Classity in Hida's theory (Q2845584)

\textit{H. Hida} developed the theory of ordinary cuspidal \(p\)-adic Siegel modular forms in ``Control theorems of coherent sheaves on Shimura varieties of PEL type'' [J. Inst. Math. Jussieu 1, No. 1, 1--76 (2002; Zbl 1039.11041)]. It is conjectured that any \(p\)-adic modular form is classical, i.e., comes from a usual Siegel modular form, if its weight \((k_1,\dots,k_g)\) is large enough. The main result of this paper is to effectively prove this conjecture under a certain condition (on the prime number \(p\), the weight and the level of \(f\)). More precisely, the author proves that an ordinary cuspidal \(p\)-adic Siegel modular form of weight \((k_1,\dots,k_g)\) is classical if \(p>g(g+1)/2\), \(k_1>\dots>k_g>g+1\), \(\sum_i(k_i-k_g)<p-g(g+1)/2\) and \(f\) is of principal level at in integer \(n>12\) not divisible by \(p\).NEWLINENEWLINEA by-product of the proof of the main theorem is to uniquely lift cuspidal Siegel modular forms from \(\mathbb{F}_p\) to \(\mathbb{Z}[1/n]\), where \(n\) is as in the main theorem (i.e., integer \(n>12\) not divisible by \(p\)). The main ingredients to prove this lifting are: {\parindent=6mm \begin{itemize}\item[(1)] A theorem of Shepherd-Barron saying that \(\omega^k(-D)\) is ample, where \(\omega\) is the canonical bundle over a specific compatification \(\bar{\mathcal{A}_g}\) of the Siegel modular scheme over \(\mathbb{Z}[1/n]\), \(D\) is the boundary and \(k>n/12\). \item[(2)] The Berstein-Gelfand-Gelfand complex \(DR(\mathcal{V}_{\lambda})\) with filtration \(F^*DR\). The graduation complex of the BGG complex is then canonically quasi-isomorphic to a split complex associated with the sheaf of Siegel modular forms. \item[(3)] Ogus' vanishing theorem which claims that higher cohomologies of certain sheaves over \(\bar{\mathcal{A}}_{g,\mathbb{F}_p}\) vanish. It applies to the sheaf of Siegel modular forms twisted by \(-D\) by the quasi-isomorphism in (2). NEWLINENEWLINE\end{itemize}} Another auxiliary result to prove the main theorem is the vanishing of \(T_p\cdot f\bmod p\) on the non-ordinary locus of \(\mathcal{A}_g(\bar{\mathbb{F}}_p)\). This will give an extension result of ordinary Siegel modular forms mod \(p\) on the ordinary locus to the whole \(\bar{\mathcal{A}}_{g,\mathbb{F}_p}\). Then, with the lifting theorem and Hida's theory we can conclude.