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Consider a tower of curves $\dots \to C_n \to C_{n-1} \dots \to C_1$ over $\mathbb F_q$ where $C_n: f(x^{\ell^n},y) = 0$.We can look at the multiset $S_n$ of eigenvalues of the Frobenius $\sigma_q$ on $H^1(\overline{C_n},\mathbb Z_\ell)$ and also the characteristic polynomial $g_n(x)$.

We will have a factorization $g_n(x) = \prod_{i=1}^n f_i(x)$ and we can ask how $f_i(x)$ varies with $i$. This could be called a version of Iwasawa theory over function fields, has it been considered in the literature before?

I have found papers that study how the class group/class number changes with $i$ but that's a little less information than the characteristic polynomial itself.

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  • $\begingroup$ Do such towers exist? $\endgroup$
    – Will Sawin
    Commented Jan 22, 2021 at 2:48
  • $\begingroup$ Yes, you can consider $f(x^{\ell^n},y) = 0$ for instance $\endgroup$
    – Asvin
    Commented Jan 22, 2021 at 2:52
  • $\begingroup$ (I have been thinking about this problem and want to make sure that my results are new.) $\endgroup$
    – Asvin
    Commented Jan 22, 2021 at 2:58
  • $\begingroup$ That has Galois group $\mathbb Z_\ell \rtimes \mathbb Z_\ell$, at least if $q$ contains the $\ell$th roots of unity. $\endgroup$
    – Will Sawin
    Commented Jan 22, 2021 at 3:00
  • $\begingroup$ Sorry, could you explain more? You get the automorphisms that send $x \to x\zeta_{\ell^n}$ on $C_n$ but what other automorphisms do you get? Are you including the Frobenius? If so, I guess I meant the geometric Galois group should be $\mathbb Z_\ell$ - sorry for being unclear. $\endgroup$
    – Asvin
    Commented Jan 22, 2021 at 3:04

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