Supercongruences and Complex Multiplication

We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of \lambda, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Rodriguez-Villegas for the \lambda=1 case and confirm some other supercongruence conjectures at special values of \lambda.Comment: 19 page

Complex Multiplication Tests for Elliptic Curves

We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminant of the endomorphism ring of the curve.Comment: 13 pages, 2 tables, 1 appendi

Difficulties in Complex Multiplication and Exponentiation

During my study of the iteration of functions of the form $f(z)=z^{\alpha}+c$, where z,c \in \mathbbC, and $\alpha$ is a rational non-integer larger than 2 (\cite{s1}), I encountered a fundamental difficulty in the exponentiation of a complex number. This paper will explore this difficulty and the problems encountered in trying to resolve it using a Riemann surface which is the direct generalization of the polar form of the complex plane. This paper will also answer two questions raised by Robert Corless in his \emph{E.C.C.A.D.} presentation \cite{co}: "Can a Riemann surface variable be coded? What will the operations be on it?" Unfortunately, the addition operation will be incompatible with the Riemann surface structure.Comment: 17 pages, 9 figures (.ps format

Some Mirror partners with Complex multiplication

In this note we provide examples of families of Calabi-Yau 3-manifolds over Shimura varieties, whose mirror families contain subfamilies over Shimura varieties. Therefore these original families and subfamilies on the mirror side contain dense sets of complex multiplication fibers. In view of the work of S. Gukov and C. Vafa this is of special interest in theoretical physics.Comment: 7 page

Rational Conformal Field Theories and Complex Multiplication

We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT's corresponding to T^2 target and identify the Cardy branes with geometric branes. The T^2's leading to RCFT's admit ``complex multiplication'' which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFT's on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.Comment: 39 pages, 6 figures, harvmac; references adde