Geometric modular forms and the cohomology of torsion automorphic sheaves (Q2900284)

From the geometric point of view, modular forms of level \(\Gamma\) can be interpreted as sections of certain holomorphic line bundles over the compactified modular curve \(X_{\Gamma}\), which in turn can be related with the cohomology with coefficients in \(\mathbb{C}\) of the open modular curve \(Y_{\Gamma}\), say via the Eichler-Shimura isomorphism. These objects admit nice integral models, which make it possible to define modular forms in mixed characteristics (not just \(\mathbb{C}\)). This partly explain the arithmetical nature of modular forms.NEWLINENEWLINEThis setting can be generalized to higher dimension upon replacing (1) the modular curve by Shimura varieties together with their compactifications, (2) the Eichler-Shimura isomorphism by Faltings dual BGG spectral sequence, together with its degeneracy due to mixed Hodge theory. On the other hand, people know that the torsion part in the cohomology of Shimura varieties can have arithmetical significance. Hence it is of utmost importance to develop a mixed-characteristics theory in higher dimension, at least for Shimura varieties of PEL-type and ``good'' residual characteristic \(p\). This is the main concern of the first part of the survey article under review.NEWLINENEWLINEThe author succeeded in developing a theory comprising (i) integral models for Shimura varieties of PEL-type, (ii) the toroidal compactifications thereof, (iii) automorphic sheaves, which permit to define modular forms, and (iv) canonical extensions of the automorphic sheaves; all of them work for ``good'' mixed characteristics \(p\). In particular, this provides an important extension of the work of \textit{G. Faltings} and \textit{C.-L. Chai} [Degeneration of abelian varieties. Berlin etc.: Springer-Verlag (1990; Zbl 0744.14031)].NEWLINENEWLINEIn the second part, the author studies the vanishing, freeness, and liftability of the space of such modular forms. One of the crucial ingredients thereof is the vanishing theorem in characteristic \(p\) of \textit{H. Esnault} and \textit{E. Viehweg} [Lectures on vanishing theorems. Notes, grew out of the DMV-seminar on algebraic geometry, held at Reisensburg, October 13--19, 1991. Basel: Birkhäuser Verlag (1992; Zbl 0779.14003)]. Such results are surely crucial in concrete applications, for example the liftability of mod \(p\) modular forms, cf. [\textit{K.-W. Lan} and \textit{J. Suh}, Int. Math. Res. Not. 2011, No. 8, 1870--1879 (2011; Zbl 1233.11042)].NEWLINENEWLINEAltogether, this survey article gives a friendly introduction to the geometric theory of modular forms, with plenty of references and helpful remarks. It also serves as a nice summary to the author's long thesis on the arithmetic compactifications of PEL-type Shimura varieties.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].