Explicit transformation of an intersection of two quadrics to an elliptic curve in Weierstrass form

This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular mapping) between a rationally defined smooth intersection of two quadrics in projective three-space and an elliptic curve in Weierstrass form which maps a distinguished rational point to the point at infinity. The usual approach of transforming a smooth plane cubic to a curve in Weierstrass form by mapping an inflection point to the point at infinity in a particular way is not applicable in our setting, because there may be no inflection point defined over the rationals. This difficulty is overcome by a construction dating back to Nagell. The results are exemplified in two situations of number-theoretical interest: Euler's problem of concordant forms and the occurrence of four rational squares in arithmetic progressions.