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Let $E$ be a supersingular curve over a field of characteristic $p$ with endomorphism ring $\mathcal O_D$ which is a maximal order in a division ring $D$ over $Q$ ramified at $p$ and $\infty$.

The norm defines a (positive definite) quadratic form in $4$ variables, let us call it $Q(x,y,z,w)$. Given an integer $n$, is it known what the number of solutions to $Q(x,y,z,w) = n$ is? I would like this to depend only on $p$, $n$ and the size of the unit group in the maximal order.

For instance, if $D$ were the Hamitonian quaternions (over $\mathbb Q$), then this is classical and Wikipedia has a formula essentially in terms of $\sigma(n)$, the sum of divisors on $n$ here. They don't give the formula for the maximal order but I am really interested in the maximal order and even just the case of $n$ prime might be good enough.

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    $\begingroup$ If $p$ is large then there are many nonisomorphic maximal orders in $D$, and the representation numbers are usually going to be different, so I am not sure how you would even formulate the answer. In particular your goal of having a number that depends only on $p$, $n$ and the size of the unit groups is probably impossible to fulfill. $\endgroup$
    – Aurel
    Commented Dec 15, 2019 at 0:31
  • $\begingroup$ Would a uniform bound be more plausible? $\endgroup$
    – Asvin
    Commented Dec 15, 2019 at 0:34
  • $\begingroup$ Yes, but probably not very useful. For instance, for $n$ prime and $p$ large, the best lower bound will probably be $0$. $\endgroup$
    – Aurel
    Commented Dec 15, 2019 at 0:37
  • $\begingroup$ I was more interested in upper bounds! $\endgroup$
    – Asvin
    Commented Dec 15, 2019 at 0:38
  • $\begingroup$ Won't the generating function be a level $N$ weight $2$ Eisenstein series ? Thus of the form $\sum_{\gamma \in SL_2(Z/NZ)} C_\gamma E_2(\gamma(z))$ so you want to know $N$ and an upper bound for $\sum |C_\gamma |$ $\endgroup$
    – reuns
    Commented Dec 15, 2019 at 15:14

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