EPSRC DTP PhD studentship: Realisable Galois Module Classes for the Square Root of the Inverse Different

University of Exeter - College of Engineering, Mathematics and Physical Sciences

Main Supervisor: Prof Nigel Byott (University of Exeter)

This project is at the interface of algebraic number theory and abstract algebra.

Let L/K be a finite Galois extension of number fields with Galois group G. Then the ring of integers OL is a module for the integral group ring Z[G], and also over OK[G].  One may ask whether OL is free over either of these rings, or more generally, whether it is locally free (that is, free after passing to completions at any prime number p, respectively any prime ideal of OK. It turns out that OL is locally free over Z[G] or OK[G] if and only if L/K is at most tamely ramified, and that whether or not it is free over Z[G] is related to analytic data (the constants in the functional equations of the Artin L-functions) attached to L/K.

The failure of a locally free OK[G]-module to be free is detected by its class in a finite abelian group Cl(OK[G]) of OK[G]. The classes in Cl(OK[G]) given by the rings of integers OL as L runs through all tamely ramified Galois extensions of K with Galois group G, form the subset R(OK[G]) of realisable Galois module classes. In a series of papers in the 1980’s, L. McCulloh characterised R(OK[G]) when G is abelian, in particular showing that it is a subgroup of Cl(OK[G]) in that case. Several more recent papers of N. Byott, B. Sodaigui and others have obtained similar results for certain nonabelian groups G.

If L/K is a Galois extension of odd degree with Galois group G, there is a fractional ideal AL/K of OL whose square is the inverse different of L/K. This ideal is special as it is the only ideal which is self-dual with respect to the trace map. When L/K is at most tamely ramified (or, more generally, at most weakly ramified), AL/K is locally free over OK[G]. One can therefore consider the subset of classes in Cl(OK[G]) given by the ideals AL/K as L runs through tamely (or weakly) ramified Galois extensions of K with Galois group G. A recent paper of C. Tsang (Journal of Number Theory, 2016) does so when G is abelian, showing that the set of classes obtained is a subgroup of Cl(OK[G]). The aim of this PhD project is to obtain a more explicit description of this subset in the abelian case, and to obtain similar results for some nonabelian Galois groups G.

The student's role would be to prove theorems on the Galois module structure of the square root of the inverse different, for various classes of Galois group. This would be done under the guidance and direction of the lead supervisor. The student would initially need to familiarise themselves thoroughly with the background to the project, including the work of Tsang, and various papers of Byott and Sodaigui on realisable classes of rings of integers. They would then seek to adapt the techniques of these papers to obtain new results in various cases, such as for elementary abelian, dihedral and tetrahedral Galois groups. These results may involve restrictions on the ground field, for example in relation to the roots of unity it contains.

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South West England