2019/06/24 by Hagen Knaf, Knaf, Hagen, Erich Selder +3
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Polynomial and algebraic computation #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.1906.10230
Care was taken to explain all construction steps both geometrically and arithmetically, to richly illustrate these steps with suitable diagrams, and to provide explicit formulas which can be easily implemented in any programming language. Thus the emphasis was on clarity, not on conciseness. $\phantom{a}$
arxiv created 2020/03/24 · arxiv updated 2020/03/26
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.