On p-adic regulators of number fields

In this thesis, we study the p-adic regulator of number fields, which is an arithmetic invariant introduced by Leopoldt. We aim to derive conjectural results regarding the distribution of p-adic regulators for families of number fields, which until now has been exclusively done for number fields of odd prime degree. Furthermore, we investigate the existence of formulas for p-adic regulators of number fields in terms of subfields, which supplements previous research for other arithmetic invariants.
We obtain conjectures for families of quadratic number fields by computing images of logarithm maps. More generally, for a family of number fields, we establish a principle that states that the distribution of the valuations of p-adic regulators equals the distribution of specific random modules. We provide principles for normal (extensions of) number fields and non-normal number fields. One of the main results, and foundation for the principles, grants for every prime number p and for every number field the existence of a finite module that has an order linked to the p-adic regulator. Additionally, we prove formulas for expected distributions of p-adic regulators for a range of normal number fields, e.g., with cyclic Galois group. The formulas are derived by utilizing an approach that involves random matrices and group determinants.
Moreover, we obtain formulas for the valuation of the p-adic regulator for normal number fields in terms of subfields and a generalization for normal extensions of number fields.
To show these formulas, we use the concepts of scalar norm relations, or rather Brauer relations, and cohomological Mackey functors. Additionally, we derive a formula for p-adic L-functions of a number field and its subfields.
Lastly, we discuss open questions concerning distributions of p-adic regulators and provide computational data that supports the formulated principles for a range of examples.
In conclusion, the principles extend previous conjectures on distributions of p-adic regulators, and the proven formulas for the valuation of the p-adic regulator for normal number fields allow more efficient computations by exploiting subfields.