Rationality of trace of endomorphism of Iwasawa-thing

Question

Let $n$ be a positive integer, and $p$ a prime number. Let $K_i$ be the cyclotomic field containing exactly the $np^i$th roots of unity. Let $H$ be the inverse limit of $p$-power torsion of the class groups of the $K_i$. Let $V$ be the $\mathbb{Q}_p$ - vector space $H \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$. It is a result of Iwasawa theory that $V$ is finite-dimensional.

For any integer $q$ that is relatively prime to $np$, we have an automorphism $f_q$ of $\bigcup_{i = 1}^{\infty} K_i$ which, for each root of unity $\omega$, brings $\omega^q$ to $\omega$. This induces an automorphism of $H$ and thus an automorphism of $V$, say $f^*_q$.

What I'm interested in is whether $\mathrm{Tr}(f^*_q) \in \mathbb{Q}_p$ is rational, and related questions. Are there situations where we know this is the case, besides obvious cases like $q = 1$ or $V$ is trivial etc.? Are there situations where we know this isn't the case? Are there related operators on $V$ that nontrivially have rational trace? Can we ever say something weaker like $\mathrm{Tr}(f_q)$ is algebraic over $\mathbb{Q}$?

The reason I ask: Given a smooth projective curve $C$ over a finite field $K$ with algebraic closure $\overline{K}$, the $q$th power Frobenius morphism on $C$ induces the $\overline{K}$-linear Frobenius morphism on $\overline{C} = C \otimes_{K} \overline{K}$, which induces an endomorphism of the $\ell$-adic cohomology vector space $H^1(\overline{C}, \mathbb{Q}_{\ell})$. The traces of powers of this morphism yield information about the zeta function of the curve, but the only reason this works is that these traces are rational numbers (more specifically, integers) so they can be seen as both $\ell$-adic numbers or as real numbers, the latter being what we need for applications to the zeta function.

To relate this to the original question, I don't think these $f_q$ morphisms are analogous to powers of the $\overline{K}$-linear Frobenius, but rather a closely related morphism, which could be briefly described as the "inverse Frobenius on coefficients", and whose powers still induce endomorphisms of the $\ell$-adic cohomology vector spaces which have rational traces which give us essentially the same information as the $\overline{K}$-linear Frobenius. So basically I want to know if a similar thing holds in the number field setting, with $V$ playing the role of $H^1(\overline{C}, \mathbb{Q}_{\ell})$, in hopes that it might yield interesting information pertaining to non-p-adic L-functions (although I don't expect anything nearly as straightforward as in the function field case).

Answer 1

This is not really an answer, just a (very!) long comment. Everything I write is obvious for people working in Iwasawa theory, and I apologize for the trivialities.

Let me start by your final paragraph, where you discuss the analogy with curves over finite fields and the action of Frobenius: I guess you are aware that this was one of the motivations for Iwasawa theory, and this is discussed a bit in Washington's book, Chapter 13 (both in the introductory paragraph and in 13.6). But my feeling is that $V$ is somehow analogous to a $p$-adic theory (like crystalline, or overconvergent), rather than étale $l$-adic: in the analogy, we consider $K_\infty/K_0$ as the constant field extension, which has only one Frobenius. What changes in the rationality business are the coeffients: here, your coefficients are always the same, and $p$-adic, and you change "Frobenius".

Passing to your question, anyhow, let me stick for notational ease to the case $n=p$ and write $K_\infty=\varinjlim \mathbb{Q}(\zeta_{p^{k+1}})$. The Galois group $\operatorname{Gal}(K_\infty/K_0)$ is procyclic, isomorphic to $1+p\mathbb{Z}_p$ through the cyclotomic character $\kappa\colon\operatorname{Gal}(K_\infty/K)\to 1+p\mathbb{Z}_p$ (or rather its inverse, to be consistent with your choice): let me fix a topological generator $\gamma_0$ of this Galois group which, for notational ease, I suppose to be sent via $\kappa$ to the element $(1+p)\in(1+p\mathbb{Z}_p)$. Introduce the logarithm $\mathcal{L}$ "in base $1+p$" $$ \mathcal{L}(u)=\frac{\log_p(u/\omega(u))}{\log_p(1+p)}%\quad\text{ and }\quad (\mathbb{Z}/p)^\times\ni\ell\colon x\mapsto 1\in\mathbb{Z}/(p-1) $$ where $\log_p$ is Iwasawa's $p$-adic logarithm, $\omega$ is the$\mod{p}$ Teichmüller character, and $u\in\mathbb{Z}_p^\times$. Then, for every integer $q\not\equiv 0\pmod{p}$ we have $f_q=\gamma_0^{%-\ell(\omega(q))+ \mathcal{L}(q)}$.

Now, if the action of $\gamma_0$ is semi-simple, then the matrix of $\gamma_0$ acting on $V\otimes \overline{\mathbb{Q}}_p$ is $$ M_{\gamma_0}=\begin{pmatrix}\alpha_1&&\\&\ddots&\\ &&\alpha_{\lambda}\end{pmatrix}, $$ where the $\alpha_i$'s are the roots of the characteristic polynomial, and then $\operatorname{Tr}(\gamma_0)=\alpha_1+\dots+\alpha_\lambda$. By the above remark, $\operatorname{Tr}(f_q^\ast)=\alpha_1^{%\omega^{-1}(q) \mathcal{L}(q)}+\dots+\alpha_\lambda^{%\omega^{-1}(q) \mathcal{L}(q)}$ and to address your question it seems natural to start studying these roots $\alpha_i$'s themselves. The first thing we can observe is that $\alpha_i\equiv 1\pmod{\mathfrak{m}_{\mathbb{C}_p}}$ because $\gamma_0$ is topologically unipotent and thus $\lim_{n}\alpha_i^{p^n}=1$. Moreover, by definition, $$\alpha_i^{\mathcal{L}(q)}=\exp_p(\log_p(\alpha_i)\mathcal{L}(q))=\exp_p\Bigl(\log_p(\alpha_i)\frac{\log_p(q/\omega(q))}{\log_p(1+p)}\Bigr); $$ I am not an expert in $p$-adic transcendence, but granted that $q/\omega(q),(1+p)$ and $\alpha_i$ are algebraically independent (for non-pathological $\alpha_i$), I would be surprised if the above expression were algebraic.

To go further, observe that $V$ is actually a direct sum $V^+\oplus V^-$ of two subrepresentations, which are the $\pm$-eigenspaces for the action of the complex-conjugation $c\in\operatorname{Gal}(K_\infty/K_0)$. A well-known conjecture by Greenberg predicts that $V^+=0$ (this has been checked numerically in many cases), so it is natural to restrict only to the subspace $V^-$. Greenberg himself has proven in his paper On a certain $l$-adic representation (Invent. Math., 1973) that the action of $\gamma_0$ on $V^-$ is semi-simple and that its minimal polynomial is $f_{\gamma_0}^-(T)=(T-1)^sg(T)$ where $s$ is the number of primes above $p$ in $K_0^+$ which split in $K_0$: so, in our case $n=p$ we have $s=0$ and $\alpha_i\neq 1$ for all $i$. The roots of $f_{\gamma_0}^-(T)$ (again, conjecturally $f_{\gamma_0}^-(T)=f_{\gamma_0}(T)$ because $V^+$ should be $0$) are connected, via the Main Conjecture of Iwasawa Theory (now a theorem), to the zeros of the $p$-adic $L$-function $L_p(s,\chi)$ of Kubota--Leopoldt, where $\chi$ runs through all even characters of $\operatorname{Gal}(K_0/\mathbb{Q})$. This goes as follows: the representation $V^-$ can be further decomposed as $V^-=\oplus_{\chi}V(\chi^{-1}\omega)$ where $\chi$ are the even characters of $\operatorname{Gal}(K_0/\mathbb{Q})$ and $V(\chi^{-1}\omega)$ is the subspace on which the action of $\operatorname{Gal}(K_0/\mathbb{Q})$ is given by $\chi^{-1}\omega$: accordingly, $f_{\gamma_0}^-(T)=\prod_\chi f(T,\chi)$. Then Mazur--Wiles and Rubin have proven that $L_p(s,\chi)=f((1+p)^s,\chi)$ and thus if $\alpha_i$ is a root of $f_{\gamma_0}^-(T)$, then $f(\alpha_i,\chi)=0$ for some $\chi$ and $L_p(\mathcal{L}(\alpha_i),\chi)=0$; and conversely, if $L_p(s_0,\chi)=0$ then $(1+p)^{s_0}$ is a root of $f(T,\chi)$ and hence of $f_{\gamma_0}(T)$.

Now comes the sad side of the story, namely that (in Washington's words, see the Remark following Theorem 5.11 in his book Introduction to Cyclotomic Fields) "the nature of the zeros of the $p$-adic $L$-function is not yet understood". Computations by Ernvall and Metsänkylä show that, in general, the $\alpha_i$ are not rational numbers, because the $p$-adic valuation of $\alpha-1$ is non-integral (see Propositions 4 and 5 in their paper Computation of the zeros of $p$-adic $L$-functions, Math. Comp., 58, 1992). It is worth quoting the final sentence of the paper:

On the basis of the numerical data computed so far it seems natural to expect that the zeros $T_j$ (corresponding to my $\alpha_j-1$) and $s_i$ (corresponding to my $\mathcal{L}(\alpha_i)$) are distributed randomly as regards their $p$-adic value (within the prescribed limits) and their inclusion in various extensions of $\mathbb{Q}_p$. Also the $p$-adic expansions of the zeros fail to show any regularity.

Remark I have chosen a highly non-standard normalization, so if you look at the litterature, pay attention: in particular, what is called $f(T)$ is normally the characteristic polynomial of $\gamma_0-1$ rather than of $\gamma_0$.