hi, I was trying to implement this on SAGE. I honestly couldn't understand how exactly Kohel's thesis deal with the following? (1) How are endomorphism being represented and sampled? (2) How to check the rank of endomorphisms? (3) How to check the algebra ramifies at {p,inf}? (4) How to check an endomorphism generates an maximal ideal? I checked out the Chapter 7 but just couldn't make any such connection.

No, the argument does not hold. even if the triple sum contains all terms $(A_{ji}x_i)$, it does the square of not necessarily be greater. Since extra terms might cancel out with original terms.

The first equation is in general wrong since it should be $$ \sup_{\Vert x\Vert=1}\sum_{j=1}^m\left|\sum_{i=1}^na_{ji}x_i\right|^2 $$

The problem I'd like to solve algorithmically is to take input $e_1,...,e_r$ and outputs $\alpha_1^{e_1}...\alpha_r^{e_r}x.$

It is the graph with vertex set $X$ and $x_1,x_2$ connected when $\alpha_i x_1=x_2,$ for some $\alpha_i$