p-Adic Interpolation of Automorphic Periods for GL_{2}

We give a new and representation theoretic construction of $p$-adic interpolation
series for central values of self-dual Rankin-Selberg $L$-functions for
$GL_{2}$ 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 $GL_{2}$ 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 $GL_{2}$-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.

2010 Mathematics Subject Classification: Primary 11F67; Secondary 11F41, 11F33, 11F70

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