Finite flatness of torsion subschemes of Hilbert-Blumenthal abelian varieties (Q2706825)

Let \(K\) be the quotient field of a discrete valuation ring \(A\), and let \(X\) be a Hilbert-Blumenthal Abelian variety (HBAV) defined over \(K\), with real multiplication by the ring of integers of a totally real number field of degree \(d\). The author constructs a set \(S\) of \(d\) Hilbert modular forms associated to \(X\), and proves that if an integer \(n\) satisfies a certain divisibility condition phrased in terms of \(S\), then the torsion subscheme \(X[n]\) of the \(K\)-scheme \(X\) extends to a finite and flat group scheme over \(A\). If \(X\) is an elliptic curve, the corresponding result is due to \textit{J. Tate} [in: Elliptic curves, modular forms, and Fermat's last theorem, Proc. Conf. elliptic curves modular forms, Hong Kong 1993, Ser. Number Theory 1, 162-184 (1995; Zbl 1071.11508)]; in this case, the sufficient divisibility condition takes the form: \(n|\Delta\), where \(\Delta\) is the minimal discriminant of \(X\). According to the author, his ``ultimate aim is to investigate Diophantine equations associated to moduli spaces of HBAV's. The current work can be seen as the geometric portion of this arithmetic-geometric problem''. The author mentions, in particular, the generalised Fermat equation \(x^4+y^2=z^p\) considered in his Harvard thesis, whose solutions, ``under certain 2-adic conditions on \(x,y,z\), would produce non-modular HBAV's''.