Section: New Results

Number and function fields

Participants : Athanasios Angelakis, Karim Belabas, Pieter Rozenhart.

In joint work with R. Scheidler and M. Jacobson, P. Rozenhart has generalized Belabas's algorithm for tabulating cubic number fields to cubic function fields [17] . This generalization required function field analogues of the Davenport-Heilbronn Theorem and of the reduction theory of binary cubic and quadratic forms. As an additional application, they have modified the tabulation algorithm to compute 3-ranks of quadratic function fields by way of a generalisation of a theorem due to Hasse. The algorithm, whose complexity is quasi-linear in the number of reduced binary cubic forms up to some upper bound $X$, works very well in practice. A follow-up article [35] describes how to use these results to compute 3-ranks of quadratic function fields, in particular yielding examples of unusually high 3-rank.

In 1976, Onabe discovered that, in contrast to the Neukirch–Uchida results that were proved around the same time, a number field $K$ is not completely characterised by its absolute abelian Galois group $A_K$. The first examples of non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a classification by Kubota of idele class character groups in terms of their infinite families of Ulm invariants, and did not yield a description of $A_K$. In [21] , A. Angelakis and P. Stevenhagen provide a direct “computation” of the profinite group $A_K$ for imaginary-quadratic $K$, and use it to obtain many different $K$ that all have the same minimal absolute abelian Galois group.

On March 29–April 2, 2010, a meeting was organized by J.-M. Couveignes, D. Bertrand, Ph. Boalch and P. Debes, at the Luminy CIRM (France) on geometric and differential Galois theories, witnessing the close ties these theories have woven in recent years. The volume [18] collects the proceedings of this meeting. The articles gathered in this volume cover the following topics: moduli spaces of connections, differential equations and coverings in finite characteristic, liftings, monodromy groups in their various guises (tempered fundamental group, motivic groups, generalised difference Galois groups), and arithmetic applications.

Using Galois theory of extension rings, J.-M. Couveignes, R. Lercier and T. Ezome have proposed a new pseudo-primality test in [13] . For every positive integer $k\le logn$, this test achieves the security of $k$ Miller-Rabin tests at the cost of $k^{1/2}+o\left(1\right)$ Miller-Rabin tests. The implementation in Magma shows that this test is competitive for primes with a few thousands digits.