Klein’s formulas and arithmetic of Teichmüller modular forms
HTML articles powered by AMS MathViewer
- by Takashi Ichikawa
- Proc. Amer. Math. Soc. 146 (2018), 5105-5112
- DOI: https://doi.org/10.1090/proc/14244
- Published electronically: September 17, 2018
- PDF | Request permission
Abstract:
We apply the arithmetic theory of Teichmüller modular forms to calculating constants in relations, which are connected with Klein’s (amazing) formulas, between certain invariants of canonical curves of genus $g = 3, 4$.References
- B. Brinkmann and L. Gerritzen, The lowest term of the Schottky modular form, Math. Ann. 292 (1992), no. 2, 329–335. MR 1149038, DOI 10.1007/BF01444624
- L. Gerritzen, Equations defining the periods of totally degenerate curves, Israel J. Math. 77 (1992), no. 1-2, 187–210. MR 1194792, DOI 10.1007/BF02808017
- Takashi Ichikawa, Theta constants and Teichmüller modular forms, J. Number Theory 61 (1996), no. 2, 409–419. MR 1423061, DOI 10.1006/jnth.1996.0156
- Takashi Ichikawa, Generalized Tate curve and integral Teichmüller modular forms, Amer. J. Math. 122 (2000), no. 6, 1139–1174. MR 1797659, DOI 10.1353/ajm.2000.0046
- Jun-ichi Igusa, Schottky’s invariant and quadratic forms, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 352–362. MR 661078
- Jun-ichi Igusa, Problems on theta functions, Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 101–110. MR 1013168, DOI 10.1090/pspum/049.2/1013168
- Felix Klein, Zur Theorie der Abel’schen Functionen, Math. Ann. 36 (1890), no. 1, 1–83 (German). MR 1510611, DOI 10.1007/BF01199432
- Gilles Lachaud and Christophe Ritzenthaler, On some questions of Serre on abelian threefolds, Algebraic geometry and its applications, Ser. Number Theory Appl., vol. 5, World Sci. Publ., Hackensack, NJ, 2008, pp. 88–115. MR 2484050, DOI 10.1142/9789812793430_{0}005
- Gilles Lachaud, Christophe Ritzenthaler, and Alexey Zykin, Jacobians among abelian threefolds: a formula of Klein and a question of Serre, Math. Res. Lett. 17 (2010), no. 2, 323–333. MR 2644379, DOI 10.4310/MRL.2010.v17.n2.a11
- Marco Matone and Roberto Volpato, Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2575–2587. MR 3056547, DOI 10.1090/S0002-9939-2012-11526-6
- David Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272. MR 352106
- G. Salmon, Traité de géométrie analytique à trois dimensions, Troisième partie, Ouvrage traduit de l’anglais sur la quatrième édition, Paris, 1892.
- Jean-Pierre Serre, Two letters to Jaap Top, Algebraic geometry and its applications, Ser. Number Theory Appl., vol. 5, World Sci. Publ., Hackensack, NJ, 2008, pp. 84–87. MR 2484049, DOI 10.1142/9789812793430_{0}004
- Shigeaki Tsuyumine, Thetanullwerte on a moduli space of curves and hyperelliptic loci, Math. Z. 207 (1991), no. 4, 539–568. MR 1119956, DOI 10.1007/BF02571407
Bibliographic Information
- Takashi Ichikawa
- Affiliation: Department of Mathematics, Graduate School of Science and Engineering, Saga University, Saga 840-8502, Japan
- MR Author ID: 253584
- Email: ichikawn@cc.saga-u.ac.jp
- Received by editor(s): January 22, 2018
- Received by editor(s) in revised form: April 18, 2018
- Published electronically: September 17, 2018
- Communicated by: Benjamin Brubaker
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5105-5112
- MSC (2010): Primary 11F46, 14H10
- DOI: https://doi.org/10.1090/proc/14244
- MathSciNet review: 3866850