ARITHMETIC OF p ‐IRREGULAR MODULAR FORMS: FAMILIES AND p ‐ADIC L ‐FUNCTIONS

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