Timo Keller

We report on work in progress with Mulun Yin.
Castella--Gross--Lee--Skinner recently proved Perrin-Riou's Heegner
point main conjecture for modular abelian varieties at odd primes $p$ of
good reduction for which the mod-$p$ Galois representation $\rho_p$ is
reducible (``Eisenstein primes''). They have the restriction that the
characters in the semisimplification of $\rho_p$ are non-trivial on
$\operatorname{Gal}_{\mathbb{Q}_p}$. For example this excludes the case when there is a
non-trivial $p$-torsion point.

We are working on removing this restriction and generalize the result to
newforms of higher weight, allowing us to also treat bad multiplicative
reduction using Hida theory. As a consequence, we get the $p$-part of the
Birch--Swinnerton-Dyer conjecture for analytic rank 1 (and 0 by
Castella--Grossi--Skinner) and a $p$-converse theorem.

Combining this with previous results of Kato, Skinner--Urban, Skinner,
$\ldots$, Castella--Ciperiani--Skinner--Sprung, we get the strong BSD
conjecture in analytic rank 0 and 1 for squarefree level $N$ under a mild
condition on the discriminant except maybe for the 2-part.