Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

     

On a new characterization of Demuskin groups

Inventiones mathematicae -     

Computing Stark units for totally real cubic fields

A method for computing provably accurate values of partial zeta functions is used to numerically confirm the rank one abelian Stark Conjecture for some totally real cubic fields of discriminant less than 50000. The results of these computations are used to provide explicit Hilbert class fields and some ray class fields for the cubic extensions.

     

An extension of Iwasawa's theorem on finitely generated modules over power series rings

This paper describes a structure theorem for finitely generated modules over power series rings O[[T]], where O is a maximal order in a semisimple Q_p-algebra of finite dimension over Q_p, extending Iwasawa's structure theorem (the case O=?p). A particular case of such power series ring is the ring Λ[Δ], where Λ is the power series ring ?_p?T? and Δ is a finite group of order prime to p. Several applications are given, including a new proof of a result of Iwasawa important for the relationship between Hecke characters and certain Galois representations for CM fields.     

Cm Elliptic Curves and Bicolorings of Steiner Triple Systems

It is shown that a necessary condition for the existence ofa bicolored Steiner triple system of order n is that n can bewritten in the form A²+3B² for integers A and B. In the casewhen n=q is either a prime congruent to 1 mod 3, or the squareof a prime congruent to 2 mod 3, it is shown that the numbersof colored vertices in the triple system would be unique, andare given by the number of points on specific twists of theCM elliptic curve y²=x³–1 over the finite field F_q. 2000Mathematics Subject Classification 05B07, 11G20, 14G15 (primary);11G15, 14K22 (secondary).