Sophisticated Diagram in LaTeX/Tikz

As pointed out by Torbjørn T. there is no real challenge. At least it is very much simpler to draw such diagrams than to understand Galois theory.

\documentclass{article}
\usepackage{tikz-cd}
\usepackage{amsmath}
\usepackage{amsfonts}
\DeclareMathOperator{\Frob}{Frob}
\usepackage{dsfont} % I personally like this better than mathbb from AMS
\begin{document}
\begin{tikzcd}[column sep=large]
 T_\ell E(\mathds{F}_p) \arrow[loop below,"\Frob_p"] 
 \arrow[r,"\underset{\sim}{\varphi}"]
 \arrow[r,bend left,"\underset{\sim}{\varphi'}"]
 & (\mathds{Z}_\ell)^2 
 \arrow[loop below,"\psi\circ\Frob_q\circ\psi^{-1}"] 
 \arrow[loop above,"\begin{array}{@{}c@{}}
 \varphi'\circ\Frob_q\circ\varphi^{\prime-1}\\
 \varphi\circ\Frob_q\circ\varphi^{-1}
 \end{array}"] 
 & T_\ell E(\mathds{F}_q) \arrow[loop below,"\Frob_q"] 
 \arrow[l,"\underset{\sim}{\psi}"']
\end{tikzcd}

\bigskip
\begin{tikzcd}[column sep=large]
 T_\ell E(\mathbb{F}_p) \arrow[loop below,"\Frob_p"] 
 \arrow[r,"\underset{\sim}{\varphi}"]
 \arrow[r,bend left,"\underset{\sim}{\varphi'}"]
 & (\mathbb{Z}_\ell)^2 
 \arrow[loop below,"\psi\circ\Frob_q\circ\psi^{-1}"] 
 \arrow[loop above,"\begin{array}{@{}c@{}}
 \varphi'\circ\Frob_q\circ\varphi^{\prime-1}\\
 \varphi\circ\Frob_q\circ\varphi^{-1}
 \end{array}"] 
 & T_\ell E(\mathbb{F}_q) \arrow[loop below,"\Frob_q"] 
 \arrow[l,"\underset{\sim}{\psi}"']
\end{tikzcd}
\end{document}

This is a starting point....

\documentclass[12pt]{article}
\usepackage{mathtools}

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow[r, "\beta"] \arrow["\alpha"', loop, distance=2em, in=305, out=235] \arrow[r, "\gamma", bend left=49] & B \arrow["\delta"', loop, distance=2em, in=305, out=235] \arrow[loop, distance=2em, in=305, out=235] \arrow[loop, distance=2em, in=305, out=235] \arrow[loop, distance=2em, in=305, out=235] \arrow["\overset{p}{q}", loop, distance=2em, in=55, out=125] & G \arrow[l, "\kappa"'] \arrow["u", loop, distance=2em, in=235, out=305]
\end{tikzcd}
\end{document}

ADDENDUM:

....and here there is my final...destination...

\documentclass[12pt]{article}
\usepackage{mathtools}
\usepackage{amssymb}


\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
 T_{l} E(\mathbb{F}_p)  \arrow[r, "\underset{\sim}{\varphi}"] \arrow["\text{Frob}_q"', loop, distance=2em, in=305, out=235] \arrow[r, near start, "\underset{\sim}{\varphi'}", bend left=40] & (\mathbb{Z}_{l})^2  \arrow["\psi\,\circ\,\text{Frob}_q\,\circ\,\psi^{-1}"', loop, distance=2em, in=305, out=235] \arrow["\overset{\scriptstyle \varphi\,\circ\,\text{Frob}_q\,\circ\,\varphi'^{-1}}{\scriptstyle \varphi\,\circ\,\text{Frob}_q\,\circ\,\varphi^{-1}}", loop, distance=2em, in=55, out=125] &  T_{l} E(\mathbb{F}_q) \arrow[l, "\underset{\sim}{\psi'}"'] \arrow["\text{Frob}_q", loop, distance=2em, in=235, out=305]
\end{tikzcd}
\end{document}