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Kobayashi formulated the analog of the main conjecture in Iwasawa's theory for elliptic curves which are supersingular at a prime p. This makes use of the $\pm$ Selmer groups which are shown to be cotorsion with the use of the $\pm$ Coleman maps, and the definition of $\pm$ L-functions (with bounded coefficients) due to Pollack. For cuspidal eigenforms for which $a_p=0$, Lei (2011) generalized these maps to formulate an analog of this conjecture.

What is the state of the art on these conjectures today in the non CM case?

Also, has there been any attempt at relaxing Lei's $a_p=0$ condition?

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    $\begingroup$ Xin Wan and myself have a proof of the Main Conjecture for eigencuspforms of weight $k≥2$ under a few technical assumptions on the residual representation (which in particular exclude the CM case). In particular, there is no assumption on the reduction type at $p$ or on the value of $a_p$. $\endgroup$
    – Olivier
    Commented May 1, 2019 at 9:07

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Theorem (Xin Wan): If $E$ is an elliptic curve of square-free conductor $N$, and $p \ge 3$ is a prime such that $p \nmid N$ and $a_p(E) = 0$, then Kobayashi's $\pm$ Iwasawa main conjectures are true for $E$ (for both choices of sign).

The proof is in this paper. As the introduction to the paper notes, the assumption of $N$ being squarefree could potentially be relaxed, but I don't know if anyone has pushed this further.

Supersingular elliptic curves with $a_p \ne 0$ can only occur for $p = 2$ or $p = 3$. The case $p = 3$ has been proved by Sprung in this paper, using an extension of Wan's methods.

As for higher weight modular forms, Lei formulated a main conjecture for $a_p = 0$ in his PhD thesis as you apparently know, and there is a more general formulation covering the $a_p \ne 0$ case due to Lei, Zerbes and myself (2010). This conjecture has been proved under some technical assumptions by Wan in this paper.

So the case of good ordinary and good supersingular reduction is getting to be quite well understood now. On the other hand, for bad (additive) reduction we know almost nothing

In the bad additive reduction case, there is (AFAIK) no satisfactory way to define a p-adic L-function in this case, hence no way to formulate a main conjecture in the form "p-adic L-function = char. ideal of torsion Selmer group". However, are other approaches to formulating a main conjecture: in particular, there is Kato's formulation from his 2004 Asterisque paper, relating the size of an $H^2$ to the index of a special element in an $H^1$. Under a modest "big Galois image" assumption, Kato proves one inclusion in this formulation, and I learned from Olivier's comment that he and Wan have proved the other inclusion under similar hypotheses, so Kato's main conjecture is now a theorem in this case as well.

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  • $\begingroup$ I'm aware that $a_p=0$ for supersingular elliptic curves and that this follows from the Weil bound, my second question is directed at modular forms of weight>2 or with field of coefficients larger than $\mathbb{Q}$, thanks for the references. $\endgroup$
    – user130124
    Commented May 1, 2019 at 6:51
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    $\begingroup$ On the other hand, for bad (additive) reduction we know almost nothing I cannot quite agree $\endgroup$
    – Olivier
    Commented May 1, 2019 at 9:08
  • $\begingroup$ Thank you, have some of these results been extended to the noncommutative Iwasawa theory setting? $\endgroup$
    – user130124
    Commented May 1, 2019 at 13:40
  • $\begingroup$ @Olivier: You are right, of course, I was thinking too narrowly (failing to take into account formulations without p-adic L-functions). $\endgroup$
    – David Loeffler
    Commented May 1, 2019 at 13:42
  • $\begingroup$ @AnweshRay No, and this seems very far off indeed with current technology. $\endgroup$
    – David Loeffler
    Commented May 1, 2019 at 13:43

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