Finite Flatness of Torsion Subschemes of Hilbert-Blumenthal Abelian Varieties Jordan S. Ellenberg February 29, 2000 Contents 1 HBAV’s and Hilbert modular forms 4 1.1 Elliptic curves with multiplicative reduction ............ 4 1.2 Definitions and examples ....................... 6 1.3 Semi-HBAV’s ............................. 8 1.4 The Tate HBAV and q-expansion .................. 9 1.5 Uniformization of HBAV’s ...................... 12 2 Discriminantal sets of Hilbert modular forms 15 2.1 Vertices and edges .......................... 15 2.2 The main theorem .......................... 19 2.3 Existence of discriminantal sets ................... 23 2.4 Examples ............................... 25 Abstract Let E be a totally real number field of degree d over Q. We give a method for constructing a set of Hilbert modular cuspforms f 1 ,... ,f d with the following property. Let K be the fraction field of a complete dvr A, and let X/K be a Hilbert-Blumenthal abelian variety with mul- tiplicative reduction and real multiplication by the ring of integers of E. Suppose n is an integer such that n divides the minimal valuation of fi (X) for all i. Then X[n ′ ]/K extends to a finite flat group scheme over A, where n ′ is a divisor of n with n ′ /n bounded by a constant depending only on f 1 ,... ,f d . When E = Q, the theorem reduces to a well-known property of f 1 =Δ. In the cases E = Q( √ 2) and E = Q( √ 5), we produce the de- sired pairs of Hilbert modular forms explicitly and show how they can be used to compute the group of N´ eron components of a Hilbert-Blumenthal abelian variety with real multiplication by E. 1