Hyperelliptic Simple Factors of J₀(N) with Dimension at Least 3

Abstract

We develop algorithms for three problems. Starting with a complex torus of dimension g ≥ 2, isomorphic to a principally polarized, simple abelian variety A/C, the first problem is to find an algorithmic solution of the hyperelliptic Schottky problem: Is there a hvperelliptic curve C of genus g whose jacobian variety I_c is isomorphic to A over C? Our solution is based on [Poor 1994]. If such a hyperelliptic curve C exists, the next problem is the construction of the Rosenhain model C : y² = X(X−1)(X−λ₁)(X−λ₂)…(X−λ_2g−1) for pairwise distinct numbers λ_j E ∈ C \ {0, 1}. Applying the theory of hyperelliptic theta functions we show that these numbers λ_j can easily be computed by using theta constants with even characteristics. If the abelian variety A is defined over a field k (this field could be the field of rational numbers, an algebraic number field of low degree, or a finite field), we show only in the case k = Q for simplicity, how the method in [Mestre 1991] can be generalized to get a minimal equation over Z[½] for the hyperelliptic curve C with jacobian variety I_c ≅_c A. This is our third problem. For some hyperelliptic, principally polarized and simple factors with dimension g = 3, 4, 5 of the jacobian variety J₀(N) = I_X0(N) of the modular curve X₀(N) we compute the corresponding curve equations by applying our algorithms to this special situation.