HYPERELLIPTIC SIMPLE FACTORS OF J_{0}(N) WITH DIMENSION AT LEAST 3

HERMANN-JOSEF WEBER

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 hyperelliptic curve C of genus g whose jacobian variety 𝒥_{C} is isomorphic to A over ℂ? Our solution is based on an article by Poor. If such a hyperelliptic curve C exists, the next problem is the construction of the Rosenhain model C : Y^{2} = X (X − 1)(X − λ_{1})(X − λ_{2}) … (X − λ_{2g−1}) for pairwise distinct numbers λ_{j} ∈ 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 a method of Mestre can be generalized to get a minimal equation over ℤ [\frac{1}{2}] for the hyperelliptic curve C with jacobian variety 𝒥_{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 J_{0}(N) = 𝒥_{X_{0}(N)} of the modular curve X_{0}(N) we compute the corresponding curve equations by applying our algorithms to this special situation.