Mathematical Research Letters

Volume 22 (2015)

Number 5

On the conjectures of Gross and Leopoldt

Pages: 1509 – 1540



Soogil Seo (Department of Mathematics, Yonsei University, Seoul, Korea)


We show that the Leopoldt conjecture implies the so-called Hilbert’s theorem 90 for compact modules made from the $p$-units over the cyclotomic $\mathbb{Z}_p$-extension $k^{\mathrm{cyc}}_{\infty}$ of $k$. And under the generalized Gross conjecture, we show that Hilbert’s theorem 90 for the compact modules above is equivalent to the affirmation of the Leopoldt conjecture.

Accepted 13 August 2014

Published 13 April 2016