ELLIPTIC CURVES OVER Q(i)
نویسندگان
چکیده
A study of the diophantine equation v2 = 2u4 − 1 led the authors to consider elliptic curves specifically over Q(i) and to examine the parallels and differences with the classical theory over Q. In this paper we present some extensions of the classical theory along with some examples illustrating the results. The well-known diophantine equation v2 = 2u4 − 1 , has, ignoring signs, only two integer solutions, namely (u, v) = (1, 1) and (13, 239). This is not an easy result to prove (e.g. see [1]). The change of variable x = 2iv − 2 u2 , y = −4(v + i) u3 , transforms this equation into the elliptic curve y2 = x3 + 8x . The two integer solutions are transformed as follows: (1, 1) → (−2 + 2i,−4− 4i), (13, 239) → ( 2(−1 + 239i) 132 , −4(239 + i) 133 ) . Thus integer solutions to the diophantine equation become Gaussian rational points on the elliptic curve. This observation motivated us to look at elliptic curves specifically over Q(i) to examine the parallels and differences with the classical theory.
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