## Tohoku Mathematical Journal 2013 September SECOND SERIES VOL. 65, NO. 3

 Tohoku Math. J. 65 (2013), 441-465

Title ON THE TWO-VARIABLES MAIN CONJECTURE FOR EXTENSIONS OF IMAGINARY QUADRATIC FIELDS

Author Stéphane Vigué

(Received November 21, 2011, revised September 25, 2012)
Abstract. Let $p$ be a prime number at least 5, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two conjugate primes. Let $k_\infty$ be the unique ${\boldsymbol Z}_p^2$-extension of $k$, and let $K_\infty$ be a finite extension of $k_\infty$, abelian over $k$. We prove that in $K_\infty$, the characteristic ideal of the projective limit of the $p$-class group coincides with the characteristic ideal of the projective limit of units modulo elliptic units. Our approach is based on Euler systems, which were first used in this context by Rubin.

Mathematics Subject Classification. Primary 11G16; Secondary 11R23, 11R65.

Key words and phrases. Elliptic units, Euler systems, Iwasawa theory.