  Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to
$p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique
$p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when
its Hecke polynomial at $p$ admits a single repeated root. In particular, we
study $p$-adic weight families through $f$ and its base-change to an imaginary
quadratic field $F$ where $p$ splits, and prove that the respective eigencurves
are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic
$L$-function over a Coleman family through $f$, and a three-variable $p$-adic
$L$-function over the base-change of this family to $F$. We relate the two- and
three-variable $p$-adic $L$-functions via $p$-adic Artin formalism. These
results are used in work of Xin Wan to prove the Iwasawa Main Conjecture in
this case.
  In an appendix, we prove results towards Hida duality for modular symbols,
constructing a pairing between Hecke algebras and families of overconvergent
modular symbols and proving that it is non-degenerate locally around any cusp
form. This allows us to control the sizes of (classical and Bianchi) Hecke
algebras in families.
