Which cases of Beilinson-Bloch-Kato for elliptic motives are known?

Question

Let $V$ be a semisimple geometric Galois representation of a number field. Then the Bloch-Kato conjectures state that $$ \operatorname{ord}_{s=0}{L(V^*(1),s)} = \operatorname{dim}{H^1_f(G_k,V)}-\operatorname{dim}{H^0(G_k,V)}. $$

Beilinson has similar conjectures relating the LHS to algebraic K-theory rather than Selmer groups.

If $E$ is an elliptic curve, and we set $V=h_1(E)=h^1(E)(1)$, then the conjecture above is equivalent to the statement that the analytic rank is the same as the rank of the $p$-adic Selmer group. Beilinson's conjecture in this case is equivalent to the assertion that the analytic rank equals the Mordell-Weil rank.

For $V$ of non-negative weight, the conjecture simply asserts that the Selmer group vanishes. Assuming the conjectured properties of $L$-functions, the conjectures for $V$ and $V^*(1)$ are equivalent (in particular, the case of weight $\le -2$ follows from the case of weight $\ge 0$).

My question is: which cases of this conjecture are known for $V=\operatorname{Sym}^k{h^1(E)}(n)$ for $E$ an elliptic curve? I know that many cases are known when $k=n=1$ due to the theory of Heegner points, Gross-Zagier, etc, but I'd like to know what's known outside that range. (I'm especially interested in cases where $w=k-2n=-2, -3, -4$).

Feel free to give reference, or even better, specific elliptic curves in LMFDB.

Answer 1

There are three approaches I know of to studying $H^1_{\mathrm{f}}(K, V)$, where $V = Sym^k(h^1(E))(n))$. All rely on $E$ being modular, so let me assume this henceforth (of course, this is no assumption if $K = \mathbf{Q}$, or for some other small-degree fields).

• Via "anticyclotomic" Euler systems, such as Heegner points (and the closely-related method of "arithmetic level-raising"). This works extremely well when $k = n = 1$, and $K$ is totally real (or $K$ is CM and $E$ is base-extended from $K^+$); under these hypotheses we know the BK conjecture holds, for any $p$, whenever the analytic rank is 0 or 1 (Zhang, Nekovar). More generally, this might potentially be accessible for any $n$ and $k = 2n-1$, although huge amounts of work would be needed to carry that out. However, it's entirely impossible to generalise this approach beyond the case of motivic weight $w = -1$.

• Via modularity-lifting theorems. This gives a way of studying Selmer groups of representations that have the shape $W \otimes W^*$, where $W$ is irreducible. More generally, if $W$ has some extra structure (e.g. self-duality) which forces $W \otimes W^*$ to be reducible, then you can get some information about the cohomology of the pieces. This gives you very nice control over $Sym^k(h^1(E))(n)$ for $k = 2$ and $n = 1$ (or by duality $n = 2$) (Diamond--Flach--Guo). More generally, one should be able to get some information about general $n$ and $k = 2n$ or $2n-2$ using the recent work of Newton--Thorne; Theorem 5.6 of this paper tells you something about $W\otimes W^*$ where $W = Sym^k(h^1(E))$, and this representation breaks up as a sum of $Sym^{2m}(h^1(E))(m)$ for $0 \le m \le k$. [Caveat: I'm not sure exactly what their method gives; it's possible that you need to twist by an odd quadratic character at some point.] However, this is again restricted to specific values of $w$; it won't tell you anything unless $w = 0$ or $w = -2$.

• Via "cyclotomic" Euler systems, such as Kato's Euler system. This method has the advantage that it can tell you something about general motivic weights (i.e. a fixed $k$ and any $n \in \mathbf{Z}$). However, it gives you a criterion for vanishing of the $H^1$ in terms of p-adic L-functions, and these are only indirectly related to complex $L$-functions outside the critical range (i.e. away from $w = -1$ in your case). Nonetheless, these p-adic L-functions are computable, so you can check explicitly whether they vanish in examples. For elliptic curves over $\mathbf{Q}$, an Euler system for $Sym^k E$ exists for $k = 1$ due to Kato, and for $k = 2$ [*] and $k = 3$ due to Zerbes and myself (building on work of lots of other people). So, for example, if $K = \mathbf{Q}$, and $k = 1, 2, 3$, this would give an approach to proving the vanishing of $H^1_{\mathrm{f}}(\mathbf{Q}, Sym^k(h^1(E))(n))$ for your favourite elliptic curve and a specific but arbitrary value of $n$ (and $p$), using only a finite amount of computation.

[*] Actually there is a caveat here -- embarrassingly, I forgot the statement of my own theorem! -- so the result as published only applies to $Sym^2(E)$ twisted by a non-trivial Dirichlet character. But the un-twisted case might also be accessible with some extra work.