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We construct liftings of reduction maps from complex multiplication (CM) points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras. [ABSTRACT FROM PUBLISHER]

In this Letter we propose complex modular multiplication of two N-bit negabinary complex numbers by adopting digital partitioning and weighting-shifting techniques. The module (-2)N - 1 is introduced for its simple arithmetic. Both complex numbers are treated as two N-bit negabinary real numbers. All four real numbers are decomposed into several short bytes; a corresponding complex product modular (-2)N - 1 is achieved by cyclic convolution. The subtraction in complex multiplication is converted to addition by weighting shifting. The algorithms can be implemented in a fully parallel manner, especially for optics. Both pre- and postprocessing are simple. The proof-of-principle experiment is performed by an incoherent optical correlator, and results are given. [ABSTRACT FROM AUTHOR]

We show that any abelian surface with multiplication by the quaternion ℚ-algebra of discriminant 6, with field of moduli ℚ and which is a Jacobian in characteristic 2 and 3, has infinitely many primes of superspecial reduction. This is done by examining complex multiplication points in characteristic 0 and p and the values of a certain j-function on the associated moduli space at these points. [ABSTRACT FROM AUTHOR]