AI 中文总结
该研究建立自守周期显式公式，为推广Bertolini-Darmon-Prasanna公式，证明适配p进插值的GU(1,1)周期公式，引入p亏格Hecke算子以关联自守周期与p进Abel-Jacobi映射像，服务于p进Rankin-Selberg L-函数特殊值研究。
AI 中文摘要
本文建立自守周期的显式公式，将用于后续研究p进Rankin-Selberg L-函数的特殊值。我们的动机是将Bertolini-Darmon-Prasanna公式推广到阿基米德局部符号与原设定相反的情形。为此，我们证明了从GSO(2)到GSp₄的θ提升的GU(1,1)周期公式，该公式适配p进插值。我们还为该θ提升引入了p亏格Hecke算子，它将在关联自守周期与p进Abel-Jacobi映射的像中起关键作用。
英文摘要
In this paper, we establish an explicit formula for automorphic periods which will be used in a sequel to study special values of $p$-adic Rankin-Selberg $L$-functions. Our motivation is to extend the Bertolini-Darmon-Prasanna formula to the case where the archimedean local sign is opposite to that in the original setting. To this end, we prove a formula for the $\mathrm{GU}(1,1)$-period of a theta lift from $\mathrm{GSO}(2)$ to $\mathrm{GSp}_4$, which is adapted to $p$-adic interpolation. We also introduce a $p$-depletion Hecke operator for this theta lift, which will play a crucial role in relating the automorphic period to the image of the $p$-adic Abel-Jacobi map.
