Tohoku Mathematical Journal

On the two-variables main conjecture for extensions of imaginary quadratic fields

Stéphane Vigué

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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.

Article information

Tohoku Math. J. (2), Volume 65, Number 3 (2013), 441-465.

First available in Project Euclid: 12 September 2013

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Mathematical Reviews number (MathSciNet)

Primary: 11G16: Elliptic and modular units [See also 11R27]
Secondary: 11R23: Iwasawa theory 11R65: Class groups and Picard groups of orders

Elliptic units Euler systems Iwasawa theory


Vigué, Stéphane. On the two-variables main conjecture for extensions of imaginary quadratic fields. Tohoku Math. J. (2) 65 (2013), no. 3, 441--465. doi:10.2748/tmj/1378991025.

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