Bounded cohomology of the \(p\)-adic upper half plane (Q2906516)

Starting with the fundamental works of \textit{Yu. I. Manin} [Mat. Sb., N. Ser. 92(134), 378--401 (1973; Zbl 0293.14007)], \textit{Y. Amice} and \textit{J. Velu} [Astérisque 24--25, 119--131 (1975; Zbl 0332.14010)] and \textit{M. M. Višik} [Math. USSR, Sb. 28(1976), 216--228 (1978; Zbl 0369.14010)], the theory of \(p\)-adic distributions, defined as continuous functionals on a suitable space of analytic or locally analytic functions over a \(p\)-adic domain, has shown many fruitful arithmetic applications. Among those, we mention the construction of \(p\)-adic \(L\)-functions for modular forms (this was Manin's, Amice-Velu's and Višik's original motivation), as well as the related study of general \(\mathcal{L}\)-invariants (see Section 2 in [\textit{S. Dasgupta} and \textit{J. Teitelbaum}, University Lecture Series 45, 65--121 (2008; Zbl 1153.14021)] for a description of these applications). The survey under review presents in a slighlty innovative perspective the theory of bounded cohomology on Drinfeld's \(p\)-adic upper half plane connecting it to a space of distributions.NEWLINENEWLINEIn more detail, fix a prime number \(p\) and let \(K\) be a \(p\)-adic field, whose completion of an algebraic closure we denote by \(\mathbb{C}_p\). The space \(\mathcal{X}=\mathbb{P}^1(\mathbb{C}_p)\setminus\mathbb{P}^1(K)\) introduced by \textit{V. G. Drinfel'd} in [Funct. Anal. Appl. 10, 107--115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29--40 (1976; Zbl 0346.14010)] plays a crucial role in the \(p\)-adic uniformization of Shimura curves (see [\textit{J.-F. Boutot} and \textit{H. Carayol}, Astérisque 196--197, 45--158 (1991; Zbl 0781.14010)]) and its de Rham cohomology \(H^1_{dR}(\mathcal{X})=\Omega/d\mathcal{O}\) is isomorphic to the space of harmonic \(1\)-cochains on the Bruhat-Tits tree \(\mathcal{T}\) of \(G=\mathrm{SL}_2(K)\). There is a notion, both for a harmonic cochain and for a differential form, of being \textit{bounded}, and we append a superscript \(^{\mathrm{bnd}}\) to denote subspaces of bounded elements. The group \(G\) acts naturally both on forms in \(\Omega\) and on \(1\)-cochains and respects bounded subspaces: moreover, there is a ``residue'' \(G\)-isomorphism NEWLINE\[NEWLINE \mathrm{res}:H^1_{\mathrm{dR}}(\mathcal{X})\overset{\cong}{\longrightarrow}C^1_{\mathrm{har}} NEWLINE\]NEWLINE given by interpreting an oriented edge \(e\) of the tree \(\mathcal{T}\) as an annulus in \(\mathcal{X}\) and defining, for \(\omega\in\Omega\), \(c_\omega\) as the cochain \(e\mapsto \mathrm{res}_e(\omega)\). Given any finite dimensional \(G\)-representation \(M\) over \(K\), one can tensor the above objects to find analogous spaces \(\Omega(M),C^1_{\mathrm{har}}(M),H^1_{\mathrm{dR}}(\mathcal{X};M)\dots\) and likewise their bounded subspaces. In this generality, there is a filtration \(F^\cdot \Omega(M)\) of the rigid forms \(\Omega(M)\) by \(G\)-stable submodules, and the main result of the paper is a refinement of the residue isomorphism as follows:NEWLINENEWLINE{Theorem 1.1.} The residue isomorphism induces an isomorphism NEWLINE\[NEWLINE F^n\Omega(M)\cap \Omega(M)^{\mathrm{bnd}}\cong C^1_{\mathrm{har}}(M)^{\mathrm{bnd}} NEWLINE\]NEWLINE where \(n\) is the biggest integer such that \(F^n\Omega(M)\neq 0\). In other words, every bounded \(M\)-valued harmonic cochain is of the form \(c_\omega\) for some \(\omega\in F^n\Omega(M)\cap\Omega(M)^{\mathrm{bnd}}\) and if a bounded \(\omega\) which lies in the last step of the filtration is exact, then it is zero.NEWLINENEWLINEAs the author himself comments, the result is not entirely new, although the approach he takes might be original. The connection with distributions hinted at in the first paragraph of the review comes from Morita duality which gives a perfect pairing NEWLINE\[NEWLINE\mathcal{C}(M)/\mathcal{R}(M)\times\Omega(M)\longrightarrow \mathbb{C}_p \tag{1}NEWLINE\]NEWLINE where \(\mathcal{C}(M)\) are suitable ``meromorphic'' functions with values in \(M\) and \(\mathcal{R}(M)\) are rational functions with controlled poles; in particular, (1) allows one to interpret \(\Omega(M)\) as a space of distribution, and the proofs of injectivity and surjectivity in Theorem 1.1 both come from interpreting the involved steps in the filtration on \(\Omega(M)\) in terms of distribution.NEWLINENEWLINEThe paper is remarkably well-written in an informal and readable form. It is basically self-contained, although basic notions about the Bruhat-Tits tree \(\mathcal{T}\) and \(p\)-adic distributions are required. The proof of surjectivity rests on the results of Amice and Vélu, while injectivity relies on explicit computations of norms in the relevant \(p\)-adic Banach spaces. Section 1 is mainly notational, although it introduces in some detail much of the needed tools. Section 2 concerns Morita duality and its compatibility with the filtrations, while Section 3 contains the proof of Theorem 1.1.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00048].