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I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what it means that “all Euler systems come from cyclotomic units in an easy way”; what does this mean and what is that easy way?

Also, what is the extend to which similar thing can be done if the field Q of rationals is replaced by an imaginary quadratic field?

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    $\begingroup$ I have no idea what this question means, and I've been working on Euler systems for half my career. Can you make it a bit more precise, maybe with some links and/or references? $\endgroup$
    – David Loeffler
    Commented Aug 14, 2020 at 18:14
  • $\begingroup$ The last paragraph of page 7 of Seo's "Circular Distributions and Euler Systems" states that "there are almost no other Euler systems except the cyclotomic units". How do I prove this? re there any other examples apart from the cyclotomic units? $\endgroup$
    – Ash
    Commented Aug 14, 2020 at 23:19
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    $\begingroup$ With the commonly understood definition of "Euler system" (as in Karl Rubin's classic book "Euler Systems"), this assertion of Seo is simply untrue. Presumably Seo is using some other, more specific definition -- maybe he means only Euler systems for the trivial representation and $K = \mathbf{Q}$ -- but even so, the use of "almost" suggests this is supposed to be a rough guideline rather than a rigorous theorem. Soogil Seo appears to be currently mathematically active, so why don't you ask him directly what the phrase meant? $\endgroup$
    – David Loeffler
    Commented Aug 15, 2020 at 7:03
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    $\begingroup$ The assertion is not precise enough to have a well-defined truth value. Maybe what Seo is aiming at is the assertion that the module of Euler systems for $K = \mathbf{Q}$ and $T = \mathbf{Z}_p(1)$ is generated by the image of the cyclotomic units under the Kummer map; this sounds plausible, and I think it may follow from the Mazur--Wiles theorem (former Iwasawa main conjecture), but I'd have to check. But there are lots of other Galois representations for which it is interesting to study Euler systems, and these other Euler systems have nothing much to do with cyclotomic units. $\endgroup$
    – David Loeffler
    Commented Aug 15, 2020 at 9:56
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    $\begingroup$ As this question came up again today, let me add that Coleman's conjecture has been proven in this article by Bullach, Burns, Daoud, Seo. Probably this answers this vague question. $\endgroup$
    – Chris Wuthrich
    Commented Jan 15 at 10:01

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