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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)
##### Keywords:
modular forms; algebraic and arithmetic geometry
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##### 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, Trait\'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
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