On Z₂-extensions of real quadratic fields with four odd ramified prime numbers

In this paper, we study the Iwasawa -invariant of the cyclotomic Z ₂ -extension of a real quadratic field Q (p₁p₂p₃p₄), where p₁, p₂, p₃ and p₄ are distinct odd prime numbers satisfying certain arithmetic conditions. We give a new infinite family of real quadratic fields for which Greenberg’s conjecture holds.