I am looking at a result of Peyre, and he says for a certain variety, the number of rational points of height less than $B$ is:

$$ N(B) \sim \frac{1}{3} \color{#3DB08E}{\prod_p \left( 1 - \frac{1}{p}\right)^4\left( 1 + \frac{4}{p} + \frac{1}{p^2} \right) }B (\log B)^3 $$

The manifold is some sort of Veronese del Pezzo type variety surface:

$$ V(\mathbb{Q}) = \big\{ \big( (x_0:y_0), (x_1:y_1),(x_2:y_2) \big) : x_0 x_1 x_2 = y_0 y_1 y_2 \big\} \subset \text{P}^1_\mathbb{Q}\times \text{P}^1_\mathbb{Q} \times \text{P}^1_\mathbb{Q} $$

The height is some sort of exotic height -- probably a Weil height of some kind.

$$ H( ...) = \sup \big(|x_0|, |y_0|\big)\sup \big(|x_1|, |y_1|\big) \sup \big(|x_2|, |y_2|\big)$$

This is the height associated to the anti-canonical divisor $\omega_V^{-1} = \mathcal{O}_V(1,1,1) $. He has removed some exceptional divisors which come from variables being zero, $x = 0$ or $y = 0$. So that $U = V \,\backslash \bigcup\{ x_i = y_j = 0 \}$.

What do we know about this constant? Is it a period? It looks like the Euler product of some type of L-function of some scheme. An algebraic geometer, might try to read the product over primes as the "Adelic" part and the $B \log^3 B$ part as places over some ring of polynomials.

Do the Cohen-Lenstra heuristics predict anything here? Those are more about estimating ranks of class groups, or Galois groups, but they momentarily look like a probability.


The variety $V$ is a del Pezzo surface of degree $6$ (this has nothing to do with Veronese varieties). One has

$$\#V(\mathbb{F}_p) = 1 + 4p + p^2,$$ which is how the Euler factors naturally arise (the factor $(1 - 1/p)^4$ is just a convergence factor). The constant has an interpretation as an adelic volume; all this is explained in the cited paper of Peyre.

This has nothing to do with the Cohen-Lenstra heuristics. (I have no idea why you thought these might be relevant here).

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    $\begingroup$ This is a completely separate question. In papers by Tschinkel (and perhaps also Peyre), I have seen a claim that the "Batyrev-Manin philosophy" also includes an upper bound on the least height of a rational point related to an inverse Tamagawa number. Do you happen to know a good reference for this claim? $\endgroup$ – Jason Starr Nov 24 '17 at 11:48
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    $\begingroup$ @Jason: There are unfortunately not many precise heuristics/conjectures concerning this problem. Certainly the Batyrev-Manin conjecture does not give such a statement. In fact the original conjecture of Tschinkel on this problem was false. You can read more about this story in the following papers of Elsenhans and Jahnel: uni-math.gwdg.de/jahnel/Arbeiten/kl_pt/kleinster_punkt4f.pdf and uni-math.gwdg.de/jahnel/Arbeiten/klpt_8b.pdf $\endgroup$ – Daniel Loughran Nov 24 '17 at 11:55
  • $\begingroup$ Great! I will look at the papers of Elsenhans and Jahnel. Thanks so much. $\endgroup$ – Jason Starr Nov 24 '17 at 12:10

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