I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0". I'm surely missing something really simple here, and I'd appreciate any comments to help clarify the matter.


Specifically, it is in the first equality of Formula (39), the computation of $$\sum\_{E \in F : H'(E) < X} (\#S\_3(E) - 1);$$ here, $F$ is a "large family" of elliptic curves (I only really care about the set of *all* rational elliptic curves, which sure enough is "large"), $H'(E)$ is the (naive) height of $E$, and $S\_3(E)$ is the 3-Selmer group of $E$.


Here is my question:



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> How does the first equality of Formula (39) follow from Theorem 24 and Propositions 35 and 38?
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Let me now try to explain why I'm confused. 


In the notation of Theorem 24 of loc. cit. (the "weighted congruence counting theorem"), the above sum is equal to $N\_\phi(S(F); X)$, where $S(F)$ is the set of *locally soluble* integral ternary cubic forms associated to the large family $F$, and $\phi$ is the weight function $1/m(f)$ defined in the run-up to Proposition 35. However, the formula in Theorem 24 is about $N\_\phi(V\_\mathbb{Z}; X)$, and not $N\_\phi(S(F); X)$.



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> When $\phi$ is equal to $1/m(\cdot)$, do we have $N\_\phi(S(F); X) = N\_\phi(V\_\mathbb{Z}; X)$? More generally, in Theorem 24, are we *really* counting orbits in a subset $S$ of $V\_\mathbb{Z}$ which is defined by (infinitely many) congruence conditions?
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But even if that is true, there is still a problem. Propositions 35 and 38 seem to be saying that our weight function is *acceptable*, so we can indeed apply Theorem 24 to $N\_\phi(S(F); X)$. But applying Theorem 24 would give us $$\sum\_{E \in F : H'(E) < X} (\#S\_3(E) - 1) = N(V\_\mathbb{Z}; X)\prod\_p\int\_{f \in V\_{\mathbb{Z}\_p}}\frac{1}{m\_p(f)}df + o(X^{5/6}),$$ while the authors claim that we have $$\sum\_{E \in F : H'(E) < X} (\#S\_3(E) - 1) = N(V\_\mathbb{Z} \cap S\_\infty(F); X)\prod\_p\int\_{f \in S\_p(F)}\frac{1}{m\_p(f)}df + o(X^{5/6}).$$



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> Is it true that $N(V\_\mathbb{Z}; X) = N(V\_\mathbb{Z} \cap S\_\infty(F); X)$? And why can we restrict the integration to only those forms in $S\_p(F)$?
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Finally, in the second equality, the authors seem to be claiming $$N(V\_\mathbb{Z} \cap S\_\infty(F); X) = \frac{4}{9}{\tt{Vol}}(PGL\_3(\mathbb{Z})\backslash PGL\_3(\mathbb{R}))M\_\infty(V,F;X),$$ which doesn't seem to be explained anywhere in the paper.



On the number of 3-Selmer elements of rational elliptic curves
I think that you apply Theorem 24 to the function
$$ \phi(f) = \frac{1}{m(f)} I\_{S(F)} $$
where $I\_{S(F)}$ means the characteristic function of $S(F)$.
The point of Propositions 35 and 38 is that *this* function is acceptable, with
$$ \phi\_p(f) = \frac{1}{m\_p(f)} I\_{S\_p(F)} $$
and this is why you get integration over $S\_p(F)$ in your second question.


I am not sure about what happens at infinity; with a quick glance, I have been unable to find the definition of $S\_\infty(F)$. But it might be that $V\_{\mathbb{Z}} \cap S\_\infty(F)$ is either $V\_{\mathbb{Z}}^+$ or $V\_{\mathbb{Z}}^{-{}}$ or their union.


