Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 060, 11 pages      arXiv:1811.10476      https://doi.org/10.3842/SIGMA.2024.060

Some Differential Equations for the Riemann $\theta$-Function on Jacobians

Robert Wilms
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen-Normandie, BP 5186, 14032 Caen Cedex, France

Received March 08, 2024, in final form June 26, 2024; Published online July 02, 2024

Abstract
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.

Key words: $\theta$-functions; Riemann surfaces; Jacobians; differential equations.

pdf (354 kb)   tex (14 kb)  

References

1. Arakelov S.J., An intersection theory for divisors on an arithmetic surface, Math. USSR Izv. 8 (1974), 1167-1180.
2. de Jong R., Gauss map on the theta divisor and Green's functions, in Modular Forms on Schiermonnikoog, Cambridge University Press, Cambridge, 2008, 67-78, arXiv:0705.0098.
3. de Jong R., Theta functions on the theta divisor, Rocky Mountain J. Math. 40 (2010), 155-176, arXiv:math.AG/0611810.
4. Faltings G., Calculus on arithmetic surfaces, Ann. of Math. 119 (1984), 387-424.
5. Fay J.D., Theta functions on Riemann surfaces, Lect. Notes Math., Vol. 352, Springer, Berlin, 1973.
6. Guàrdia J., Analytic invariants in Arakelov theory for curves, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 41-46.
7. Mumford D., Tata lectures on theta. I, Prog. Math., Vol. 28, Birkhäuser, Boston, MA, 1983.
8. Mumford D., Tata lectures on theta. II. Jacobian theta functions and differential equations, Prog. Math., Vol. 43, Birkhäuser, Boston, MA, 1984.
9. Wilms R., New explicit formulas for Faltings' delta-invariant, Invent. Math. 209 (2017), 481-539, arXiv:1605.00847.