On Iwasawa invariants of modular forms with reducible and non-$p$-distinguished residual Galois representations

In the present paper, we study the $p$-adic $L$-functions and the (strict) Selmer groups over $\mathbb{Q}_{\infty}$, the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, of the $p$-adic weight one cusp forms $f$, obtained via the $p$-stabilization of weight one Eisenstein series, under the assumption that a certain Eisenstein component of the $p$-ordinary universal cuspidal Hecke algebra is Gorenstein. As an application, we compute the Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with the same residual Galois representations as the one of $f$, which in our setting, is reducible and non-$p$-distinguished. Combining this with a result of Kato \cite[Theorem~17.4.2]{kato04}, we prove the Iwasawa main conjecture for these forms. Also, we give numerical examples that satisfy the Gorenstein hypothesis. The crucial point on the analytic counter part is that under the Gorenstein hypothesis, we are able to define, following Greenberg--Vatsal, the $p$-adic $L$-functions of $p$-adic weight one forms $f$ as an element in the one-dimensional Iwasawa algebra by using Mazur--Kitagawa two-variable $p$-adic $L$-function and then, to compute them explicitly via local explicit reciprocity law. On the algebraic counter part, we compute the (strict) Selmer groups of $f$ over $\mathbb{Q}_{\infty}$ via the knowledge of the Galois representations of $f$ studied in \cite{BDP}.