ON A TWO-PARAMETER FAMILY OF TROPICAL EDWARDS CURVES

抄録

In this paper, a certain two-parameter family of plane embeddings of the Edwards elliptic curve E_a ∶ x ² + y ² = a ² (1 + x ²y ²) is introduced to provide explicitly computed tropical curves corresponding to degeneration as a → 1. Applying the theta uniformization of E_a with the method of ultradiscretization given by Kajiwara et al. (Kyushu J. Math. 63 (2009), 315–338), we give a formula for the coordinate functions that traces the cycle part of the tropical elliptic curve. We also illustrate how one can recover the whole part of the tropical curve as a quotient of the Bruhat–Tits tree after Speyer’s algebraic approach in smooth cases.