mathematical Physics
Local ground fields in algebraic geometry, 14H10, 14H15, 14C40, 81T30, moduli space of algebraic curves, Ramanujan delta function, FOS: Physical sciences, Arithmetic ground fields for curves, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Mathematical Physics (math-ph), Families, moduli of curves (analytic), Mathematics - Algebraic Geometry, normalized Mumford form, FOS: Mathematics, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Theta functions and curves; Schottky problem, Families, moduli of curves (algebraic), Arithmetic varieties and schemes; Arakelov theory; heights, Polyakov string measure, Algebraic Geometry (math.AG), Mathematical Physics
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