I am looking for a book or lecture notes on applications of riemannian geometry to diophantine equations. The main motivation is the following : suppose i have a quadratic form $q$ on $\mathbf{Q}^n$ of signature $(0,n-1,1)$, then the $\frac{1}{2}$-hyperboloids $\{q=\pm 1\}$ can be endowed with a riemannian variety structure (it's the hyperbolic space) and the distance associated is linked with $q$ by the identity $d(x,y)={\rm argch}(-q(x,y))$. I believe this point of view can be very useful to study the diophantine properties of $q$, for instance to count the numbers of rational points of $\{q=\pm 1\}$ of small height, do you have references of this ?
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$\begingroup$ Are you asking for a reference for a very specific and speculative application of Riemannian geometry to a certain type of quadratic form equation? Likely the answer is no. I would suggest you look at papers which study quadratic form equations in many variables, and then see if there’re any possibilities of improving any of the results using some other tools in mathematics like Riemannian geometry. $\endgroup$– Stanley Yao XiaoNov 7 at 2:33
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