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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?

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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.

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    $\begingroup$ 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. $\endgroup$ Apr 20 at 7:11
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    $\begingroup$ @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.) $\endgroup$ Apr 23 at 23:57
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If you want to solve a particular pair of these equations locally, and you know the intersection isn't singular, then you can always create a genus one model for the intersection on Magma and use the function IsLocallySoluble. See the the magma documentation for genus one models for more information.

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