I am trying to understand the statement of the equivariant BSD conjecture.
Let $E/\mathbb{Q}$ be an elliptic curve. Let $\rho$ be a finite-dimensional irreducible Artin representation, and let $K/\mathbb{Q}$ be any finite Galois extension such that $\rho$ factors through $\mathrm{Gal}(K/\mathbb{Q})$.
I have seen the equivariant BSD conjecture stated as:
\begin{equation}\tag{1} \mathrm{ord}_{s=1}L(E,\rho,s)=\mathrm{dim}_\mathbb{C}\,E^{(\rho)}, \end{equation}
where $E^{(\rho)}$ is the $\rho$-isotypical component of $E(K)\otimes\mathbb{C}$. However, I have also seen the conjecture stated as: \begin{equation}\tag{2}\mathrm{ord}_{s=1}L(E,\rho,s)= \text{multiplicity of $\rho$ in $E(K)\otimes\mathbb{C}.$} \end{equation}
Here is my confusion: If the dimension of $\rho$ is not 1, aren't these different statements?
For some context, in the representation theory textbooks I have read, the multiplicity and isotypical components are defined as follows:
Given a finite-dimensional complex representation V (of a finite group G), we can canonically decompose V into a direct sum of its irreducible components $$V=\bigoplus_{I\in\mathrm{\,IrrReps}(G)}I^{\oplus\, m(I,V)}\quad\quad\text{(with $m(I,V)=0$ for almost all $I$)}.$$ Then, for each $I$, they define:
The $I$-isotypical component of $V$ is the summand $V^{(I)}:=I^{\oplus\, m(I,V)}$.
The multiplicity of $I$ in $V$ is the exponent $m(I,V)$.
In particular, we have $$\mathrm{dim}_\mathbb{C} V^{(I)} = m(I,V) \,\mathrm{dim}_\mathbb{C} I.$$
So, going back to conjectures (1) and (2), if $\mathrm{dim}_\mathbb{C} \rho\neq 1$, the statements are not the same. Which one is correct? Have I misunderstood something?
I suspect some authors mean a different thing when they say "$\rho$-isotypical component", which may be the source of my confusion.
Having worked in this area a fair bit myself, I can confirm that the correct statement of equivariant BSD is statement (2), namely
$\operatorname{ord}_{s = 1} L(E, \rho, s) = \operatorname{dim} \operatorname{Hom}_{\operatorname{Gal}(K / \mathbf{Q})}(\rho, E(K) \otimes \mathbf{C})$
for any $K$ such that $\rho$ factors through $\operatorname{Gal}(K / \mathbf{Q})$.
As for the terminology: "multiplicity space" is probably the more correct term for this Hom-space, while "$\rho$-isotypic component" should be the sum of the images of all the Homs in this space. But some authors do indeed use "isotypic component" to mean the Hom-space itself (even though it is not a subspace of $E(K) \otimes \mathbf{C}$); it's just a quirk of the literature one needs to get used to.