A new method of constructing $p$-adic $L$-functions associated with modular forms

We give a new method of constructing admissible $p$-adic measures associated with modular cusp eigenforms, starting from distributions with values in spaces of modular forms. A canonical projection operator is used onto the characteristic subspace of an eigenvalue $\alpha$ of the Atkin–Lehner operator $U_p$. An algebraic version of nearly holomorphic modular forms is given and used in constructing $p$-adic measures.