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I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields.

As Soogil Seo explains in his paper "Circular Distributions and Euler Systems" any Euler system over abelian extensions of the rationals can be obtained from cyclotomic units. However, I am struggling with understanding if the same statement is true about imaginary quadratic extensions i.e. if all Euler systems over abelian extensions of imaginary quadratic fields come from elliptic units (which are the analogous of cyclotomic units in the imaginary quadratic case).

I understand how elliptic units can be made into Euler systems and how they are used by Rubin in proving the main conjecture in the imaginary quadratic extension. But I find it difficult to understand how elliptic units can generate the module of Euler systems in abelian extensions the imaginary quadratic fields case. Any help would be appreciated.

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  • $\begingroup$ Your question is rather vague, and thus difficult to answer definitively; it amounts to "I find it difficult to understand X", which is arguably not really a question at all. What precisely is it that you are not understanding? (You have now asked a long string of questions, all focussing on the same very specific and rather obscure piece of mathematics. Have you been given this as a thesis problem? If so, then shouldn't you be discussing it with your advisor, rather than trying to crowd-source a thesis from MathOverflow?) $\endgroup$
    – David Loeffler
    Jan 19, 2021 at 9:25
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    $\begingroup$ @DavidLoeffler This is a maths discussion room and we are supposed to share our knowledge and understanding of this beautiful subject and if anyone is stuck on understanding (not superficial understanding, but in fact a deep understanding) of one piece then others who have more experience can perhaps help. I don't think if the way I asked the question would imply that I am trying to crowd-source a thesis from MathOverflow, but instead I have done my job of research and come up with my own understanding of this part of mathematics, but as this website allows me to communicate with $\endgroup$
    – Ash
    Jan 19, 2021 at 10:18
  • $\begingroup$ @DavidLoeffler (following from previous comment) experts in this field (people like yourself) the I take the opportunity to ask them to support me with having a better understanding. Perhaps, I should have expressed my question in a better way or explained more about it, but I am very good at typing maths on MathOverflow (I tried it once and it didn't look good at all) so I can't be specific on what I don't understand, that's why the questions seems to be very general and vague. $\endgroup$
    – Ash
    Jan 19, 2021 at 10:22
  • $\begingroup$ @DavidLoeffler I hope this clears the misunderstanding that the above question has caused to you. $\endgroup$
    – Ash
    Jan 19, 2021 at 10:24
  • $\begingroup$ @DavidLoeffler What I find useful about MathOverflow is that it provides a platform for us to learn about a particular piece of mathematics from individual perspectives. Hence, I was hoping to see individuals' point of view about the particular area of mathematics that was referred to in my questions, otherwise the question of whether all Euler systems of an abelian extension of an imaginary quadratic extension comes from elliptic units" is more of a conjecture and there hasn't been any answer to it. $\endgroup$
    – Ash
    Jan 19, 2021 at 10:33

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