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Given a number field $F$ and a prime $p$, it is natural to study Iwasawa theory over the cyclotomic $\mathbb{Z}_p$-extension of $F$, i.e., the unique $\mathbb{Z}_p$-extension of $F$ which is contained in $F(\mu_{p^\infty})$. Much work has been done on the study of the Iwasawa theory of $p$-primary Selmer groups associated to elliptic curves $E_{/F}$. There is much interest in the study of such questions over other pro-$p$ extensions of $F$, like the anticyclotomic $\mathbb{Z}_p$-extension, or a class of $p$-adic Lie extensions that contain the cyclotomic $\mathbb{Z}_p$-extension (known as strongly admissible extensions). These are infinite Galois extensions $F_\infty$ such that $G=\operatorname{Gal}(F_\infty/F)$ has the structure of a $p$-adic Lie group and such that $F_\infty$ contains $F^{cyc}$, the cyclotomic $\mathbb{Z}_p$-extension of $F$. It is also required that $G$ has no nontrivial $p$-torsion and that there are only finitely many primes that are ramified in $G$. Such extensions arise from the $p^\infty$ torsion of abelian varieties over $F$ for instance.

There has been recent interest in the study of analogous questions over a global function field $F$, i.e., a finite extension of $\mathbb{F}(x)$, where $\mathbb{F}$ is a finite field of characteristic $p>0$. There are some recent results in this direction, in which is established a dichotomy between $2$ cases of interest. First, is the study of Iwasawa theory of pro-$p$ extensions of $F$, for instance $\mathbb{Z}_p^d$-extensions. The other case is that of pro-$\ell$ extensions for some prime $\ell\neq p$, for instance $\mathbb{Z}_\ell^d$-extensions. There is a canonical $\widehat{\mathbb{Z}}$-extension, namely, setting $F_\infty:=F\cdot \bar{\mathbb{F}}$. This is however, not a good analog of the cyclotomic $\mathbb{Z}_p$-extension in the number field case.

There are infinitely many independent $\mathbb{Z}_p$-extensions. These can be seen to arise from the Galois fields generated by Drinfeld modules over $F$ of rank $1$. Consider the special case when $F=\mathbb{F}_p(x)$. The Carlitz module is a Drinfeld module of rank 1 over $F$ which gives rise to infinitely many independent $\mathbb{Z}_p$-extensions, one for each monic irreducible polynomial in $\mathbb{F}_p[x]$. These enjoy properties similar to the cyclotomic $\mathbb{Z}_p$-extension over a number field since no primes are infinitely decomposed in these extensions.

Questions:

  1. Do all $\mathbb{Z}_p$-extensions arise from Drinfeld modules of rank $1$ or as a quotient of $F\cdot \bar{\mathbb{F}}$? Are there any $\mathbb{Z}_p$-extensions in which some primes can be infinitely decomposed (as is the case with the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field).

  2. What are some natural examples of $\mathbb{Z}_\ell$-extensions for primes $\ell\neq p$, other than the unique $\mathbb{Z}_\ell$ extension that arises as a quotient of $F\cdot \bar{\mathbb{F}}$? What is the splitting behavior of primes in these extensions.

  3. What is a good analog of the term "admissible Lie extension" in the function field context? One would like to perform devissage arguments by relating the Iwasawa theory of a $p$-adic Lie group $G$ to certain abelian quotients of $G$. In the number field context one insists that $G$ contains the cyclotomic $\mathbb{Z}_p$-extension. The $p$-adic Lie extensions I am interested in are those arising from Drinfeld modules of rank $>1$. What are natural examples of $\ell$-adic Lie extensions to consider for $\ell\neq p$?

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    $\begingroup$ "This is however, not a good analog of the cyclotomic $\mathbb{Z}_p$-extension in the number field case." -- I would have thought that it is a good analogue. I think $\mathbb{Z}_p$-extensions of number fields were studied by Iwasawa in analogy to precisely that $\hat{\mathbb{Z}}$-extension in the characteristic $p$ case. Since everything becomes "geometric" over this extension, i.e. algebraic varieties over an algebraically closed field, it is the most reasonable intermediate step of "devissage". So I wonder why you think it is not a good analogue. $\endgroup$
    – Chris Wuthrich
    Commented Jun 10, 2022 at 8:35
  • $\begingroup$ The field extension $\bar{\mathbb{F}}$ is a good analog of a cyclotomic extension of $\mathbb{F}$. However, when we consider $F\cdot \mathbb{F}$, the resulting extension is unramified at all primes. The extensions cut out by the Carlitz module have similar ramification properties to extensions cut out by $p$ power roots of unity. For instance, the extension $\mathbb{Q}(\mu_{p^n})$ is unramified away from $p$. Furthermore, the prime above $p$ in $\mathbb{Q}(\mu_p)$ becomes totally ramified in the extensions $\mathbb{Q}(\mu_{p^n})$. Compare this to the extensions that arise from the Carlitz... $\endgroup$
    – Anwesh Ray
    Commented Jun 10, 2022 at 9:33
  • $\begingroup$ ... module. See for instance, Propn 12.7 in Rosen's book "Number theory in Function fields". Given an irreducible monic polynomial $P$, and an integer $e\geq 1$, we have the extension $K_{P^e}$, which is unramified at every prime ideal $(Q)$ not equal to $(P)$, and the prime above $(P)$ in $K_{P^e}$ is totally ramified in $K_{P^e}$. The degree of the extension is $\Phi(P^e)$, which uncannily similar to the cyclotomic case. I would contend that these properties make the infinitely generated extension arising from the Carlitz module the correct analog of the cyclotomic extension... $\endgroup$
    – Anwesh Ray
    Commented Jun 10, 2022 at 9:38
  • $\begingroup$ ... $\cup_n\mathbb{Q}(\mu_n)$ from the number field setting. $\endgroup$
    – Anwesh Ray
    Commented Jun 10, 2022 at 9:39
  • $\begingroup$ @Chris Wuthrich .... Another property that cyclotomic extensions have in the number field context is that every abelian extension is contained in $\mathbb{Q}(\mu_n)$ for some $n$. This property is shared by the extensions that arise from Drinfeld modules. I am quoting Rosen here, see p. 192 in the chapter "Cyclotomic Function fields" where he states that these ideas were used to realize explicit global class field theory in the function field setting following the work of Drinfeld and Hayes, giving a solution of Hilbert's 9th problem. $\endgroup$
    – Anwesh Ray
    Commented Jun 10, 2022 at 9:45

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