完备上同调与Beilinson-Kato系的分解
Une factorisation de la cohomologie complétée et du système de Beilinson-Kato
AI总结：
本文证明模符号$(0,\infty)$在经典点处插值Kato的Euler系，从而将Beilinson-Kato系分解为两个模符号的乘积，并利用$p$-进局部Langlands对应及完备上同调的Kirillov模型给出新证明与精细化。
摘要：
我们证明，模符号$(0,\infty)$作为Emerton完备上同调的对偶中的元素，在经典点处插值Kato的Euler系，并由此推出Beilinson-Kato系可以分解为两个符号$(0,\infty)$的乘积（这是Rankin方法的代数类比）。证明使用了${\mathbf GL}_2({\mathbf Q}_p)$的$p$-进局部Langlands对应以及Emerton关于模曲线塔的完备上同调的分解，对此我们给出了一个新的证明，该证明依赖于为完备上同调构造Kirillov模型，并且我们通过在经典点处施加条件对这一分解进行了精细化；这种精细化的存在体现了模形式的$p$-进周期所具有的解析性。
英文摘要：
We show that the modular symbol $(0,\infty)$, considered as an element of the dual of Emerton's completed cohomology, interpolates Kato's Euler system at classical points, and we deduce from this a factorisation of Beilinson-Kato's system as a product of two symbols $(0,\infty)$ (an algebraic analog of Rankin's method). The proof uses the $p$-adic local Langlands correspondence for ${\mathbf GL}_2({\mathbf Q}_p)$ and Emerton's factorization of the completed cohomology of the tower of modular curves for which we provide a new proof resting upon the construction of a Kirillov model for the completed cohomology, and which we refine by imposing conditions at classical points; the existence of such a refinement is a manifestation of an analyticity property for $p$-adic periods of modular forms.
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