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Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(K_\rho/\mathbb{Q})$. The equivariant BSD conjecture states that $$\mathrm{ord}_{s=1}L(E\otimes\rho,s) \; = \; \text{multiplicity of $\rho$ in $E(K_\rho)\otimes\mathbb{C}$}.$$

When $\rho$ is the trivial representation, the above conjecture reduces to the classical BSD conjecture. In this case, as far as I understand, the conjecture is almost completely proven for rank 0 and 1, while for higher ranks we have computational evidence.

My question/reference request is: Do we also have (computational) evidence for the above conjecture when the dimension of $\rho$ is $\ge2$ and the multiplicity is also $\ge2$? I would be happy to see any numerical examples, papers, or even code. I'm having a hard time even finding elliptic curves in which a non-trivial representation appears twice in $E(K_\rho)\otimes\mathbb{C}$. I guess they exist, but as in the classical conjecture, higher multiplicity might be extremely rare.

As a special case, I'm particularly interested in examples where $\rho$ is a 2-dimensional odd irreducible Artin representation (so it is associated to a weight 1 eigenform). I have been learning about the subject, and I see that many of the known results concern such representations. In this setting, one can use the Rankin-Selberg method to study the $L$-function, potentially leading to some very interesting examples.

On a side note: I would be pleased to know if someone has already implemented the objects on the right-hand side of the conjecture in SageMath (or in some other software).

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    $\begingroup$ Good question. I don't think there is a large amount of examples like this ever been computed or written down. I might have some, I will have to search for them. There is a folklore conjecture which states that ranks should be 100% of the time minimal (0 or 1 imposed by parity conditions), which will tell you that indeed it is rare to find examples of your kind. But over $\mathbb{Q}$ we also find rank $2$ curves. $\endgroup$
    – Chris Wuthrich
    Commented Jun 13 at 15:45
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    $\begingroup$ The sage question: Sage has Denis Simon's gp script for calculating the rank of elliptic curve over number fields, but the search for points is often more efficient using magma. It turns out this is often the hardest part of evaluating your right hand side. Once you have the points it is not hard to decompose it into irreducible factors using characters. But none is systematically implemented. $\endgroup$
    – Chris Wuthrich
    Commented Jun 13 at 15:48
  • $\begingroup$ @ChrisWuthrich Thank you for pointing that out! I went to check Simon's script, which is also used by the method '.gens()' in SageMath. But I'm puzzled by the following comment in SageMath's documentation: "Contrary to what the name of this method suggests, the points it returns do not always generate a subgroup of full rank in the Mordell-Weil group, nor are they necessarily linearly independent." So I wonder, can we still compute the RHS with that? Perhaps, in some cases, we can guarantee (by some other reason) that such is list of points is complete, and then we focus on these cases? $\endgroup$
    – J M T P
    Commented Jun 17 at 10:59
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    $\begingroup$ Yes, there is the method .saturation. That works if the algorithm has found points that generate a subgroup of finite index. However over a large degree number field, it can very well happen that the heights of the generators are so large that the search won't find them. $\endgroup$
    – Chris Wuthrich
    Commented Jun 17 at 11:27

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You might want to study the work of Darmon--Lauder--Rotger, notably this paper: https://web.mat.upc.edu/victor.rotger/docs/DLR1.pdf

They study cases of the equivariant BSD conjecture where $\rho$ is a tensor product $\rho_g \otimes \rho_h$ of modular 2-dimensional representations and $L(E, \rho, s)$ vanishes at $s = 1$ to positive even order, hence $\ge 2$. They formulate some rather precise conjectures about constructing points by $p$-adic analytic methods, and prove those in certain cases, but (more relevantly for this question) also amass an impressive amount of numerical data confirming their conjectures for particular elliptic curves.

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  • $\begingroup$ Wow, thank you for the reference! That is what I wanted to see. Many interesting examples. $\endgroup$
    – J M T P
    Commented Jun 17 at 11:02

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