# A question about Iwasawa Theory

I am just reading about Iwasawa theory about Coates and Sujatha's book on Iwasawa Theory. I was wondering that since Iwasawa thought about the whole theory from the analogy of curves over finite fields, so what should be the analog of the module $U_\infty$/$C_\infty$ in the curve case (if there is any) where $U_\infty$ is the inverse system of local units and $C_\infty$ is the cyclotomic units.

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An interview with Iwasawa: math.washington.edu/~greenber/IwInt.html – Thomas Riepe Jun 21 '10 at 20:46
Ok that changes some things. But I still will like to know if there is an analogy of the above module in the curve case – Arijit Jun 21 '10 at 21:55
A historial article and a book by Greenberg: math.washington.edu/~greenber/iwhi.ps math.washington.edu/~greenber/book.pdf – Thomas Riepe Jun 28 '10 at 12:16
Thanks a lot, Thomas. – Arijit Jun 28 '10 at 16:56

There is a very close analogy but to unravel it requires some work.

So take $X$ a smooth curve over $\mathbb F_{\ell}$ (more generally you could take $X$ a scheme over $\mathbb F_{\ell}$) and let $\mathscr F$ be a smooth sheaf of $\mathbb Q_{p}$-vector spaces on $X$ (you could be much more general in your choice of coefficient ring, and indeed, I think you might need to consider more general coefficient rings in order to really grasp the analogy, but that will do for the moment). Moreover, we will assume for simplicity that $\mathscr F$ comes from a motive over $X$, a sentence which will remain vague but aims to convey the idea that $\mathscr F$ has geometric origin.

Then the cohomology complex $R\Gamma(X,\mathscr F)$ is a perfect complex so it has a determinant $D$. This complex fits in a exact triangle $$R\Gamma(X,\mathscr F)\rightarrow R\Gamma(X\otimes\bar{\mathbb F}_{\ell},\mathscr F)\rightarrow R\Gamma(X\otimes\bar{\mathbb F}_{\ell},\mathscr F)$$

Here the very important fact to understand in order to grasp the analogy is that the second arrow is given by $Fr(\ell)-1$. This exact triangle induces an isomorphism $f$ of $D$ with $\mathbb Q_{p}$.

There is conjecturally another such isomorphism. Assume that the action of the Frobenius $Fr(\ell)$ acts semi-simply on $H^{i}(\bar{X},\mathscr F)$ for all $i$ (this is widely believed under our hypothesis on $\mathscr F$). Then degeneracy of a the spectral sequence $H^{i}(\mathbb F_{\ell},H^{j}(X\otimes\bar{\mathbb F}_{\ell},\mathscr F))$ gives an isomorphism $g$ between $D$ and $\mathbb Q_{p}$.

Now, consider $gf^{-1}(1)$. This happens to be the residue at 1 of the zeta function of $X$.

What has all this to do with units in number fields? Change setting a bit and take $X_{n}=\operatorname{Spec}\mathbb Z[1/p,\zeta_{p^{n}}]$ and $\mathscr F=\mathbb Z(1)$. We would like to carry the same procedure as above but we can't, because we are lacking crucially the exact triangle involved in the definition of the isomorphism $f$. Nonetheless, there is a significantly more sophisticate way to construct a suitable $f_{n}$ for all $n$ and it turns out that this construction will crucially involve $U_{\infty}/C_{\infty}$.

So to sum up, the analog of $U_{\infty}/C_{\infty}$ for curves over finite fields is none other than the Frobenius morphism $Fr(\ell)-1$. You may know that the cyclotomic units form an Euler system, that is to say that they satisfy relations involving corestriction and the characteristic polynomial of the Frobenius morphisms. This fact is I believe what led K.Kato to describe the analogy above.

You can read about all this in much much greater details in the contribution of Kato in the volume Arithmetic Algebraic Geometry (Springer Lecture Notes 1553).

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The link to Kato's lecture etc.: mathoverflow.net/questions/6928/how-do-we-study-iwasawa-theory/… – Thomas Riepe Jun 22 '10 at 10:40
I guess I dont have enough background to read that book. So I dont know whether its a relevant question or not. So what is the analog of Coleman's power series map in the case that you explained. I guess a better but a vague question: what is the correct way to look at the Coleman map? – Arijit Jun 22 '10 at 16:17
I think this is stretching it, but the Coleman map is the collections of the fn, so in a sense the analog of the Coleman map in the geometric case is the isomorphism f. What is the correct way to look at the Coleman map is an excellent question, which admits a precise albeit technical question: the Coleman map is an instance of the so-called epsilon morphism. You could read for instance Fukaya-Kato on this. It is natural to feel intimated by the level of Fukaya-Kato or Kato's lecture, but you should give it a try once in a while: you will learn a lot from them. – Olivier Jun 22 '10 at 17:12
Thanks a lot Olivier. – Arijit Jun 22 '10 at 23:33