Jonathan W. Sands

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.

Computation of Iwasawa Lambda invariants for imaginary quadratic fields

A method for computing the Iwasawa lambda invariants of an imaginary quadratic field is developed and used to construct a table of these invariants for discriminants up to 1,000 and primes up to 20,000.

An algorithm for testing Leopoldt's conjecture

Abstract Based on an elementary version of Leopoldts conjecture due to Iwasawa and Sands we develop an algorithm for testing this conjecture in an arbitrary algebraic number field for any prime p . Using this algorithm we are able to prove Leopoldts conjecture for several pure fields of degree 5 and 7. We also discuss relations with class numbers.

VALUES AT s = -1 OF L-FUNCTIONS FOR MULTI-QUADRATIC EXTENSIONS OF NUMBER FIELDS, AND THE FITTING IDEAL OF THE TAME KERNEL

Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z_2-extension of F, and describe a class of biquadratic extensions of F=Q that satisfy this condition.

Numerical verification of the Stark-Chinburg conjecture for some icosahedral representations

In this paper, we give 14 examples of icosahedral representations for which we have numerically verified the Stark-Chinburg conjecture.

On the l -adic Iwasawa l-invariant in a p -extension

For distinct primes I and p, the Iwasawa invariant λ − l stabilizes in the cyclotomic Z p -tower over a complex abelian base field. We calculate this stable invariant for p = 3 and various I and K. Our motivation was to search for a formula of Riemann-Hurwitz type for λ − l that might hold in a p-extension. From our numerical results, it is clear that no formula of such a simple kind can hold. In the course of our calculations, we develop a method of computing λ − l for an arbitrary complex abelian field and, for p = 3, we make effective Washingtons theorem on the stabilization of the l-part of the class group in the cyclotomic Z p -extension. A new proof of this theorem is given in the appendix.

Leopoldt’s conjecture in parameterized families

For each fixed prime p

Numerical evidence for higher order Stark-type conjectures

4 5, we prove Leopoldts conjecture in two infinite families of fields of degree five whose normal closure has Galois group over the rationals isomorphic to S5. The units of these fields were determined by Maus [4]; we develop and apply a simple reformulation of Leopoldts conjecture to obtain the result. We also observe that Leopoldts conjecture in one field can imply the same in a second field related by congruence conditions.

Computing annihilators of class groups from derivatives of

We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Starks conjecture over

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