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In 1989, Manin and his collaborators formed a series of conjectures on the asymptotic behaviour of the number of solutions to diophantine equations. Let $X\subset \mathbb{P}^n$ is be a fano variety (that is $-K_X$ is ample) under its anticanonical embedding, and let $H$ be the associated height function. Then it is expected that there exists a zariski open subset $U \subset X$ such that the number of rational points of height less than $B$ (e.g. the number of solutions in an expanding ball or box) is asymptotic to $c_X B(\log B)^{r_X},$ as $B \to \infty$ for some constants $c_X$ and $r_X$.

This result is true in some cases, for example complete intersections with many variables and small degree via the circle method, quadratic forms, toric varieties, flag varieties and also for some del Pezzo surfaces. But there are counter-examples showing that it is not true for all fano varieties (namely one expects $r_X=\textrm{rank } \textrm{Pic}(X)-1$, but this is not true in general).

At any rate, Peyre formed a conjecture on the leading constant $c_X$ which occurs in the asymptotic formula. It is very close to what you describe. One defines a measure on the set of adelic points on $X$, and then the leading constant is essentially the volume of the closure of $X(\mathbb{Q})$ inside the adeles. This is really an adelic integral and not a real integral, but for suitable varieties (namely those which satisfy weak approximation), the local factors at the primes come out as the $c_p$ in the way you describe. In general though one needs to introduce convergence factors to insure that the product over the $c_p$ converges. These come from an Artin L-function associated to the Picard group.

There are however some extra factors $\alpha$ and $\beta$ present in the constant, related to the position of the anticanonical divisor in the effective cone and the Brauer group of $X$. For conics with a rational point, we have $\alpha=1/2$ and $\beta =1$. This might explain your missing factor of two.

Papers:

J. Franke, Y. I. Manin and Y. Tschinkel, Rational Points of Bounded Height on Fano Varieties. Invent. Math. 95, 421--435 (1989).

E. Peyre, Hauteurs et measures de Tamagawa sur les variétiés de Fano. Duke Math. J., 79(1), 101--218 (1995).

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In 1989, Manin and his collaborators formed a series of conjectures on the asymptotic behaviour of the number of solutions to diophantine equations. Let $X\subset \mathbb{P}^n$ is a fano variety (that is $-K_X$ is ample) under its anticanonical embedding, and let $H$ be the associated height function. Then it is expected that there exists a zariski open subset $U \subset X$ such that the number of rational points of height less than $B$ (e.g. the number of solutions in an expanding ball or box) is asymptotic to $c_X B(\log B)^{r_X},$ as $B \to \infty$ for some constants $c_X$ and $r_X$.

This result is true in some cases, for example complete intersections with many variables and small degree via the circle method, quadratic forms, toric varieties, flag varieties and also for some del Pezzo surfaces. But there are counter-examples showing that it is not true for all fano varieties (namely one expects $r_X=\textrm{rank } \textrm{Pic}(X)-1$, but this is not true in general).

At any rate, Peyre formed a conjecture on the leading constant $c_X$ which occurs in the asymptotic formula. It is very close to what you describe. One defines a measure on the set of adelic points on $X$, and then the leading constant is essentially the volume of the closure of $X(\mathbb{Q})$ inside the adeles.

HoweverThis is really an adelic integral and not a real integral, there but for suitable varieties (namely those which satisfy weak approximation), the local factors at the primes come out as the $c_p$ in the way you describe. In general though one needs to introduce convergence factors to insure that the product over the $c_p$ converges. These come from an Artin L-function associated to the Picard group.

There are also however some extra factors $\alpha$ and $\beta$ present in the constant, related to the position of the anticanonical divisor in the effective cone and the Brauer group of $X$. For conics with a rational point, we have $\alpha=1/2$ and $\beta =1$. This might explain your missing factor of two.

Papers:

J. Franke, Y. I. Manin and Y. Tschinkel, Rational Points of Bounded Height on Fano Varieties. Invent. Math. 95, 421--435 (1989).

E. Peyre, Hauteurs et measures de Tamagawa sur les variétiés de Fano. Duke Math. J., 79(1), 101--218 (1995).

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In 1989, Manin and his collaborators formed a series of conjectures on the asymptotic behaviour of the number of solutions to diophantine equations. Let $X\subset \mathbb{P}^n$ is a fano variety (that is $-K_X$ is ample) under its anticanonical embedding, and let $H$ be the associated height function. Then it is expected that there exists a zariski open subset $U \subset X$ such that the number of rational points of height less than $B$ is asymptotic to $c_X B(\log B)^{r_X},$ as $B \to \infty$ for some constants $c_X$ and $r_X$.

This result is true in some cases, for example complete intersections with many variables and small degree via the circle method, quadratic forms, toric varieties, flag varieties and also for some del Pezzo surfaces. But there are counter-examples showing that it is not true for all fano varieties (namely one expects $r_X=\textrm{rank } \textrm{Pic}(X)-1$, but this is not true in general).

At any rate, Peyre formed a conjecture on the leading constant $c_X$ which occurs in the asymptotic formula. It is very close to what you describe. One defines a measure on the set of adelic points on $X$, and then the leading constant is essentially the volume of the closure of $X(\mathbb{Q})$ inside the adeles.

However, there are also some extra factors $\alpha$ and $\beta$ present, related to the position of the anticanonical divisor in the effective cone and the Brauer group of $X$. For conics with a rational point, we have $\alpha=1/2$ and $\beta =1$. Hopefully this should This might explain your missing factor of two.

Papers:

J. Franke, Y. I. Manin and Y. Tschinkel, Rational Points of Bounded Height on Fano Varieties. Invent. Math. 95, 421--435 (1989).

E. Peyre, Hauteurs et measures de Tamagawa sur les variétiés de Fano. Duke Math. J., 79(1), 101--218 (1995).