The non-existence of stable Schottky forms (Q2877497)

As it is well-known, the moduli space \(M_g\) of smooth complex projective curves of genus \(g\) can be embedded into the moduli space \(A_g\) of principally polarized abelian varieties of dimension \(g\) by mapping a curve to its Jacobian variety. The image of \(M_g\) under this embedding \(j_g:M_g\hookrightarrow A_g\) is called the Jacobian locus in \(A_g\), and the problem of its precise description is the famous Schottky problem.NEWLINENEWLINE The classical Satake compactification \(A^S_g\) of \(A_g\) and its boundary \(A^S_g\setminus A_g= A^S_{g-1}\) are well understood since more than fifty years. In particular, \(A^S_g\) is a normal projective variety with an ample \(\mathbb{Q}\)-line bundle \({\mathcal L}\), the so-called Hodge bundle, whose spaces \(H^0(A^S_g,{\mathcal L}^k)\) of global sections are just the spaces \([\Gamma_g,k]\) of modular forms of weight \(k\) on the Siegel upper half-space \(\mathbb{H}_g\), \(k\geq 1\), with respect to the Siegel modular group \(\Gamma_g= \text{Sp}(2,\mathbb{Z})\). Therefore one obtains a natural homomorphism \(\Phi:[\Gamma_g k]\to[\Gamma_{g-1}, k]\), the so-called Siegel operator, which is induced by restricting sections of \({\mathcal L}^k\) to the boundary of \(A^S_g\).NEWLINENEWLINE In this context, a classical result by \textit{E. Freitag} [Math. Ann. 230, 197--211 (1977; Zbl 0359.10026)] says that for \(g\geq 2k+2\), the Siegel operator is an isomorphism, and that the according space \([\Gamma_\infty,k]\) of stable modular forms can be explicitly described.NEWLINENEWLINE In the paper under review, the authors' main theorem is the following: If \(F\in[\Gamma_{g+1}, k]\) is a Siegel modular form on \(A_{g+1}\) that vanishes with multiplicity at least \(m\geq 1\) along the Jacobian locus \(j_{g+1}(M_{g+1})\subset A_{g+1}\), then \(\Phi(F)\) vanishes with multiplicity at least \(m+1\) along \(j_g(M_g)\).NEWLINENEWLINE Furthermore, it is shown that this result (Theorem 1.3.) has several consequences. First of all, the main theorem implies that there are no non-trivial stable forms vanishing on the Jacobian locus \(j_g(M_g)\) for every genus \(g\). On the other hand, if \(M^S_g\) denotes the closure of the Jacobian locus \(j_g(M_g)\) in the Satake compactification of \(A_g\), then \(A^S_g\) can be regarded as a subvariety of \(A^S_{g+m}\) for all pairs \((g,m)\in\mathbb{N}^2\), and \(M^S_{g+m}\cap A^S_g= M^S_g\) contains the \(m\)th order infinitesimal neighborhood of \(M^S_g\) inside \(A^S_g\) (Theorem 1.1.). This second main result of the paper shows that the intersection \(M^S_{g+m}\cap A^S_g\) is indeed far from being transverse. Actually, Theorem 1.1. may be regarded as a far-reaching generalization of a recent result by \textit{S. Grushevsky} and \textit{R. Salvati Manni} [Am. J. Math. 133, No. 4, 1007-1027 (2011; Zbl 1225.81114)], who proved that the genus 5 Schottky form, a difference of certain theta series, does not vanish along the Jacobian locus in \(A_5\).NEWLINENEWLINE It should be pointed out that the first author's Ph.D. thesis [Satake compactifications, lattices and Schottky problem. Cambridge: University of Cambridge, Selwyn College (2013)] contains further results in this direction. This thesis can be downloaded from his web site [\url{https://www.dpmms.cam.ac.uk/~gc438/}].