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

1906.10230Jun 24, 2019Care 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. $\phantom{a}$

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

Hagen Knaf, Erich Selder, Karlheinz Spindler

Algebraic Geometry Number Theory

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Abstract

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.

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The paper addresses: 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.

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Author comment: 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. $\phantom{a}$

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