Abstract
We describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.
Highlights
This survey article intends to describe developments that originate in classical algebraic number theory and have established intimate connections with modern arithmetic, involving elaborate concepts and deep far-reaching conjectures
We describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group
Keywords Class groups · Fitting ideals · Cohomology · Iwasawa theory
Summary
This survey article intends to describe developments that originate in classical algebraic number theory and have established intimate connections with modern arithmetic, involving elaborate concepts (cohomology, derived categories) and deep far-reaching conjectures (equivariant Tamagawa number conjectures, main conjectures, . . . ). Even though the statement of goal (4) is until now the haziest, it is the most realistic and the most promising This is what we will focus on; the invariants to be studied are the so-called Fitting ideals, introduced by Hans Fitting around 1936 (by the way, his main field was group theory, not module theory or number theory). At the end of this article we discuss a result which might look weak at first glance It establishes, without appealing to unproved conjectures, that a certain generalized Stickelberger element lies in the Fitting ideal of the Pontryagin dual of the class group. Without appealing to unproved conjectures, that a certain generalized Stickelberger element lies in the Fitting ideal of the Pontryagin dual of the class group This result due to Dasgupta and Kakde is extremely strong, since it gives an almost completely general proof of the Brumer conjecture. The author would like to thank Alessandro Cobbe and Sören Kleine for a lot of extremely helpful comments
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