Coherent cohomology and Galois representations
The authors study the coherent cohomology of Shimura varieties of Hodge type, using the integral \(p\)-adic ``strange'' models of the minimal compactifications of these varieties introduced by \textit{P. Scholze} in his preprint [``On torsion in the cohomology of locally symmetric varieties'', Preprint, \url{arXiv:1306.2070}] and the existence, as soon as the level is ramified enough at \(p\), of certain torsion modular forms which commute with the action of the Hecke algebra of level prime to \(p\). These so-called ``Hodge-Tate classes'' advantageously constitute a linear system without base point of the suitably modified Hodge sheaf, which is ample over the strange model. A computation ``à la'' Chech then allows to represent the coherent cohomology classes by sections defined on open sets of non-vanishing of certain Hodge-Tate classes, which can be \(p\)-adically approximated, while respecting the Hecke action, by classical modular forms. The main result asserts that higher coherent cohomology classes are \(p\)-adic limits of cuspidal modular forms. In the first two sections of the paper, the authors take up again Scholze's construction of a formal model of the minimal compactification, first of Siegel varieties of level \(p^n\), then of Shimura varieties of Hodge type, in order to build a formal Hodge-Tate period map between minimal and toroidal compactifications. The point is that automorphic coherent sheaves are defined over the Kottwitz toroidal compactifications, whereas the proof of the main theorem requires to work with the strange models of the minimal compactification. Once the bridge is built, the authors can associate Galois representations to some automorphic representations appearing in the coherent cohomology of Shimura varieties. More precisely, let \(p\) be a prime number, \(G\) a reductive group over \(\mathbb Q\), \(K=K_pK^p\) an open compact subgroup of \(\mathrm{GL}(\mathbb A_f)\). Given a Shimura data for \(G\), suppose that the associated Shimura variety \(X_G(K)\) is of Hodge type, defined over a number field \(E\). Fix an embedding \(E\hookrightarrow\mathbb C_p\) and consider \(X_G(K)\) as a scheme over \(\mathbb C_p\) . Let \(X_G(K)^{\mathrm{tor}}\) be a toroidal compactification and let \(\omega^\kappa\) be an automorphic coherent sheaf of weight \(\kappa\) over \(X_G(K)^{\mathrm{tor}}\). Let \(D\) be the divisor of \(X_G (K)^{\mathrm{tor}}\backslash X_G (K)\). Let \(\mathbb T\) be the \(\mathbb Z_p\)-Hecke algebra generated by the operators with level prime to \(K\) and to \(p\), which acts on the coherent cohomology groups \(H^i(X_G(K)^{\mathrm{tor}}\), \(\omega^\kappa)\) or \(H^i(X_G (K)^{\mathrm{tor}}\), \(\omega^\kappa(-D))\) for all \(i\geq 0\). Let \(\mathbb T^{p-ad}\) the \(p\)-adic Hecke algebra of level \(K^p\), i.e., the inverse limit over the levels \(K'= K'_pK^p\) and the finite sets \(\{\kappa_1,\dots,\kappa_r\}\) of the \(\mathbb Z_p\)-algebras. \(\mathbb T(K',\{\kappa_1,\dots,\kappa_r\})=\mathrm{Im}(T\to\mathrm{End}(H^0(X_G (K')^{\mathrm{tor}}\), \(\omega^{\kappa_1}(-D)\oplus\dots\oplus\omega^{\kappa_r}(-D)))\). The authors' main result then reads: The action of \(\mathbb T\) on the coherent cohomology groups \(H^i(X_G(K)^{\mathrm{tor}}\), \(\omega^\kappa)\) or \(H^i(X_G(K)^{\mathrm{tor}}\), \(\omega^\kappa(-D))\) factors to give a continuous action of \(T^{p-ad}\) for all \(i\geq 0\). Given a system \(\lambda\) of eigenvalues for the action of \(\mathbb T\) on a coherent cohomology group, if one could associate a system of Galois representations to each cuspidal holomorphic eigenform of regular weight, then one could associate to \(\lambda\) a system of Galois representations which verifies the local-global compatibility at the non-ramified places. This is the case for \(\mathrm{GSp}_4\), or an unitary group over a CM field, or \(\mathrm{Sp}_{2g}\) over a totally real field, assuming the stabilization of the twisted trace formula (which is supposed to always hold in the latter example), for which the authors give a detailed description of the Galois representations which appear.
scientific article; zbMATH DE number 6676669
scientific article