In his expanded lecture notes *Rational points on varieties*, Bjorn Poonen writes the following:



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> REMARK 2.5.3: There is an algorithm that, given a local field $k$ of characteristic $0$ and a $k$-variety $X$, decides whether $X(k)$ is nonempty. (...)
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> 


I would like to see if this algorithm can be applied to determine the existence of rational points on a very specific family of varieties over $\mathbb{Q}\_p$. Unfortunately, I could not find anything related on the web so far. Could you point to some literature on this matter, or explain some of the ideas?



Rational points on varieties over local fields
Let us assume that $X$ is smooth and projective for simplicity, given by a
number of polynomial equations with coefficients in the ring of integers
$\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$.
Reduce the equation moduloe the maximal ideal to get the reduced variety
$\bar X$. Enumerate the $\kappa$-points of $\bar X$ (this is a finite
problem). If $\bar X(\kappa) = \emptyset$, then $X(k)$ is empty.
If there is a smooth point $\bar P \in \bar X(\kappa)$
(and $\dim \bar X = \dim X$), then it lifts by Hensel's Lemma to a point
on $X(k)$, so $X(k)$ is non-empty. Otherwise, consider the lifts of
each point to points modulo the square of the maximal ideal. This comes
down to looking at equations $f(\dots, a\_i + \pi x\_i, \dots)$,
where $f$ runs through the equations defining an affine patch of $X$
containing a lift of $\bar P$, $a\_i$ are lifts
of the coordinates of $\bar P$ and $x\_i$ are variables; divide the equations
by the highest possible power of $\pi$, which is a generator of the
maximal ideal, and reduce mod $\pi$. This gives equations for the reduction
mod $\pi$ of a different model of $X$. Smoothness of $X$ guarantees
that the process will terminate after finitely many lifting steps.


If $X$ is not smooth, consider its singular locus $X'$ first (recursively).
If $X'(k)$ is non-empty, so is $X(k)$. Otherwise, run the previous
procedure on $X$. Since $X'(k)$ is empty, all points that lift indefinitely
will end up on the smooth part of $X$, which again gives termination.


Magma has an implementation (IsLocallySoluble). See
Nils Bruin, *Some ternary Diophantine equations of signature (n, n, 2)*,
In Bosma and Cannon, *Discovering Mathematics with Magma*,
Springer-Verlag, Heidelberg, 2004. 


