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Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity.

My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots of articles which dealing Euler system of elliptic units. For me, it(elliptic units) is just a set of complicatedly defined global units which allow us to do computations. Even I understood some procedures of computation, I cannot catch the reason why people developed such things.

Could anyone give me some heuristics behind elliptic units?

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    $\begingroup$ Do you know cyclotomic units and how they can be used to prove the main conjecture for the ideal class group? $\endgroup$
    – Chris Wuthrich
    Commented Dec 23, 2013 at 17:46
  • $\begingroup$ There is a certain circularity in this question: you state that there are many articles that use elliptic units; why not read those articles and see what they use them for? $\endgroup$
    – David Loeffler
    Commented Dec 23, 2013 at 21:53
  • $\begingroup$ @ChrisWuthrich I don't know about it. Could you recommend some references for that? $\endgroup$
    – Kevin.lijh
    Commented Dec 23, 2013 at 22:16
  • $\begingroup$ @DavidLoeffler I tried to read them, but it was really hard to find some 'ideas' behind the messy computation. $\endgroup$
    – Kevin.lijh
    Commented Dec 23, 2013 at 22:18

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Given that you have not seen cyclotomic units, I think you should start with them. Rubin's appendix to Lang's book(s) on cyclotomic fields is one place or the book by Coates and Sujatha. Then for elliptic units, I like Rubin's part in the Cetraro notes "Arithmetic of ellitpic curves".

An Euler system is a stepping stone between L-values and the Selmer groups. On the one hand, your elliptic units are linked to the values of L-functions associated to the Grössencharacter (and hence to the elliptic curve), see for instance theorem 7.17 in Rubin. On the other hand you can bound a part of the ideal class group with them (theorem 9.5). This gives you great results like, if the L-functions does not vanish then the rank is zero and the Tate-Shafarevich group is bounded. Furthermore you can do these arguments in a $\mathbb{Z}_p$-extension and you get one part of the main conjecture, (theorem 11.7), which then bounds the rank in general by the order of vanishing of some $p$-adic L-functions.

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  • $\begingroup$ I am sorry that I am vague, but the question is. I hope nonetheless this may help you. $\endgroup$
    – Chris Wuthrich
    Commented Dec 25, 2013 at 12:13

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