We construct and study two series of curves whose Jacobians admit complex multiplication. The curves arise as quotients of Galois coverings of the projective line with Galois group metacyclic groups Gq,3 of order 3q with q ≡ 1 mod 3 an odd prime, and Gm of order 2 . The complex multiplications arise as quotients of double coset algebras of the Galois groups of these coverings. We work out the C...

In accordance with Fermat's Variation Principle, a ray path connecting two arbitrary points in a scene via multiple reflectors is given by a non-linear system. If we fix one of the two points and let the other change, the system can be considered as a function relating the reflection points along the path to the varying point. In this paper, we present the concept of path Jacobians for such a r...

1. Complex multiplication. We say an abelian variety A of dimension g over a field K admits sufficiently many Complex Multiplications (smCM) if D = End(A) contains a commutative semi-simple algebra Λ ⊂ D of rank dimQ(Λ) = [Λ : Q] = 2g. In this case we say A is a CM abelian variety. A point z ∈ Ag(k) is called a CM point if z is the moduli point of a polarized abelian variety z = [(A,μ)] such th...

Carl Gustav Jacob Jacobi (1804–1851), one of the nineteenth century Germany’s most accomplished scientists, developed the theory of determinants and transformations into a powerful tool for evaluating multiple integrals and solving differential equations. Since then, the Jacobian (determinant) has played a critical role in multidimensional analysis and related fields, including nonlinear elasti...

For any Lagrangean Kähler submanifold M ⊂ T Cn, there exists a canonical hyper Kähler metric on T ∗M . A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, a class of pseudo h...

The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Mani...

The flux-corrected transport (FCT) methodology is generalized to implicit finite element schemes and applied to the Euler equations of gas dynamics. For scalar equations, a local extremum diminishing scheme is constructed by adding artificial diffusion so as to eliminate negative off-diagonal entries from the high-order transport operator. To obtain a nonoscillatory low-order method in the case...

Fix a squarefree integer N , divisible by an even number of primes, and let Γ be a congruence subgroup of level M , where M is prime to N . For each D dividing N and divisible by an even number of primes, the Shimura curve X(Γ0(N/D)∩Γ ) associated to the indefinite quaternion algebra of discriminant D and Γ0(N/D) ∩ Γ -level structure is well defined, and we can consider its Jacobian J(Γ0(N/D) ∩...