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$p$-Adic Interpolation of Automorphic Periods for GL$_2$

机译：GL $ _2 $的$ p $-自守周期的Adic插值

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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.
机译：我们给出CM域二面塔中自对偶兰金-塞尔伯格$ L $函数的GL $ _2 $中心值的$ p $ -adic插值级数的新的表示理论构造，自构周期。这种结构的主要新颖之处在于其工作的普遍性，它完全是本地的。尽管可以采用类似的方法，但我们在此处给出了在与$ mathfrak {p} $-平常Hilbert形式的平行权重2和平凡字符相对应的完全实数域上GL $ _2 $的尖峰自构表示的构造，尽管可以采用类似的方法在可以选择基础GL $ _2 $表示形式的任何设置中，采用离散估值环中的值。借助Martin-Whitehouse和File-Martin-Pitale的定理，向量的特定选择使我们能够建立精确的插值公式。这样的插值公式是Bertolini-Darmon在以前的作品中推测的。我们的构造还为Cornut-Vatsal不变的定理提供了概念框架，因为它描述了潜在的theta元素。为了强调后一点，我们描述了当根数一般等于$ -1 $时，构造如何在并行权重两个设置中扩展以给出中心导数的$ p $ -adic插值序列，在这种情况下，公式可以使用元-张-张的坐标来给出四元离子Shimura曲线上CM点的高度的插值公式。

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