# Finding rational points on intersection of quadrics in affine 3-space

Consider the subvariety of Spec $$\mathbb{Q}[x,y,z]$$ cut out by the equations $$\begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\ f_2 : a_2x^2 - z^2 - b_2^2 & = & 0 \end{eqnarray*}$$ where $$a_1,a_2,b_1,b_2$$ are positive integers, lets assume all distinct. By Legendre's theorem, the equation $$f_i = 0$$ has a solution in $$\mathbb{Q}$$ iff and only if $$-1$$ is a square modulo $$SF(a_i)$$, where $$SF$$ denotes the square-free part of an integer.

Is it possible to give a reasonable description of how the solution set of the system is affected by the properties of $$a_1,a_2,b_1,b_2$$? Or at least, is it possible to say anything interesting/nontrivial in this direction?

The closure in $$\mathbb{P}^3$$ of the intersection of your two quadrics is a smooth curve of genus $$1$$. If it has a rational point, it acquires the structure of an elliptic curve, and we know that determining the set of rational points on elliptic curves over $$\mathbb{Q}$$ is a very hard problem in general.
• The hardest part here is to determine if there is a rational point. This curve $C$ represents an element of order two in $H^1(\mathbb{Q},Jac(C))$. Even if it has points locally it may be hard to determine if it is trivial in the Tate-Shafarevich group though the Cassels pairing can help. These curves are exactly those appearing in 2-descent. See chapter X in Silverman. Apr 20 at 7:11
• @ChrisWuthrich Thank you for the reference to Silverman, though in his examples his method for dealing with solutions in local fields seems rather ad-hoc. Is there a more-or-less systematic way to see if $C$ has a point over a given local field? (Sorry for the very naive question.) Apr 23 at 23:57