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Inverse values of the modular \(j\)-invariant and homotopy Lie theory. (English) Zbl 1431.11055

Summary: The goal of this article is to give a simple arithmetic application of the enhanced homotopy (Lie) theory for algebraic varieties developed by the second and third authors. Namely, we compute an inverse value of the modular \( j\)-invariant by using a deformation theory for period matrices of elliptic curves based on homotopy Lie theory. Another key ingredient in our approach is J. A. Carlson and P. A. Griffiths’ explicit computation regarding infinitesimal variations of Hodge structures [in: Journées de géométrie algébrique d’Angers (juillet 1979). Rockville, MD: Sijthoff & Noordhoff, 51–76 (1980; Zbl 0479.14007)].

MSC:

11F03 Modular and automorphic functions
11Y99 Computational number theory
14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations

Citations:

Zbl 0479.14007
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Full Text: DOI

References:

[1] Berndt, Bruce C.; Chan, Heng Huat, Ramanujan and the modular \(j\)-invariant, Canad. Math. Bull., 42, 4, 427-440 (1999) · Zbl 0990.11022
[2] Carlson, James A.; Griffiths, Phillip A., Infinitesimal variations of Hodge structure and the global Torelli problem. Journ\'ees de G\'eometrie Alg\'ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, 51-76 (1980), Sijthoff &Noordhoff, Alphen aan den Rijn-Germantown, Md. · Zbl 0479.14007
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