Analytic continuation on Siegel varieties
The article studies the rigid-analytic continuation of overconvergent modular forms on Siegel modular threefolds, and gives a criterion for such forms to be classical. This is an important contribution to our understanding of congruences of Siegel modular forms; the analogous results were crucial in the study of forms on modular curves, and have interesting arithmetic consequences. Fix a prime \(p\) and an integer \(N\geq 3\) not divisible by \(p\). Let \(\mathcal{O}_K\) be the ring of integers of a finite extension \(K/\mathbb{Q}_p\). Let \(X\) denote the moduli space of principally polarized abelian surfaces over \(\mathcal{O}_K\) with principal level-\(N\) structure and Iwahori level structure at \(p\) (the latter means a choice of totally isotropic subgroups \(H_1\subset H_2\) of the \(p\)-torsion group). Given a weight \(\kappa=(k_1\geq k_2)\in\mathbb{Z}^2\), the Hodge bundle is a locally free sheaf \(\omega^\kappa\) on \(X\), constructed from the sheaf of relative differentials. The space of classical Siegel modular forms under consideration is the space of global sections \(M(\kappa, X)=H^0(X,\omega^\kappa)\). If \(\bar{X}\) denotes a toroidal compactification of \(X\), then \(\omega^\kappa\) has a canonical extension to \(\bar{X}\) and the Koecher principle says that \(H^0(\bar{X},\omega^\kappa)=H^0(X,\omega^\kappa)\). There are at least three natural ways of defining overconvergent forms in this setting, and it is not known whether they agree (they coincide when \(g=1\)). Let \(X_{\text{rig}}\) be the rigid-analytic fiber of the formal completion of \(X\) along its special fiber; similarly get \(\bar{X}_{\text{rig}}\) from \(\bar{X}\). The ordinary locus \(Y\subset X_{\text{rig}}\) is the open formal subscheme classifying abelian surfaces with ordinary reduction and \(H_2\cong\mu_p^2\). The analogous consideration gives the ordinary locus \(\bar{Y}\subset\bar{X}_{\text{rig}}\). The various spaces of overconvergent forms are built from \(Y\) or \(\bar{Y}\) as follows: \(M(\kappa, X)^\dagger\) is the space of rigid-analytic sections of \(\omega^\kappa\) on a strict neighborhood of \(Y\) in \(X_{\text{rig}}\). \(M(\kappa, X, \bar{X})^\dagger\) is the space of sections on a strict neighborhood of \(Y\) in \(\bar{X}_{\text{rig}}\). \(M(\kappa, \bar{X})^\dagger\) is the space of sections on a strict neighborhood of \(\bar{Y}\) in \(\bar{X}_{\text{rig}}\). It is known that \[ M(\kappa, X)\hookrightarrow M(\kappa, \bar{X})^\dagger\hookrightarrow M(\kappa, X, \bar{X})^\dagger\hookrightarrow M(\kappa, X)^\dagger \] (The first inclusion is due to the author [Bull. Soc. Math. Fr. 140, No. 3, 335--400 (2012; Zbl 1288.11063)]). The natural question is to give a good description of the image of the classical space \(M(\kappa,X)\) inside \(M(\kappa, X)^\dagger\). This is the main result of the article (Theorem 1.2): If \(F\in M(\kappa, X)^\dagger\) is an eigenvector of the operator \(U_p\) with eigenvalue \(a_p\) and \(k_2 > v(a_p)+3\), then \(F\) is classical. As an immediate consequence, any \(p\)-adic cuspidal ordinary form of weight \(k_1\geq k_2\geq 4\) is classical. The main result generalizes the well-known criterion of \textit{H. Hida} [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 2, 231--273 (1986; Zbl 0607.10022)] and \textit{R. F. Coleman} [Invent. Math. 124, No. 1--3, 215--241 (1996; Zbl 0851.11030)] for \(g=1\); this was later reproved by \textit{K. Buzzard} [J. Am. Math. Soc. 16, No. 1, 29--55 (2003; Zbl 1076.11029)] and \textit{P. L. Kassaei} [Duke Math. J. 132, No. 3, 509--529 (2006; Zbl 1112.11020)], who employ techniques of rigid-analytic continuation. It is this latter approach that the author uses in establishing his theorem. Buzzard's insight was to rewrite the identity \(U_p F= a_p F\) as \(a_p^{-1} U_p F=F\) (for \(a_p\neq 0\)), and to view it as a functional equation that relates the value of \(F\) at \(x\in X_{\text{rig}}\) to its values at the finitely many points associated to \(x\) under the Hecke correspondence \(U_p\). This idea allows one to extend the original domain of definition of \(F\) by successive applications of \(U_p\). In order to realize this program, the author starts by studying the dynamical system defined by \(U_p\) (and similar operators) on the space \(X_{\text{rig}}\). He does this for general \(g\geq 2\) and identifies the obstruction to analytic continuation. In the second part of the paper, the author performs a delicate analysis of the Kottwitz-Rapoport stratification of the special fiber of \(X\), and the interaction of the strata and the correspondence \(U_p\). He then obtains the desired analytic continuation, first to an formal open subscheme whose complement has codimension \(2\), and finally to all of \(X_{\text{rig}}\).
scientific article; zbMATH DE number 5883141
scientific article