Thank you! Given that this proof also uses a class number formula and the passage via the units, maybe my hunch that this should be avoidable is simply false. This proof certainly has the merit that it avoids Lee's classification, although I cannot see straightaway whether one can go all the way to the full strength of (1) without it, and also whether one can avoid saying anything about the Galois module structure of the units in the real case.

...is $3$. For the details see arxiv.org/abs/0904.2416, particularly the table in Example 6.4, which lists the possible $\mathbb{Z}[S_3]$-module structure of $U_L$ and the corresponding value for a certain quotient of class numbers.

@A.Maarefparvar: In the special case that $L/\mathbb{Q}$ is a real $S_3$-extension, my last paragraph applies to the relative extension, but you can also sometimes exploit the extra structure. Indeed, if you know the $\mathbb{Z}_3[S_3]$-module structure of $U_L$, then you also know the $\mathbb{Z}_3[C_3]$-module structure. For the $S_3$-structure, there are only finitely many possibilities, and they can sometimes be read off from the class numbers of the intermediate extensions. In particular, if a certain quotient of class numbers is equal to $1/9$, then you will know that the norm index...

It's worth noting that the K-groups that (conjecturally) appear as zeta-values are Quillen K-groups, while Kato's article concern Milnor K-groups.

Firstly, it is not true that ${\rm dim}{\rm Hom}_k(A,B)$ always has rank $2g$, rather it has rank at least $2g$ (and at most $(2g)^2$). For example the endomorphism ring of a supersingular elliptic curve is an order in a quaternion algebra. And secondly, for a basis as you demand to exist, all $f_i$ would have to have the same (minimal) degree, which is clearly not always possible. If, for example, $E$ is an ordinary elliptic curve with CM by $K$, then the only way you can have a basis for ${\rm End} E$ with both elements of degree $1$ is if $K=\mathbb{Q}(i)$.

In fact, for every single elliptic curve over $\mathbb{Q}$ of rank at least $2$, the finiteness of its Tate-Shafarevich group is an open problem.

Just take an infinite sequence $X_n$ of independent random draws from $\Omega$ and let $E_n$ be the event $X_n\in [0,1/n]$.

As Emanuele Tron said below, in mathematics one would expect all authors of a paper to contribute substantial ideas, not just the problem statement and some reading suggestions. The latter warrants a grateful mention in the acknowledgements, rather than coauthorship. Of course, depending on your relationship with the other person, this decision may not be down to you. And the other person might not even see their contribution in the same way as you do.