Arithmetische und Algebraische Geometrie

Cohomological Mackey functors provide a systematic way to associate modules to subgroups of a finite group and have proven useful in number theory. In this talk, we apply this formalism to S-class groups in cyclic extensions, leading to a property analogous to Hilbert's theorem 90 introduced by Quadrelli and Weigel. We first derive a class-field-theoretic criterion for this property and discuss some examples.

We then turn to Iwasawa theory, where cohomological Mackey functors have previously been used by Bley and Boltje to relate Iwasawa invariants of different number fields. In contrast, we study the Hilbert 90 property along ℤ_p-extensions. We show that, under suitable assumptions, the Hilbert 90 property in the first layer already implies that the characteristic ideal of the associated Iwasawa module is not divisible by T. This is closely connected to the Gross–Kuz'min conjecture.

If time permits, we discuss a heuristic suggesting that, for totally real number fields, the Hilbert 90 property holds for all but finitely many primes p.

Classical Iwasawa theory considers towers of number fields K ⊂ K₁ ⊂ ⋯ ⊂ K_n ⊂ ⋯ such that K_n/K is Galois with Galois group ℤ/pⁿℤ. Along these towers one considers for example the growth of p-class groups. More recently this theory was generalized to graphs. Instead of a tower of number fields we construct towers of graphs and answer the question how the number of spanning trees grows. Besides the growth formulas we will discuss further analogies between Iwasawa theory of graphs and Iwasawa theory of modular curves and number fields.

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