×

zbMATH — the first resource for mathematics

Klein’s formulas and arithmetic of Teichmüller modular forms. (English) Zbl 1454.11091
Summary: We apply the arithmetic theory of Teichmüller modular forms for calculating constants in relations, which are connected with Klein’s (amazing) formulas, between certain invariants of canonical curves of genus \( g = 3, 4\).
MSC:
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
14H10 Families, moduli of curves (algebraic)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brinkmann, B.; Gerritzen, L., The lowest term of the Schottky modular form, Math. Ann., 292, 2, 329-335, (1992) · Zbl 0747.11022
[2] Gerritzen, L., Equations defining the periods of totally degenerate curves, Israel J. Math., 77, 1-2, 187-210, (1992) · Zbl 0783.14017
[3] Ichikawa, Takashi, Theta constants and Teichm\"uller modular forms, J. Number Theory, 61, 2, 409-419, (1996) · Zbl 0920.11029
[4] Ichikawa, Takashi, Generalized Tate curve and integral Teichm\"uller modular forms, Amer. J. Math., 122, 6, 1139-1174, (2000) · Zbl 1062.14501
[5] Igusa, Jun-ichi, Schottky’s invariant and quadratic forms. E. B. Christoffel, Aachen/Monschau, 1979, 352-362, (1981), Birkh\"auser, Basel-Boston, Mass.
[6] Igusa, Jun-ichi, Problems on theta functions. Theta functions—Bowdoin 1987, Part 2, Brunswick, ME, 1987, Proc. Sympos. Pure Math. 49, 101-110, (1989), Amer. Math. Soc., Providence, RI
[7] Klein, Felix, Zur Theorie der Abel’schen Functionen, Math. Ann., 36, 1, 1-83, (1890) · JFM 22.0498.01
[8] Lachaud, Gilles; Ritzenthaler, Christophe, On some questions of Serre on abelian threefolds. Algebraic geometry and its applications, Ser. Number Theory Appl. 5, 88-115, (2008), World Sci. Publ., Hackensack, NJ · Zbl 1151.14321
[9] Lachaud, Gilles; Ritzenthaler, Christophe; Zykin, Alexey, Jacobians among abelian threefolds: a formula of Klein and a question of Serre, Math. Res. Lett., 17, 2, 323-333, (2010) · Zbl 1228.14028
[10] Matone, Marco; Volpato, Roberto, Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula, Proc. Amer. Math. Soc., 141, 8, 2575-2587, (2013) · Zbl 1277.14027
[11] Mumford, David, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math., 24, 239-272, (1972) · Zbl 0241.14020
[12] [12]S G. Salmon, <span class=”textit”>T</span>rait\'e de g\'eom\'etrie analytique \`a trois dimensions, Troisi\`eme partie, Ouvrage traduit de l’anglais sur la quatri\`eme \'edition, Paris, 1892.
[13] Serre, Jean-Pierre, Two letters to Jaap Top. Algebraic geometry and its applications, Ser. Number Theory Appl. 5, 84-87, (2008), World Sci. Publ., Hackensack, NJ · Zbl 1151.14322
[14] Tsuyumine, Shigeaki, Thetanullwerte on a moduli space of curves and hyperelliptic loci, Math. Z., 207, 4, 539-568, (1991) · Zbl 0752.14019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.