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Nigel Boston

Publications and source records attributed to Nigel Boston.

24 records · Page 2Linked to original sources
arXiv2000

Explicit computation of Galois p-groups unramified at p

In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois groups have been regarded as amongst the most mysterious objects in number theory. Very little has hitherto been discovered regarding them despite their importance in studying p-adic Galois representations unramified at p. The conjectures of Fontaine-Mazur say that they should have no p-adic analytic quotients (equivalently, the images of the Galois representations should always be finite), and there are generalizations due to the first author, suggesting that they should instead have `large' actions on certain trees. The idea is to modify the p-group generation algorithm so that as it goes along, it uses number-theoretical information to eliminate groups that cannot arise as suitable quotients of the Galois group under investigation. In the best cases we obtain a short list of candidates for the Galois group. This leads to various conjectures.

math.NT↗
arXiv2000

Tree representations of Galois groups

Much work has gone into matrix representations of Galois groups, but there is a whole new class of naturally occurring representations that have as yet gone almost unnoticed. In fact, it is well-known in various areas of mathematics that the main sources of totally disconnected groups are matrix groups over local fields AND automorphism groups of locally finite trees. It is perhaps surprising then that representations of Galois groups into the latter have been almost ignored, while at the same time Galois representations into the former have been enormously effective in resolving long-standing problems in number theory. These ``tree'' representations are important as regards topics such as the unramified Fontaine-Mazur conjecture. This conjecture states that any p-adic representation of the Galois group of an extension unramified at p (and ramified at only finitely many primes) should have finite image. In other words, p-adic representations say little about such Galois groups. This paper proposes the conjecture that these Galois groups should, on the other hand, have representations with large image (measured by Hausdorff dimension) in the automorphism group of a rooted tree. Thus, they might allow us to investigate the structure of the Galois group of infinite pro-p extensions (such as Hilbert p-class towers), something unapproachable by standard p-adic representation methods.

math.NT↗
arXiv2000

Bounding minimum distances of cyclic codes using algebraic geometry

There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH, Hartmann-Tzeng, Roos, and shift bounds and generalizations of these. In this paper we define certain projective varieties V(T,t) whose properties determine whether, if T is in the defining set, the code has minimum distance exceeding t. Thus our attention shifts to the study of these varieties. By investigating them using class field theory and arithmetical geometry, we will prove various new bounds. It is interesting, however, to note that there are cases that existing methods handle, that our methods do not, and vice versa. We end with a number of conjectures.

math.NT↗
arXiv1998

Some cases of the Fontaine-Mazur conjecture, II

This is an updated and extended version of ANT-0024. We prove more special cases of the Fontaine-Mazur conjecture regarding p-adic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it.

math.NT↗
arXiv1998

Counterexamples to a conjecture of Lemmermeyer

We produce infinitely many finite 2-groups that do not embed with index 2 in any group generated by involutions. This disproves a conjecture of Lemmermeyer and restricts the possible Galois groups of unramified 2-extensions, Galois over the rationals, of quadratic number fields.

math.NT↗