Topological and Algebraic Cycles in Kuga-Shimura Varieties.
We compute the torsion group explicitly over quadratic fields and number fields of degree coprime to 6 for a family of elliptic curves of the form , where is an integer.
We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.
Let be a totally positive algebraic integer, with the difference between its trace and its degree at most 6. We describe an algorithm for finding all such , and display the resulting list of 1314 values of which the algorithm produces.
Consider a recurrence sequence of integers satisfying , where are fixed and a₀ ∈ -1,1. Assume that for all sufficiently large k. If there exists k₀∈ ℤ such that then for each negative integer -D there exist infinitely many rational primes q such that for some k ∈ ℕ and (-D/q) = -1.