Brett A. Tangedal

The Brumer-Stark conjecture in some families of extensions of specified degree

As a starting point, an important link is established between Brumers conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if K/F is an abelian extension of relative degree 2p, p an odd prime, we prove the l-part of the Brumer-Stark conjecture for all odd primes l ≠ p with F belonging to a wide class of base fields. In the same setting, we study the 2-part and p-part of Brumer-Stark with no special restriction on F and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases.

Computing the Lead Term of an Abelian L-function

We describe the extension of the techniques implemented in [DSS] to the computation of provably accurate values for the lead term at s = 0 of abelian L-functions having higher order zeros, and provide some explicit examples. In particular we raise the question of applying the higher order extensions of the abelian Stark Conjecture to the explicit construction of an interesting field extension in a manner analogous to the applications here and in [DSS], [Ro] in the case of zeros of rank one.

Stark’s conjecture over complex cubic number fields

jjSystematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Starks 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). Starks 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.

Numerical Verification of the Brumer-Stark Conjecture

The construction of group ring elements that annihilate the ideal class groups of totally complex abelian extensions of ℚ is classical and goes back to work of Kummer and Stickelberger. A generalization to totally complex abelian extensions of totally real number fields was formulated by Brumer. Brumer’s formulation fits into a more general framework known as the Brumer-Stark conjecture. We will verify this conjecture for a large number of examples belonging to an extended class of situations where the general status of the conjecture is still unknown.

Corrigendum to “The Brumer-Stark conjecture in some families of extensions of specified degree”

Barry Smith has found an error in the statement and proof of Lemma 2.5 in our paper [GRT] (Math. Comp. 73 (2004), 297-315). This Lemma concerns a cyclic Galois extension K/E of CM fields of odd prime degree p. Towards the end of the proof, it is claimed that every root of unity in E is a norm from K. Our reasoning for this has a gap (the local part of the argument does not work at places where the quadratic extension E/E is split, where F = E is the maximal totally real subfield of E), and the statement can indeed fail, as confirmed by a concrete example calculated by Barry Smith. We will first state a corrected version of the Lemma, and then we will explain how the proof of Proposition 2.2 has to be adapted. Fortunately the statement of this Proposition (please refer to [GRT]) need not be changed at all, and therefore the rest of the paper is unaffected. (The only other place where Lemma 2.5 is used is in the proof of Proposition 2.1, but there ζp is not in K, so the relevant case of the Lemma is case (i) below, where the formula remains the same.)

Computing annihilators of class groups from derivatives of

We computationally verify that certain group ring elements obtained from the first derivatives of abelian L-functions at the origin annihilate ideal class groups. In our test cases, these ideal class groups are connected with cyclic extensions of degree 6 over real quadratic fields.

L

Classical Gauss sums are Lagrange resolvents formed from the Gaussian periods lying in a cyclic extension K over ℚ of prime conductor. Elliptic Gauss sums and elliptic resolvents (which are particular instances of Lagrange resolvents) play an important role in the theory of abelian extensions of imaginary quadratic fields. Motivated by the close relationship between the Stark units and Gaussian periods in a cyclic extension K ⊂ ℚ(ζ p ) and the analogies between Stark units over totally real fields and elliptic units over imaginary quadratic fields, we consider for the first time Lagrange resolvents constructed from Stark units over totally real fields and study the differences and similarities they share with classical Gauss sums.