Alex Bartel, University of Cambridge
Some Invariants of Integral Galois Representations

Galois groups of number fields naturally act on many interesting structures that arise naturally in number theoretic contexts. The most obvious example is the ring of integers of a Galois extension of number fields. Another very interesting example is the group of units therein. These examples have in common that they are finitely generated abelian groups, so their free parts can be regarded as finite dimensional lattices. If one wants to study the structure of the Galois action on these lattices, one is naturally led to the theory of integral representations of finite groups. This theory is vastly more difficult than that of complex representations and we have only very few general techniques available to us. In this series of talks I will describe a new such technique, which is particularly useful in the number theoretic context.

There are no prerequisites beyond an undergraduate course on algebraic number theory and on representation theory. This series of talks ties in well with the short course by Tim Dokchitser two weeks later. The topics touched upon will include:

• the regulator of a number field, the zeta function of a number field and the analytic class number formula
• arithmetic of elliptic curves over number fields, the regulator of an elliptic curve, the Birch and Swinnerton-Dyer conjecture
• the Burnside ring and the representation ring of a finite group, the kernel of the map from the one to the other
• Artin formalism for L-functions, regulator constants of Tim and Vladimir Dokchitser
• structure of the unit group in the ring of integers of a Galois extension; structure of the Mordell-Weil group of an elliptic curve as a Galois module

Everybody is welcome: no registration for participation is needed. The lectures will take place at the Centre de Recerca Matemātica. For more information about the schedule, contact with the organizers.

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