Sophisticated Diagram in LaTeX/Tikz

Question

For my thesis at university, I'd like to draw a certain diagram, but I am not sure how I should go about it. I have drawn it by hand and it should be displayed down below. I have looked at Tikz/Tikz-cd as well as other questions asked, such as How can I draw commutative diagrams in LaTeX? but mine is a little more sophisticated (as well as the fact it's not a commutative diagram; I couldn't find any other questions that were very similar), and I'm unsure how to draw this diagram due to all the specific details. Could anyone point me in the right direction?

Thank you so much in advance!

Answer 1

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}

Answer 2

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}