Abstract
This talk revolves around two focuses: complex multiplications (for elliptic curves and Abelian varieties) in connection with algebraic period relations, and (diophantine) approximations to numbers related to these periods. Our starting point is Ramanujan’s works [1], [2] on approximations to π via the theory of modular and hypergeometric functions. We describe in chapter 1 Ramanujan’s original quadratic period-quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of „ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series. These representations serve as a basis of our investigation of diophantine approximations to π and other related numbers. In Chapter 2 we look at period relations for arbitrary CM-varieties following Shimura and Deligne. Our main interest lies with modular (Shimura) curves arising from arithmetic Fuchsian groups acting on H. From these we choose arithmetic triangular groups, where period relations can be expressed in the form of hyper-geometric function identities. Particular attention is devoted to two (commensurable) triangle groups, (0,3;2,6,6) and (0,3;2,4,6), arising from the quaternion algebra over Φ with the discriminant D = 2.3.
This work was supported in part by the N.S.F., U.S. Air Force and O.C.R.E.A.E. program.
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Chudnovsky, D.V., Chudnovsky, G.V. (1988). Approximations and complex multiplication according to Ramanujan. In: Pi: A Source Book. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2736-4_63
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