A question about Iwasawa Theory

Question

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.

Answer 1

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).