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The purpose of this paper is to investigate a conjecture about the universality of the circular distribution made by Robert Coleman. The algebraic property of the universal distribution is the main ingredient in studying Euler system of Kolyvagin and Rubin. We study the universality of the circular distribution by using the Iwasawa theory and the theory of the Euler systems. The conjecture is a characterization of Euler systems in the case of number field. The results here assert that Euler syst

Motivated by the theory of circular distributions, we introduced a filtration on the global units attached to the maximal real subfield of a cyclotomic field and conjectured that the associated gradation is isomorphic, (as a Galois module) to the ideal class group. This conjecture depends on circular distributions of Coleman and a guess made by him. In this paper we show that the circular distributions of finite order cannot be constructed from cyclotomic p-units and their Galois conjugates.

Article history: Received 25 January 2012 Revised 16 May 2013 Accepted 20 May 2013 Available online xxxx Communicated by D. Burns MSC: 11R27 11R29

It is known that the order of the class group ClK of the real abelian field K is essentially equal to the order of the quotient EK/CK of the global units EK by the circular units CK of K. However, the structures of these two groups are usually very different. Motivated by the theory of circular distributions and the special units of Rubin, we introducea filtration to EK made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois mod

We study Euler systems for $\mathbb{G}_m$ over a number field $k$. Motivated by a distribution-theoretic idea of Coleman, we formulate a conjecture regarding the existence of such systems that is elementary to state and yet strictly finer than Kato's equivariant Tamagawa number conjecture for Dirichlet $L$-series at $s=0$. To investigate the conjecture, we develop an abstract theory of `Euler limits' and, in particular, prove the existence of canonical `restriction' and `localisation' sequences

The fourth author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2022R1F1A1059558).

We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.

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 Z p -extension k cyc 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.