Let $\Phi_n(X,Y)$ be the modular polynomial that are the canonical equations for the modular curve $X_0(n)$. They parameterise pairs of elliptic curves related by a cyclic isogeny of degree $n$.
I am looking for proofs for the following two results:
- If $m=p^a$, where $p$ is prime and $a>1$, then $$\Phi_m(X,Y)=\begin{cases} \dfrac{\prod_{i=1}^{\Psi(p^{a-1})}\Phi_p(X,\xi_i)}{\Phi_{p^{a-2}}(X,Y)^p}\,, & a>2\,,\\[0.5cm] \dfrac{\prod_{i=1}^{p+1}\Phi_p(X,\xi_i)}{(X-Y)^{p+1}}\,, & a=2\,, \end{cases}$$ where $\xi_i$ are the roots of $\Phi_{p^{a-1}}(X,Y)=0$, and $\Psi(m)=m\prod_{p\mid m}(1+1/p)$.
- If $p^e$ is the $p$-primary part of $n$ and if we write $n=n'p^e$, then $\Phi_n(X,Y)$ splits modulo $p$ as follows: $$\Phi_n(X,Y) = \Phi_{n'}\left( X^{p^{e-1}}, Y \right) \Phi_{n'}\left( X, Y^{p^{e-1}} \right)\prod_{i=1}^{e-1} \Phi_{n'}\left( X^{p^{e-i-1}}, Y^{p^{i-1}} \right)^{p-1}\pmod{p}\,.$$
I found the first result as (13.14) in David Cox's Primes of the form $x^2+ny^2$ but he refers the reader to Weber's Lehrbuch der Algebra.
The next result is found in Igusa's Kroneckerian Model of Fields of Elliptic Modular Functions and the reader was asked to refer to Die typen der multiplikatorenringe elliptischer funktionenkörper by Max Deuring.
I will be very happy if someone could provide a reference to proofs of these in english or direct me to english readings that would allow me to develop these proofs. THANKS!
