I asked this question at MO but did not receive an answer:
Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+b)}{\gcd(a,b)^3}\right)^2$.
A similarity $s:X\times X \rightarrow \mathbb{R}$ is defined in Encyclopedia of Distances (p. 3) as:
$s(x,y) \ge 0 \forall x,y \in X$
$s(x,y) = s(y,x) \forall x,y \in X$
$s(x,y) \le s(x,x) \forall x,y \in X$
$s(x,y) = s(x,x) \iff x=y$
A positive definite kernel $k$ is a positive definite function on some set $X$. A "kernel-similarity" is a function $f$ which is a kernel and a similiarity. One can prove that $\frac{1}{16 \left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2}$ is a kernel-similarity.
If $f$ is a "kernel-similarity" such that $f(a,a) = f(b,b) \forall a,b \in X$ then: $$d(a,b) = \sqrt{f(a,a) + f(b,b) - 2f(a,b)}$$ defines a metric and the corresponding metric space can be embedded into a Hilbert space (or for finite $X$ can be embedded to an Euclidean space).
It is conjectured that $\frac{1}{\operatorname{rad}(\frac{ab(a+b)}{\gcd(a,b)^3})}$ is a positive definite kernel over the natural numbers. (One can prove that it is a similarity over the natural numbers.). This similarity has to do with the "conductor" of the elliptic curve above. (Although I am not familiar with elliptic curves, but trying to read about them).
My 1). question is if there are other families of elliptic curves indexed by $a,b$ such that the reciprocal of the determinant over all the family of considered elliptic curves:
$$\frac{1}{\Delta(a,b)}=k(a,b)=s(a,b)$$
is a "kernel-similarity" over the natural numbers.
This would allow to apply the theory of Hilbert spaces over the family of elliptic curves.
The j-invariant is given by:
$$j(a,b) = \frac{16^2(a^2+ab+b^2)^3}{a^2b^2(a+b)^2}$$
It seems that:
$$\frac{1}{j(a,b)}$$
is positive definite.
Question: Is $$\frac{1}{j(a,b)}$$ a positive definite function over the natural numbers?