Stark's conjectures and Hilbert's twelfth problem.
Numerical verification of the Stark-Chinburg conjecture for some icosahedral representations.
On integers of the form
Regulators of rank one quadratic twists
We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.
A fast algorithm for polynomial factorization over
We present an algorithm that returns a proper factor of a polynomial over the -adic integers (if is reducible over ) or returns a power basis of the ring of integers of (if is irreducible over ). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.
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