Cyclotomic Z2-extension of real quadratic fields with cyclic Iwasawa module - Avila Artavia, Josue 1995- (2024)
For a number field K and a prime p, let K∞ denote the cyclotomic Zp-extension of K, and An denote the p-primary part of the class group of its n-th layer Kn. Greenberg conjectured that for a totally real field, the order of An becomes constant for sufficiently large n. Motivated by the work of Mouhib and Movahhedi, we focus on the case where p = 2 and K is a real quadratic field such that the Iwasawa module X∞ = lim ←An is cyclic. They determined all such fields and proved that Greenberg's conjecture holds for some cases. In this dissertation, we provide new examples of infinite families of real quadratic fields satisfying Greenberg's conjecture which were not covered completely in the work of Mouhib and Movahhedi. To achieve this, we use the theory of binary quadratic forms and biquadratic extensions to determine a fundamental system of units and the class number of the first few layers of the cyclotomic Z2-extension. Additionally, in certain cases, we can determine the size of the module X∞ and the level of the cyclotomic tower where the size of An becomes constant.