Rational 𝜃-parallelogram envelopes via 𝜃-congruent elliptic curves

Abstract

We introduce a new generalization of [Formula: see text]-congruent numbers by defining a rational [Formula: see text]-parallelogram envelope. It turns out that the existence of a rational [Formula: see text]-parallelogram envelope associated to an [Formula: see text] is connected to the property of [Formula: see text] being a [Formula: see text]-congruent number. Then, we study some problems related to the rational [Formula: see text]-parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of Ochiai, where only the case [Formula: see text] was considered. Moreover, we answer open questions raised by Ochiai in the more general setting of an arbitrary Pythagorean angle.