Katz type $p$-adic $L$-functions for primes $p$ non-split in the $\mathrm{CM}$ field

Abstract

For every triple $F$, $K$, $p$ where $F$ is a classical elliptic eigenform, $K$ is a quadratic imaginary field and $p$ is an odd prime integer which is not split in $K$, we attach $p$-adic $L$-function which interpolates the algebraic parts of the special values of the complex $L$-functions of $F$ twisted by algebraic Hecke characters of $K$ such that the $p$-part of their conductor is $p^n$, with $n$ large enough (for $p\ge 5$ it suffices $n\ge 2$). This construction extends a classical construction of N. Katz for $F$ an Eisenstein series, and of Bertolini–Darmon–Prasanna for $F$ a cuspform when $p$ is split in $K$. Moreover, we prove a Kronecker limit formula, respectively, $p$-adic Gross–Zagier formulae, for our newly defined $p$-adic $L$-functions.