In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasiprojective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper-and super-elliptic Jacobians that have large ranks over function fields of certain varieties.
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For fixed θ this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2 )x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases.1991 Mathematics Subject Classification. 11G05.
Let K = Q( √ m) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) = s/r with 0 < |s| < r and gcd(r, s) = 1. A positive integer n is called a (K, θ)-congruent number if there is a triangle, called the (K, θ, n)-triangles, with sides in K having θ as an angle and nα θ as area, where α θ = √ r 2 − s 2 . Consider the (K, θ)-congruent number elliptic curve E n,θ : y 2 = x(x + (r + s)n)(x − (r − s)n) defined over K. Denote the squarefree part of positive integer t by sqf(t). In this work, it is proved that if m = sqf(2r(r − s)) and mn = 2, 3, 6, then n is a (K, θ)-congruent number if and only if the Mordell-Weil group E n,θ (K) has positive rank, and all of the (K, θ, n)-triangles are classified in four types.
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