Let $X$ be a smooth and proper scheme over $\mathbb{Q}$ and choose integers $n,i$ such that $n>\frac{i}{2}+1$. Then we have
$$ ord\_{s=i+1-n}L(H^i(X),s)=\dim H^{i+1}\_{\mathcal{D}}(X\_\mathbb{R},\mathbb{R}(n))=\text{Ext}^1\_{\mathbb{R}-MHS}(\mathbb{R},H^i(X(\mathbb{C}),\mathbb{R}(n))$$
(see for instance 3.1.4 in [this survey](https://www.dpmms.cam.ac.uk/%7Eajs1005/preprints/d-s.pdf)). However I'm confused by the indexing: if we assume that $X$ is an elliptic curve then we hope/conjecture that $ord\_{s=1} L(H^1(X),s)$ is equal to $\dim X(\mathbb{Q})\otimes \mathbb{Q}\cong \text{Ext}\_{\mathcal{MM}}^1(1,H^1(X)(1))$. Then it would actually be natural to have $i=1,n=1$ to have
$$ord\_{s=1+1-1=1} L(H^1(X),s)=\dim \text{Ext}^1\_{\mathbb{R}-MHS}(\mathbb{R},H^1(X(\mathbb{C}),\mathbb{R}(1))\\
\overset{?}{=}\text{Ext}\_{\mathcal{MM}}^1(1,H^1(X)(1))\cong \text{rank } X(\mathbb{Q})$$
that is the Beilinson conjecture $(?)$ implies (the rank part of) the Birch/Swinnerton-Dyer conjecture (which makes sense since Beilinson's conjecture are supposed to generalize BSD). However this does not hold up as $1 \not>\frac{1}{2}+1$. Presumably my indexing is off by $1$ somewhere or this reasoning doesn't work and my question is which is it? Thanks!



Order of vanishing of $L$-function and mixed Hodge-structures
There is a good reason why *this particular form* of the Beilinson conjecture cannot possibly be valid for the particular $i$ and $n$ you mention.


The real $\mathbb{R}$-MHS $H^1(X(\mathbb{C}), \mathbb{R})$ is the same for all elliptic curves $X$: it's always a 2-dimensional pure Hodge structure of weight 1, with $(0, 1)$ and $(1, 0)$ parts 1-dimensional and interchanged by complex conjugation. So the dimension of $\operatorname{Ext}^1\_{\mathbb{R}-MHS}(\mathbb{R}, H^1(X(\mathbb{C}), \mathbb{R}(1)))$ is the same for all elliptic curves (it is always zero). However not all elliptic curves have the same rank!


A way of thinking about this is the following. Since $X$ is smooth proper, $L(H^i(X), s)$ is given by a *convergent* Euler product for $Re(s) > \tfrac{i}{2} + 1$. So for $n > \tfrac{i}{2} + 1$, the order of vanishing at $s = n$ is always 0, and hence the order of vanishing at $s = i + 1 - n$ can be read off from the shape of the functional equation (which depends only on the Hodge structure over $\mathbb{R}$). This is why it is reasonable to expect Ext groups of Hodge structures to fully capture the order of vanishing.


On the other hand, if $s = \tfrac{i + 1}{2}$ is the central value, then the order of vanishing of the L-function is a really deep, mysterious global invariant and in particular can't be read off just from knowing local information over $\mathbb{R}$. So this is why you should expect the formulation of the conjecture to be more difficult (and more interesting) in this central-value setting.


(The near-central value $s = \tfrac{i}{2} + 1$ is delicate for a different reason, because of vanishing of local Euler factors at finite primes.)


EDIT. I see you have $Ext^1\_{\mathcal{MM}}$ in your answer. There is always a map from $Ext^1\_{\mathcal{MM}}$ to $Ext^1\_{\mathbb{R}-MHS}$, but it isn't always an isomorphism. In the elliptic curve case we should always have $ord\_{s = 1}\, L(H^1(X), s) = dim\, Ext^1\_{\mathcal{MM}}$, but (if the rank is $> 0$) the map from $Ext^1\_{\mathcal{MM}}$ to $Ext^1\_{\mathbb{R}-MHS}$ will have a kernel.


