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Some Differential Equations for the Riemann $\theta$-Function on Jacobians
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in Arakelov theory of Riemann surfaces.
Keywords: $\theta$-functions, Riemann surfaces, Jacobians, differential equations. 
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     author = {Robert Wilms},
     title = {Some {Differential} {Equations} for the {Riemann} $\theta${-Function} on {Jacobians}},
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Robert Wilms. Some Differential Equations for the Riemann $\theta$-Function on Jacobians. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a59/

[1] Arakelov S.J., “An intersection theory for divisors on an arithmetic surface”, Math. USSR Izv., 8 (1974), 1167–1180 | DOI | MR

[2] de Jong R., “Gauss map on the theta divisor and Green's functions”, Modular Forms on Schiermonnikoog, Cambridge University Press, Cambridge, 2008, 67–78, arXiv: 0705.0098 | DOI | MR | Zbl

[3] de Jong R., “Theta functions on the theta divisor”, Rocky Mountain J. Math., 40 (2010), 155–176, arXiv: math.AG/0611810 | DOI | MR | Zbl

[4] Faltings G., “Calculus on arithmetic surfaces”, Ann. of Math., 119 (1984), 387–424 | DOI | MR | Zbl

[5] Fay J.D., Theta functions on Riemann surfaces, Lect. Notes Math., 352, Springer, Berlin, 1973 | DOI | MR | Zbl

[6] Guàrdia J., “Analytic invariants in Arakelov theory for curves”, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 41–46 | DOI | MR | Zbl

[7] Mumford D., Tata lectures on theta, v. I, Prog. Math., 28, Birkhäuser, Boston, MA, 1983 | DOI | MR | Zbl

[8] Mumford D., Tata lectures on theta, v. II, Prog. Math., 43, Jacobian theta functions and differential equations, Birkhäuser, Boston, MA, 1984 | DOI | MR | Zbl

[9] Wilms R., “New explicit formulas for Faltings' delta-invariant”, Invent. Math., 209 (2017), 481–539, arXiv: 1605.00847 | DOI | MR | Zbl