STNB 2016 (30è any)

On the Iwasawa-Leopoldt Conjecture


Preda Mihailescu


The structure of the $p$-part $A$ of the
$p$-th cyclotomic field, simple as it should be, is
poorly understood. The guiding expectation is
the Kummer-Vandiver conjecture, which asserts
that $A^+ = 1$. However, computations suggest
even stronger restriction: the exponent of $A$ should
be p and, in the Iwasawa tower, $A_{\infty}$ should have
linear annihilators. On the other side, Iwasawa and
Leopoldt proposed a slightly weaker conjecture than
Kummer-Vandiver, which stipulates that $A$ is $\mathbb{Z}_p[ G ]$
cyclic, with $G$ the Galois group. In this talk we discus
the implications of the various facts and conjectures and
prove the following result: Assuming the Greenberg
conjecture, $A^+$ is $\mathbb{Z}_p[ G ]$ - cyclic.


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