## Number Theory Seminar

According to a long-established analogy, the fundamental group of a number field $K$ is defined to be the Galois group $\text{Gal}(L/K)$ where $L$ is the maximal unramified Galois extension of $K$. Its maximal abelian quotient, namely $\text{Gal}(H/K)$ where $H$ is the Hilbert Class Field of $K$, is in this optic the first homology group of $K$. By Artin Reciprocity, this group is canonically isomorphic to the ideal class group of $K$, a finite abelian group, whose properties are even now poorly understood. Iwasawa pioneered the study of how class groups vary in towers of number fields with $p$-adic analytic Galois group. In this talk, I will report on joint work with Christian Maire in which we study how the exponents of class groups vary in towers contained in $L/K$ where $L$ is the universal cover of $K$. Note that according to a conjecture of Fontaine and Mazur, such towers do not have $p$-adic analytic Galois group. Our main result is that in contrast to the $p$-adic analytic case, the average exponent of $p$-class groups (a concept we define) in unramified $p$-towers can be bounded. The proof uses a combination of group theory and a key arithmetic result of Tsfasman and Vladut generalizing a classic theorem of Brauer and Siegel.