David P. Roberts

Density of cubic field discriminants

In this paper we give a conjectural refinement of the Davenport- Heilbronn theorem on the density of cubic field discriminants. We explain how this refinement is plausible theoretically and agrees very well with computa- tional data. Let an be the number of isomorphism classes of abelian cubic fields with discrim- inant n. Let bn be the number of isomorphism classes of non-abelian cubic fields with discriminant n. The numbers an are very well understood. The numbers bn have been the subject of extensive theoretical and computational study for at least sixty years, but are less well understood. The object of this note is to contribute to the study of these bn, by bringing together the theoretical and computational literature. For � ∈ {−,+}, define g�(x) =

A database of number fields

We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.

Septic fields with discriminant ±2 a 3 b

We classify septic number fields which are unramified outside of {∞, 2, 3} by a targeted Hunter search; there are exactly 10 such fields, all with Galois group S7. We also describe separate computations which strongly suggest that none of these fields come from specializing septic genus zero three-point covers.

Number fields ramified at one prime

For G a finite group and p a prime, a G-p field is a Galoisnumber field K with Gal(K/Q) ≅ G and disc(K) = ±pa for some a. Westudy the existence of G-p fields for fixed G and varying p.

Nonic 3-adic Fields

We compute all nonic extensions of Q3 and find that there are 795 of them up to isomorphism. We describe how to compute the associated Galois group of such a field, and also the slopes measuring wild ramification. We present summarizing tables and a sample application to number fields.

Timing Analysis of Targeted Hunter Searches

One can determine all primitive number fields of a given degree and discriminant with a finite search of potential defining polynomials. We develop an asymptotic formula for the number of polynomials which need to be inspected which reflects both archimedean and non-archimedean restrictions placed on the coefficients of a defining polynomial.

The tame-wild principle for discriminant relations for number fields

Consider tuples (K1;:::;Kr) of separable algebras over a com- mon local or global number eld F , with the Ki related to each other by specied resolvent constructions. Under the assumption that all ramication is tame, simple group-theoretic calculations give best possible divisibility re- lations among the discriminants of Ki=F. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramication.

Serre weights and wild ramification in two-dimensional Galois representations

A generalization of Serres Conjecture asserts that if

Mixed degree number field computations

F

Hurwitz monodromy and full number fields

is a totally real field, then certain characteristic