A correspondence between pairs of isogenies and representation numbers

Question

This question is re-posted from MSE because it didn't seem to get any traction/responses there.

This is a question from this paper about a correspondence between representation numbers of quadratic forms, and pairs of isogenies between elliptic curves.

Let $(L, Q)$ be a quadratic space over a ring $R$. For a quadratic form $F$ on $R^m$, define the representation number $R_L(F)$ as the cardinality of the set $$\{(f_i) \in L^m : Q(x_1 f_1 + \cdots + x_m f_m) = F(x_1, \ldots, x_m) \text{ for all } x \in R^m\}.$$

For a basis $\{b_i\}$ of $L$, define the diagonal of $Q$ with respect to $\{b_i\}$ to be the $n$-tuple $(Q(b_i))_i$.

The following is a quote from the paper linked above:

Such pairs $(f_1, f_2)$ correspond to representations of positive definite quadratic forms $Q(x_1, x_2) = \deg(x_1 f_1 + x_2 f_2)$, hence $$\# \{(f_1, f_2) \in \operatorname{Hom}(E, E') : \operatorname{deg} f_i = m_i\} = \sum_{Q > 0,\ \operatorname{diag} Q = (m_1, m_2)} R_{\operatorname{Hom}(E, E')}(Q).$$

Here $E$ and $E'$ are elliptic curves with complex multiplication, which correspond to an intersection point of the modular polynomials $\phi_{m_1}$ and $\phi_{m_2}$.

The way this is stated, and the fact that the rest of this paper is quite accessible, makes me think that the above correspondence is either "well-known," or trivial, but I have not encountered this before and don't see at all why it is true.

Can someone provide either a reference, or a proof/explanation of this correspondence?

Answer 1

This follows by unwinding the definitions of the quantities involved and using that $\mathrm{deg}$ is a positive definite quadratic form on isogenies.

We can take $R = \mathbb{Z}$, and the quadratic space $(L, Q) = (\mathrm{Hom}(E, E'), \mathrm{deg})$. As described in the paper/question, to a pair $f_1, f_2 \in \mathrm{Hom}(E, E')$ we can attach a quadratic form $Q_{f_1, f_2}$ on $\mathbb{Z}^2$ via $(x_1, x_2) \mapsto \mathrm{deg}(x_1 f_1 + x_2 f_2)$.

If $f_1, f_2 \in \mathrm{Hom}(E, E')$ have $\mathrm{deg}(f_i) = m_i$, then $Q_{f_1, f_2}(e_i) = m_i$ where $e_i$ is the standard basis on $\mathbb{Z}^2$, and so $Q_{f_1, f_2}$ arises in the sum on the right hand side.

Notice that $R_{\mathrm{Hom}(E, E')}(Q)$ counts the number of $(f_1, f_2) \in \mathrm{Hom}(E, E')$ such that $Q = Q_{f_1, f_2}$. In particular, such a pair has $\{\mathrm{deg}(f_i)\} = \mathrm{diag}(Q) = \{m_1, m_2\}$.

Combining these two observations shows that the sum on the right counts the cardinality of the set on the left by breaking the set up into pieces based on the quadratic form induced by the pair.