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.

  • $\begingroup$ One way to construct elements in $H^1_f(V)$ is to use the $p$-adic realization map from motivic cohomology. In you particular situation, elements in motivic cohomology have been constructed by Deninger for $n=k+1$ using torsion points of $E$ (see Higher regulators and Hecke L-series of imaginary quadratic fields I & II). One can show that these elements are linearly independent by computing the archimedean regulator (but the relation with the $L$-function is not known). I don't know if computing the $p$-adic regulator has ever been implemented. $\endgroup$ Sep 3, 2020 at 16:45
  • $\begingroup$ Recarding the case $k=1$, the motivic elements have been constructed by Beilinson. Gealy has computed their $p$-adic realisation in his PhD thesis. He also shows a result of the form ''$H^1(V)$ has rank $1$'' (as expected) but it's conditional on the Leopoldt-type conjecture that $H^2(V)$ is finite, and on Kato's main conjecture for $E$. $\endgroup$ Sep 3, 2020 at 16:53
  • $\begingroup$ You probably want some assumptions on the base number field. If it's not totally real or CM, then we know basically nothing. $\endgroup$ Sep 4, 2020 at 7:18
  • $\begingroup$ To be honest, I was really interested in elliptic curves over $\mathbb{Q}$ or maybe a quadratic imaginary field. So that's no issue $\endgroup$ Sep 6, 2020 at 2:44

1 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.

  • $\begingroup$ Thanks! What about over an imaginary quadratic field? In one case, I have a specific curve of rank $0$, and I'm interested in it over imaginary quadratic fields over which it has rank $1$. If I can show it explicitly for $k=2,3$, that's really all I need. (But I'm also interested in other cases, e.g., for certain rank $1$ curves over $\mathbb{Q}$.) $\endgroup$ Sep 6, 2020 at 2:48
  • $\begingroup$ I'd be happy to discuss this further but MO isn't really the place for extended conversations -- feel free to get in touch via my warwick.ac.uk email. $\endgroup$ Sep 6, 2020 at 7:36
  • $\begingroup$ To add to the second bullet-point, it seems that $\operatorname{Sym}^2{h^1(E)}(1)$ is covered by arxiv.org/abs/1411.7661 under some mild assumptions when the curve is modular (known in general e.g., over $\mathbb{Q}$ or a real quadratic field). $\endgroup$ Dec 30, 2020 at 22:58
  • $\begingroup$ Is it fair to say that $k=3$ would use something like Theorem D of arxiv.org/pdf/2003.05960.pdf? $\endgroup$ Jan 6, 2021 at 1:20
  • $\begingroup$ Morally yes, although there are some mildly fiddly technical issues meaning we don't have a full proof written out yet for Sym^3 of an elliptic curve. See arxiv.org/abs/2005.04786 for Sym^3 of modular forms of level 1 and big weight. $\endgroup$ Jan 6, 2021 at 9:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.