  We give a new and representation theoretic construction of $p$-adic
interpolation series for central values of self-dual Rankin-Selberg
$L$-functions for $\operatorname{GL}_2$ in dihedral towers of CM fields, using
expressions of these central values as automorphic periods. The main novelty of
this construction, apart from the level of generality in which it works, is
that it is completely local. We give the construction here for a cuspidal
automorphic representation of $\operatorname{GL}_2$ over a totally real field
corresponding to a $\mathfrak{p}$-ordinary Hilbert modular forms of parallel
weight two and trivial character, although a similar approach can be taken in
any setting where the underlying $\operatorname{GL}_2$-representation can be
chosen to take values in a discrete valuation ring. A certain choice of vectors
allows us to establish a precise interpolation formula thanks to theorems of
Martin-Whitehouse and File-Martin-Pitale. Such interpolation formulae had been
conjectured by Bertolini-Darmon in antecedent works. Our construction also
gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in
that it describes the underlying theta elements. To highlight this latter
point, we describe how the construction extends in the parallel weight two
setting to give a $p$-adic interpolation series for central derivative values
when the root number is generically equal to $-1$, in which case the formula of
Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of
heights of CM points on quaternionic Shimura curves.
