# Does Hasse-Minkowski help to produce nontrivial rational solutions?

Consider a quadratic form over $\mathbb{Q}$, say, a diagonal one in three variables $$F(X, Y,Z) = a · X^2 + b · Y^2 − c · Z^2$$ with positive integers $a,b,c$. Then $F(X,Y,Z)=0$ has a non-trivial rational solution by Hasse-Minkowski, if it has a nontrivial $p$-adic solution for the finite set of primes $p$ dividing $2 · a · b · c$. It is easy to formulate congruence conditions on $a, b, c$ such that this is the case, and Hasse-Minkowski guarantees the existence of a nontrivial rational solution. However, does the proof of Hasse-Minkowski help to produce any specific nontrivial rational solution ? I suppose this will not be easy, although by Mordell we know that the problem of finding rational and integer solutions is perfectly effective.

What can we say about explicitly finding rational solutions from the local solutions (in this case, or in general) ?

Edit: I found the following remark of B. Mazur here, on page $6$, which makes the question more precise: "But even in a concrete instance, where the Hasse-Minkowski Theorem guarantees the existence of a rational solution, there remains much yet to ponder about; for the proof of the Hasse-Minkowski Theorem does not produce any specific nontrivial rational solution easily, despite the fact that the problem of finding such solutions is perfectly effective. As Victor Miller pointed out, it would be good to give explicit bounds for the adelic version of this problem, i.e., given a finite set of prime numbers $p$ and specific $p$-adic points for these $p$, find rational points of prescribed closeness to these $p$-adic points."

• The buzzword here is "weak approximation". – Chandan Singh Dalawat Jan 10 '14 at 10:35
• See also the article ams.org/journals/mcom/2003-72-243/S0025-5718-02-01480-1/… – Chandan Singh Dalawat Jan 10 '14 at 10:38
• by Cremona and Rusin, of which the abstract reads : We present efficient algorithms for solving Legendre equations over $\mathbb Q$ (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known. – Chandan Singh Dalawat Jan 10 '14 at 10:40
• It seems you already know this reference : it is the [5] of your notes on Algebraische Zahlentheorie. – Chandan Singh Dalawat Jan 10 '14 at 10:46