## LORENZO PAGANI

PhD Graduate**PhD program:**: XXXIII

**supervisor**: G. Mondello

**advisor**: R. Schoof

**Thesis title:**Class group behaviour in cyclotomic extensions of abelian fields

The thesis is divided into two parts.
In the first part, we study Greenberg's conjecture for the ciclotomic Z_2-extension of a real quadratic field. Let F be a real quadratic field and let F_n be the n-th level of the Z2-extension. Denote by A_n the 2-part of the class group of F_n. Exploiting properties of cyclotomic units and finite Gorenstein rings, we write an algorithm to compute upper bounds for the cardinality of the groups An. As a consequence of our computations we check that Greenberg's conjecture holds for the cyclotomic Z_2-extensions of the real fields F=Q(\sqrt{f}) where f is a positive squarefree integer less than 10000.
In the second part, we fix two distinct primes p and l. Assume that the prime l is odd. Then, we study the behaviour of the l-parts A_n of the class groups of the fields K_n, where K_n is the n-th level field in the cyclotomic Zp-extension of an abelian number field K. It is known the groups A_n stabilize. By the relation between Bernoulli numbers and class number, we explicity compute a level n_0 such that the natural morphisms A_n---->A_{n_0} are isomorphisms.