September 08, 2025
Let \(N>1\) and let \(\Phi_N(X,Y)\in\mathbb{Z}[X,Y]\) be the modular polynomial which vanishes precisely at pairs of \(j\)-invariants of elliptic curves linked by a cyclic isogeny of degree \(N\). In this note we study the divisibility of the coefficients \(\Phi_N(X+J, Y+J)\) for certain algebraic numbers \(J\), in particular \(J=0\) and \(J=1728\). It turns out that these coefficients are highly divisible by small primes at which \(J\) is supersingular.
Let \(N\) be a positive integer and consider the classical modular polynomial \(\Phi_N(X,Y)\in\mathbb{Z}[X,Y]\) which vanishes precisely at pairs \((j_1,j_2)\) of \(j\)-invariants of elliptic curves linked by a cyclic \(N\)-isogeny. It has degree \[\deg_X\Phi_N(X,Y) = \deg_Y\Phi_N(X,Y) := \psi(N) = N\prod_{p|N}\left(1+\frac{1}{p}\right).\] While the coefficients of \(\Phi_N(X,Y)\) are notoriously large [1]–[4] they are also highly divisible by small primes.
Our first main result is the following.
Theorem 1. Let \(N > 1\), and write \(\Phi_N(X,Y) = \sum_{0\leq i,j \leq \psi(N)}a_{i,j}X^iY^j.\) Then for \(i+j < \psi(N)\) the following hold.
If \(2\nmid N\), then \(v_2(a_{i,j}) \geq 15(\psi(N) -i -j)\).
If \(3\nmid N\), then \(v_3(a_{i,j}) \geq 3(\psi(N) -i -j)\); moreover, \(v_3(a_{i,j}) \geq \lceil\frac{9}{2}(\psi(N) -i -j)\rceil\) if \(N\equiv 1 \bmod 3\).
If \(5\nmid N\) then \(v_5(a_{i,j}) \geq 3(\psi(N) -i -j)\).
If \(p\geq 11\) and \(p\nmid N\), then \(v_p(a_{i,j}) \geq 3(C_0(N,p)-i-j)\), where
\(C_0(N,p) := \mathrm{ord}_X (\Phi_N(X,0) \bmod p)\).
When \(p\leq 5\), this was conjectured by Wang in [5], who proved a number of special cases and showed moreover that it suffices to prove Theorem 1 for prime \(N\). The result has applications to the study of reduction types of elliptic curves, see [6].
The polynomials \(\Phi_N(X,Y)\) have important applications in cryptography and computational number theory. Theorem 1 allows us to save some space storing the coefficients of \(\Phi_N(X,Y)\), as one only needs to store the factors not predicted by Theorem 1. When \(N=5\) (see Table [tab:Phi5]) this reduces the number of decimal digits needed from 523 to 298, a \(43\%\) saving. However, for larger \(N\) the relative savings dwindle, for example when \(N=101\) we only get a reduction from \(6, 383, 216\) to \(5, 606, 370\) decimal digits, a mere \(12\%\) saving.
Given finer bounds on the sizes of individual coefficients \(a_{i,j}\), Theorem 1 might lead to tighter bounds on the Chinese Remainder Theorem primes required for CRT-based algorithms (e.g. [7]–[9]) to compute \(\Phi_N(X,Y)\).
More generally, we study the coefficients of \(\Phi_N(X+J, Y+J)\) for certain algebraic numbers \(J\), see Theorem 3 below. In particular, when \(J=1728\) we have
Theorem 2. Let \(N>1\) and write \(\Phi_N(X+1728, Y+1728) = \sum_{0\leq i,j \leq \psi(N)}a_{i,j}X^iY^j.\) Then for \(i+j < \psi(N)\) the following hold.
If \(2\nmid N\), then \(v_2(a_{i,j}) \geq 9(\psi(N) -i -j)\); moreover, \(v_2(a_{i,j}) \geq 10(\psi(N) -i -j)\) if \(N \equiv 1 \bmod 4\).
If \(3\nmid N\), then \(v_3(a_{i,j}) \geq 6(\psi(N) -i -j).\)
If \(7\nmid N\) then \(v_7(a_{i,j}) \geq 2(\psi(N) -i -j)\).
If \(p\geq 11\) and \(p\nmid N\), then \(v_p(a_{i,j}) \geq 2(C_{1728}(N,p)-i-j)\), where
\(C_{1728}(N,p) := \mathrm{ord}_{X} \big(\Phi_N(X+1728,1728) \bmod p\big)\).
When \(J=j(E)\) is a singular modulus, i.e. the elliptic curve \(E\) has complex multiplication by an order of discriminant \(D\) in an imaginary quadratic field, then by Proposition 6 below, we only get non-trivial divisibility conditions for primes \(p\) at which the reduction of \(E\) is supersingular, i.e. when \(\displaystyle\left(\frac{D}{p}\right)\neq 1\).
In these cases, we have an explicit expression for \(C_J(N,p)\) in terms of the theta series of the quaternion order \({\mathrm{End}}_{\bar{\mathbb{F}}_p}(E)\), which is positive only for primes \(p < |D|N\), see Proposition 7.
\[\begin{array}{ll} a_{0, 0} & = \mathbf{2^{90} \cdot 3^{18} \cdot 11^{9}} \cdot 5^{3} \\ a_{1, 0} & = \mathbf{2^{75} \cdot 3^{15} \cdot 11^{6}} \cdot 2^{2} \cdot 3 \cdot 5^{3} \cdot 31 \cdot 1193 \\ a_{1, 1} & = \mathbf{2^{60} \cdot 3^{12} \cdot 11^{3}} \cdot -1 \cdot 2^{2} \cdot 3 \cdot 26984268714163 \\ a_{2, 0} & = \mathbf{2^{60} \cdot 3^{12} \cdot 11^{3}} \cdot 3 \cdot 5^{2} \cdot 13^{2} \cdot 3167 \cdot 204437 \\ a_{2, 1} & = \mathbf{2^{45} \cdot 3^{9}} \cdot 2^{2} \cdot 3 \cdot 5^{4} \cdot 53359 \cdot 131896604713 \\ a_{3, 0} & = \mathbf{2^{45} \cdot 3^{9}} \cdot 2^{3} \cdot 5^{2} \cdot 31 \cdot 1193 \cdot 24203 \cdot 2260451 \\ a_{2, 2} & = \mathbf{2^{30} \cdot 3^{6}} \cdot 3^{2} \cdot 5^{4} \cdot 7 \cdot 13 \cdot 1861 \cdot 6854302120759 \\ a_{3, 1} & = \mathbf{2^{30} \cdot 3^{6}} \cdot -1 \cdot 2 \cdot 3 \cdot 5^{3} \cdot 327828841654280269 \\ a_{4, 0} & = \mathbf{2^{30} \cdot 3^{6}} \cdot 3 \cdot 5 \cdot 13^{2} \cdot 3167 \cdot 204437 \\ a_{3, 2} & = \mathbf{2^{15} \cdot 3^{3}} \cdot 2^{2} \cdot 3 \cdot 5^{3} \cdot 2311 \cdot 2579 \cdot 3400725958453 \\ a_{4, 1} & = \mathbf{2^{15} \cdot 3^{3}} \cdot 2^{5} \cdot 3 \cdot 5^{3} \cdot 12107359229837 \\ a_{5, 0} & = \mathbf{2^{15} \cdot 3^{3}} \cdot 2^{2} \cdot 3 \cdot 5 \cdot 31 \cdot 1193 \\ a_{3, 3} & = -1 \cdot 2^{2} \cdot 5^{2} \cdot 11 \cdot 17 \cdot 131 \cdot 1061 \cdot 169751677267033 \\ a_{4, 2} & = 3 \cdot 5^{3} \cdot 167 \cdot 6117103549378223 \\ a_{5, 1} & = -1 \cdot 2 \cdot 3 \cdot 5^{2} \cdot 1644556073 \\ a_{6, 0} & = 1 \\ a_{4, 3} & = 2^{5} \cdot 3 \cdot 5^{2} \cdot 197 \cdot 227 \cdot 421 \cdot 2387543 \\ a_{5, 2} & = 2^{5} \cdot 5^{2} \cdot 13 \cdot 195053 \\ a_{4, 4} & = 2^{3} \cdot 5^{2} \cdot 257 \cdot 32412439 \\ a_{5, 3} & = -1 \cdot 2^{2} \cdot 3^{2} \cdot 5 \cdot 131 \cdot 193 \\ a_{5, 4} & = 2^{3} \cdot 3 \cdot 5 \cdot 31 \\ a_{5, 5} & = -1 \\ \end{array}\]
Let \(K\) be a complete valued field with valuation \(v\), valuation ring \(A\), uniformizer \(\pi\) and algebraically closed residue field \(A/\pi\) of characteristic \(p>0\).
For \(N > 1\) and \(J\in A\), we define \[C_J(N, v) := \mathrm{ord}_{X}\big(\Phi_N(X+J, J) \bmod \pi\big).\]
We have the following general result, which implies Theorems 1 and 2 when \(p\geq 5\).
Theorem 3. Let \(E/K\) be an elliptic curve with good reduction and \(J = j(E)\in A\). Let \(N > 1\) with \(p\nmid N\). Suppose that \(p\geq 5\) or \(v(J)=0\) and let \[n_v = \left\{ \begin{array}{ll} 3 & \text{if v(J) > 0 and p\geq 5} \\ 2 & \text{if v(J - 1728) > 0 and p\geq 5} \\ 1 & \text{if v(J) = v(J-1728) = 0.} \end{array} \right.\] Then the coefficients of \(\Phi_N(X+J, Y+J) = \sum_{0\leq i,j \leq \psi(N)}a_{i,j}X^iY^j\in A[X,Y]\) satisfy \[v(a_{i,j}) \geq n_v(C_J(N,v)-i-j)\] for all \(i+j < C_J(N,v)\).
Lemma 1. Let \(f(Y) = a_0 + a_1Y + \ldots + a_dY^d \in K[Y]\). Fix \(n\in\mathbb{Z}\) and let \(y_0, y_1,\ldots, y_d\in K\) be such that
\(v(y_0)=v(y_2)=\cdots=v(y_d)=n\),
\(v(y_k-y_l) = n\) for all \(k\neq l\).
Then \[v(a_j) \geq \min_{0\leq k \leq d} v(f(y_k)) - nj \quad \text{for all j=0,1,2,\ldots, d}.\]
Conversely, if \(v(a_j) \geq B - nj\) for all \(j\), then clearly \(v(f(y_k)) \geq B\).
Proof. We solve for the coefficients \(a_j\) in the linear system \[a_0 + a_1y_k + \cdots + a_dy_k^d = f(y_k), \quad k=0,1,2,\ldots d.\] By Cramer’s rule, we get \(a_j = \frac{M_j}{V}\), where \(V = \det(y_k^i)_{0\leq k,i \leq d} = \pm\prod_{k<i}(y_k-y_i)\) is the Vandermonde determinant and \(M_j\) is the determinant where the \(j\)th column of \(V\) has been replaced by \((f(y_k))_{0\leq k\leq d}\).
By assumption, we have \(v(V) = \sum_{k<i}v(y_k-y_i) = \frac{d(d+1)}{2}n\). Factoring out suitable powers of \(\pi\) from the columns of \(M_j\), we find that \[v(M_j) \geq \left( \frac{d(d+1)}{2} - j\right)n + \min_{0\leq k \leq d} v(f(y_k)).\] The result follows. ◻
Our main tool is the following result.
Proposition 4. Let \(N > 1\) with \(p\nmid N\). Let \(n\geq 1\) and suppose that there exist elliptic curves \(E_k/K\), \(k=0,2,\ldots, \psi(N)\) satisfying the following conditions:
Each \(E_k\) has good reduction;
\(v(j(E_k)-J) = v(j(E_k)-j(E_l)) = n\) for all \(k\neq l\);
For every \(k\) and every elliptic curve \(\tilde{E}_k\) linked to \(E_k\) by a cyclic isogeny of degree \(N\), we have \[v(j(\tilde{E}_k) - j(E_k)) > 0 \Longrightarrow v(j(\tilde{E}_k) - j(E_k)) \geq n.\]
Then the coefficients of \(\Phi_N(X+J, Y+J) = \sum_{0\leq i,j \leq \psi(N)}a_{i,j}X^iy^j\in A[X,Y]\) satisfy \[v(a_{i,j}) \geq n(C_J(N,v)-i-j)\] for all \(i+j < C_J(N,v)\).
Proof. Write \[\begin{align} & \Phi_N(X+J,Y+J) = b_0(Y) + b_1(Y)X + \cdots + b_{\psi(N)-1}(Y)X^{\psi(N)-1} + X^{\psi(N)} \\ & b_i(Y) = a_{i,0} + a_{i,1}Y + \cdots + a_{i,\psi(N)}Y^{\psi(N)}, \quad i=0,\ldots, \psi(N). \end{align}\]
For each \(k\), let \(y_k = j(E_k) - J\). The roots of \(\Phi_N(X, y_k + J)\) are the \(j\)-invariants of elliptic curves \(E_{k,m}\) linked to \(E_k\) by a cyclic \(N\)-isogeny. Since \(v(N)=0\), \(E[N]\) is unramified and these elliptic curves and isogenies are all defined over \(K\).
By definition, \(C_J(N,v)\) of these roots \(j(E_{k,m})\), \(m=1,2,\ldots, C_J(N,v)\), satisfy \(v(j(E_{k,m} )-J) >0\), so \(v(j(E_{k,m} )-J) \geq v(j(E_{k,m} )-j(E_k)) \geq n\), and the rest satisfy \(v(j(E_{k,m})-J)=0\).
The coefficients \(b_i(y_k)\) of \(\Phi_N(X+J, y_k+J)\) are symmetric forms in the roots \(j(E_{k,m})-J\) which satisfy \(v(j(E_{k,m})-J) \geq n\) for \(m=1,2,\ldots, C_J(N,v)\), so it follows that \[v(b_i(y_k)) \geq n(C_J(N,v) -i), \quad i = 0, 1, \ldots, C_J(N,v).\]
Now applying Lemma 1 to the polynomials \(b_i(Y)\) with the interpolation points \(y_k\) completes the proof. ◻
Our goal now is to construct deformations \(E_k/K\) of the elliptic curve \(E/K\) satisfying the hypothesis of Proposition 4.
Proof of Theorem 3.. Let \(v(p) = e\). Suppose that \(E\) is given by a minimal Weierstrass equation \[W : y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6, \quad a_i\in A\] with \(c_4\) and \(c_6\) coefficients defined by \[\begin{align} b_2 & = a_1^2+4a_2, \quad b_4 = a_1a_3+2a_4, \quad b_6 = a_3^2+4a_6,\\ c_4 & = b_2^2 +4a_2, \quad c_6 = -b_2^3 + 36b_2b_4 - 216b_6, \\ 1728 \Delta & = c_4^3 - c_6^2, \quad J = c_4^3/\Delta, \quad v(\Delta)=0. \end{align}\] Suppose first that \(v(J)=0\), so \(v(c_4)=0\). For suitable \(\varepsilon_k\in A^*\) and \(m\geq 1\), which we will choose later, we claim there exists an elliptic curve \(E_k/K\) with \(c_4\) and \(c_6\) coefficients given by \[c_4(E_k) = c_4, \quad c_6(E_k) = c_6' = c_6 + \varepsilon_k\pi^m.\] Then \[\begin{align} & 1728\Delta_k = c_4^3 - c_6'^2 = 1728\Delta - \varepsilon_k\pi^m(2c_6 + \varepsilon_k\pi^m)\\ & j_k = j(E_k) = c_4^3/\Delta_k. \end{align}\]
We impose \(\min(v(2)+v(c_6)+m, 2m) \geq v(1728)\) to ensure that \(v(\Delta_k) = 0\) and \(E_k\) has good reduction.
When \(p\geq 5\) the existence of \(E_k\) follows from [10]. When \(p=3\), our assumption \(v(J)=0\) implies \(v(c_6)=0\), so the existence follows from [10].
When \(p=2\), we again have \(v(c_6)=0\) and furthermore \(m\geq 3e\). Then by [10], there exists \(x\in A\) such that \(v(c_6+x^2) \geq 2e\). By our assumption on \(m\), we find that also \(v(c_6' + x^2) \geq 2e\) and \(E_k\) exists by [10].
For \(\varepsilon_l \in A\), we compute \[j_k - j_l = \frac{c_4^3}{1728\Delta_k\Delta_l} \big[2c_6\pi^m(\varepsilon_k-\varepsilon_l) + \pi^{2m}(\varepsilon_k^2-\varepsilon_l^2) \big].\] Since the residue field \(A/\pi\) is infinite, we may choose arbitrarily many \(\varepsilon_k\in A^*\) such that \[\begin{align} v(j_k - j_l) & = v\big( 2c_6\pi^m(\varepsilon_k-\varepsilon_l) + \pi^{2m}(\varepsilon_k^2-\varepsilon_l^2) \big) - v(1728) \\ & = \min(v(2)+v(c_6)+m, 2m) - v(1728) \end{align}\] for all \(k\neq l\).
If \(p\geq 5\) we may choose \(m=1\) and obtain \(v(j_k - j_l) = 1\) if \(v(c_6)=0\) (i.e. \(v(J-1728)=0)\) and \(v(j_k - j_l) = 2\) otherwise.
If \(p=2,3\) our assumption is \(v(c_6)=0\), so choosing \(m=v(1728)-v(2)+1\) gives us \(v(j_k - j_l) = 1\).
We now treat the case \(v(J) > 0\) and \(p\geq 5\). In this case, \(v(c_4)>0\) and [10] gives the existence of elliptic curves \(E_k/K\) with coefficients \[\begin{align} & c_4(E_k) = c_4' = c_4 + \varepsilon_k\pi, \quad c_6(E_k) = c_6' = c_6 \\ & 1728\Delta_k = c_4'^3 - c_6^2 = 1728\Delta + \big[ 3c_4^2\varepsilon_k\pi + 3c_4\varepsilon_k^2\pi^2 + \varepsilon_k^3\pi^3 \big]. \end{align}\] For \(\varepsilon_k, \varepsilon_l \in A\) we compute \[j_k - j_l = \frac{c_6^2}{\Delta_k\Delta_l} \big[ 3c_4^2\pi(\varepsilon_l-\varepsilon_k) + 3c_4\pi^2(\varepsilon_l^2-\varepsilon_k^2) + \pi^3(\varepsilon_l^3-\varepsilon_k^3)\big].\] Again, we may choose arbitrarily many \(\varepsilon_k\in A^*\) such that \(v(j_k-j_l)=3\) for all \(k\neq l\).
Finally, setting \(\varepsilon_l=0\) in the above equations shows that \(v(j_k-J)= n_v\) in all cases.
It remains to show the claim about isogenies \(f: E_k \to \tilde{E}_k\). Since \(E_k\) has good reduction, so does \(\tilde{E}_k\). Now if \(v(j(E_k) - j(\tilde{E}_k)) > 0\) then by [11] we have \[v(j(E_k) - j(\tilde{E}_k)) \geq \frac{1}{2}\#\mathrm{Isom}_{A/\pi}(E_k, \tilde{E}_k) = \frac{1}{2}\#\mathrm{Aut}_{A/\pi}(E_k) = n_v.\] ◻
For \(p\geq 5\), the elliptic curves with \(J=0\) and \(J=1728\) have models with good reduction over \(K=\mathbb{Q}_p^\mathrm{ur}\), the maximal unramified extension of \(\mathbb{Q}_p\), and so the \(p\geq 5\) parts of Theorems 1 and 2 follow from Theorem 3. We only need to point out that \(C_0(N,5) = \psi(N)\), since \(J=0\) is the only supersingular invariant in characteristic 5, and similarly \(C_{1728}(N,7)=\psi(N)\); see Proposition 5 below.
It remains to prove divisibility by powers of 2 and 3; we do so by constructing suitable deformations and using Vélu’s isogeny formulae [12].
Proof of Theorem 1. First, let \(p=2\) and \(K=\mathbb{Q}_2^\mathrm{ur}\). For suitable \(\varepsilon_k\in A^*\), the elliptic curve \[E_k : y^2 + y = x^3 + 2\varepsilon_k x\] has discriminant \[\Delta_k = -2^9\varepsilon_k^3 -27,\] so \(v(\Delta_k) = v(27) = 0\) and \(E_k\) has good reduction. Now for \(\varepsilon_l\in A\) we compute \[j_k - j_l = \frac{2^{15}3^6(\varepsilon_k^3-\varepsilon_l^3)}{\Delta_k \Delta_l}.\] Since \(A/2A = \bar{\mathbb{F}}_2\) is infinite, we may choose sufficiently many \(\varepsilon_k\in A^*\) for which \[v(j_k-0) = v(j_k-j_l) = 15, \qquad \forall k\neq l.\] Now suppose \(f : E_k \to \tilde{E}_k\) is a cyclic isogeny of degree \(N\), which is defined over \(K\) since \(2\nmid N\). We decompose \(\ker f = S \cup (-S) \cup \{0\}\) with \(\#S = N-1\). It follows from Vélu’s formulae [12] that \(\tilde{E}_k\) has Weierstrass equation \[\tilde{E}_k : y^2 + y = x^3 + A_4x + A_6\] where \(A_4 = 2\varepsilon_k(6-5N) - 30\sum_{Q\in S}x_Q\). We see that \(v(A_4)\geq 1\) and so \(j(\tilde{E}_k)\) satisfies \[j(\tilde{E}_k) = \frac{-2^{12}3^3A_4^3}{\tilde{\Delta}_k}; \qquad v(j(\tilde{E}_k) \geq 15.\] Thus the hypothesis of Proposition 4 holds with \(n=15\).
Now let \(p=3\) and \(K=\mathbb{Q}_3(\sqrt{-3})^\mathrm{ur}\) which has ramification degree \(v(3)=e=2\) over \(\mathbb{Q}_3\). We choose the uniformizer \(\pi = 1-\omega = \frac{1-\sqrt{-3}}{2}\) with \(\pi^2=-3\omega\). For suitable \(\varepsilon_k\in A^*\) we define the elliptic curve \(E_k/K\) with Weierstrass equation \[E_k : y^2 = x^3 + \varepsilon_k\pi x^2 - \omega x.\] We compute \[\Delta_k = 2^4(4-3\varepsilon_k^2), \quad\text{so}\quad v(\Delta_k)=0\] and \[j_k = \frac{2^{12}3^3(\varepsilon_k^2-1)^3}{\Delta_k} \quad\text{so}\quad v(j_k) = 6 + 3v(\varepsilon_k^2-1).\] For \(\varepsilon_l\in A^*\), we compute \[\begin{align} j_k-j_l & = \frac{2^{16}3^3}{\Delta_k\Delta_l} \big[(1-\varepsilon_k^2)^3(4-3\varepsilon_l^2) - (1-\varepsilon_l^2)^3(4-3\varepsilon_k^2)\big] \\ & = \frac{2^{16}3^3}{\Delta_k\Delta_l} \big[-4(\varepsilon_k^6-\varepsilon_l^6) + 3(\varepsilon_k^4-\varepsilon_l^4)(4+\varepsilon_k^2\varepsilon_l^2) -3^2(\varepsilon_k^2-\varepsilon_l^2)(1+\varepsilon_k^2\varepsilon_l^2) \big]. \end{align}\] When \(N \equiv 2 \bmod 3\), we may choose arbitrarily many \(\varepsilon_k\in A^*\) with \[v(j_k - 0) = v(j_k-j_l) = 6 \quad\forall k\neq l;\] and when \(N \equiv 1 \bmod 3\), we choose \(\varepsilon_k = 1 + \varepsilon'_k\pi\in A^*\), so that \(v(1-\varepsilon_k^2) = 1\) and \[v(j_k - 0) = v(j_k-j_l) = 9 \quad\forall k\neq l.\]
Now suppose \(f : E_k \to \tilde{E}_k\) is a cyclic isogeny of degree \(N\), which is defined over \(K\) since \(3\nmid N\). We decompose \(\ker f = S \cup (-S) \cup \{0\}\) with \(S\cap(-S) = E[2]\cap C\). It follows from Vélu’s formulae [12] that \(\tilde{E}_k\) has Weierstrass equation
\[\tilde{E}_k : y^2 = x^3 + \varepsilon_k\pi X^2 + A_4x + A_6\] where \[A_4 = -15s_2 - 10\varepsilon_k\pi s_1 + 5(N-6)\omega\] and \[s_m := 2\sum_{Q\in S\smallsetminus E[2]}x_Q^m - \sum_{Q\in S\cap E[2]}x_Q^m \in A, \quad m=1,2.\] Its \(j\)-invariant is \(j(\tilde{E}_k) = C_4^3/\tilde{\Delta}_k\) where \(v(\tilde{\Delta}_k)=0\) by good reduction and \(C_4\) is given by \[C_4 = 16\pi^2\big[(\varepsilon_k^2 -1) +5(N-1) \big] + 3^2\cdot 80s_2 + 3\pi\cdot 160\varepsilon_k s_1.\] We thus have \(v(j(\tilde{E}_k)) = 3v(C_4) \geq 6\), and \(v(j(\tilde{E}_k)) \geq 9\) when \(N\equiv 1 \mod 3\) and \(v(\varepsilon_k^2 - 1) > 0\).
This satisfies the hypothesis of Proposition 4 with \(n=6\) if \(N \equiv 2\bmod 3\) and \(n=9\) if \(N \equiv 1 \bmod 3\). Finally, note that \(v_3(a_{i,j}) = \frac{1}{2}v(a_{i,j})\). This completes the proof of Theorem 1. ◻
Proof of Theorem 2 (sketch).. The proof is similar to that of Theorem 1, we give here only the Weierstrass equations of \(E_k/K\).
When \(p=2\), we let \(K = \mathbb{Q}_2(i)^\mathrm{ur}\) with ramification index \(v(2)=2\) and uniformizer \(\pi=i-1\). We define \(E_k/K\) by \[E_k : y^2+(i+1)xy+iy = x^3+ix^2+\varepsilon_k\pi^m x\] and compute \[v(j_k - 1728) = v(j_k-j_l) = 16 + 2m, \qquad \forall k\neq l,\] for suitably chosen \(\varepsilon_k\in A^*\). We choose \(m=1\) if \(N\equiv 3\bmod 4\) and \(m=2\) if \(N\equiv 1 \bmod 4\). Using Vélu’s formulas again, we find that if \(f : E_k \to \tilde{E}_k\) is a cyclic \(N\)-isogeny, then \[v(j(\tilde{E}_k) - 1728) \geq \left\{ \begin{array}{ll} 18 & \text{if N \equiv 3 \bmod 4} \\ 20 & \text{if N \equiv 1 \bmod 4}. \end{array} \right.\]
The case \(p=3\) is easier. We let \(K = \mathbb{Q}_3^\mathrm{ur}\) and define for \(\varepsilon_k\in A^*\) \[E_k : y^2 = x^3 + x + \varepsilon_k\] with \[v(j_k - 1728) = v(j_k-j_k) = 6 \quad\forall k\neq l.\] Now we can conclude the proof either with Vélu or Gross-Zagier [11], which gives \(v(j(\tilde{E}_k) - j(E_k)) \geq 6\), so \(v(j(\tilde{E}_k) - 1728) \geq 6.\) ◻
Suppose \(J = j(E)\) where \(E/K\) has good reduction and \(\mathcal{O}_{J,p} = {\mathrm{End}}_{\bar{\mathbb{F}}_p}(E)\). Then \(C_J(N,v)\) counts the number of cyclic \(N\)-isogenies of \(E\) which reduce to endomorphisms modulo \(\pi\). This depends crucially on whether the reduced elliptic curve \(E/\bar{\mathbb{F}}_p\) is ordinary or supersingular.
Proposition 5. Suppose \(p\nmid N\). We have \(C_0(N,p) = \psi(N)\) for \(p=2,3,5\); \(C_{1728}(N,p) = \psi(N)\) for \(p=2,3,7\) and \(C_5(N,13) = \psi(N)\).
Proof. The cases \(J\) and \(p\) in the statement are precisely those where \(J\) is the only supersingular invariant in characteristic \(p\). Now the result follows, since all roots of \(\Phi_N(X+J, J) \bmod \pi\) correspond to elliptic curves isogenous to \(E/\bar{\mathbb{F}}_p\) and are thus again supersingular. ◻
Proposition 6. Suppose \(p\nmid N\). Suppose \(E\) has ordinary reduction, then \(\mathcal{O}_{J,p}\) is an order of discriminant \(D\) in an imaginary quadratic field. Denote by \(\chi_D\) the associated Kronecker character.
We have \[C_J(N,v) \leq \prod_{q|N}(1 + \chi_D(q))^{v_q(N)}.\] with equality if \(\mathcal{O}_{J,p}\) is a principal ideal domain.
If \(E/K\) also has complex multiplication (necessarily by an order of discriminant \(Dp^m\) for some \(m \geq 0\)), then we find that \[C_J(N,v) = C_J(N,0) := \mathrm{ord}_{X}\big(\Phi_N(X+J, J) \in K[X]\big).\]
In case (2), \(a_{C_J(N,v),0}\) is the first non-zero coefficient of \(\Phi_N(X+J, J)\) and \(v(a_{C_J(N,v),0}) = 0\), so we get no non-trivial divisibility relations.
Proof. \(C_J(N,v)\) equals the number of principal ideals \(\mathfrak{n}\subset \mathcal{O}_{J,p}\) with \(\mathcal{O}_{J,p}/\mathfrak{n} \cong \mathbb{Z}/N\mathbb{Z}\). Such ideals exist if every prime \(q|N\) is split or ramified, and at each prime we have a choice of \(1+\chi_D(q)\) primes above \(q\). This proves (1).
If \(E/K\) has complex multiplication, then \({\mathrm{End}}_{\bar{K}}(E)\) equals \(\mathcal{O}_{J,p}\) up to a power of \(p\) in its conductor. But \(p\nmid N\), so this makes no difference. Part (2) now follows. ◻
Proposition 7. Let \(p\nmid N\). Suppose \(E\) has supersingular reduction, then \(\mathcal{O}_{J,p}\) is a maximal order in the quaternion algebra ramified exactly at \(p\) and \(\infty\).
We have \[C_J(N,v) = \frac{1}{\#\mathrm{Aut}_{\bar{K}}(E)}\sum_{d^2|N}\mu(d)\#\{f\in\mathcal{O}_{J,p} \; |\; \mathrm{nrd}(f) = N/d^2 \}.\] The cardinalities in the above sum are coefficients of the theta series associated to \(\mathcal{O}_{J,p}\).
Now suppose \(E/K\) has complex multiplication by a quadratic imaginary order \(\mathcal{O}_D\) of discriminant \(D<0\) and \(C_J(N,v) > C_J(N,0)\). Then \(p < |D|N\).
Proof. For relevant facts about orders in quaternion algebras, see [13]. Counting all elements of reduced norm \(N\) in \(\mathcal{O}_{J,p}\), not just those with cyclic quotient, gives \[\#\{f\in\mathcal{O}_{J,p} \; |\; \mathrm{nrd}(f) = N \} = \#\mathrm{Aut}_{\bar{K}}(E) \sum_{d^2|N} C_J(N/d^2,v)\] and part (1) now follows by Möbius inversion.
Now suppose the hypothesis of (2) holds. The first non-zero coefficient \(a_{C_J(N,0),0}\) of \(\Phi_N(X+J,J)\) is a product of the form \[a_{C_J(N,0),0} = \prod_{\tilde{E}\to E}(j(\tilde{E})-J)\] where \(\tilde{E}\) ranges over elliptic curves linked to \(E\) by a cyclic \(N\)-isogeny, but for which \(j(\tilde{E})\neq J\). By assumption, this product reduces to 0 modulo \(\pi\), so for one of these elliptic curves we have \(j(\tilde{E})\neq j(E)\) and \(\tilde{E}\cong E \bmod \pi\). This \(\tilde{E}\) has complex multiplication by an order \(\mathcal{O}_{Df^2}\) of discriminant \(Df^2\) for some \(f|N\).
If \(p\) divides the conductor of \(\mathcal{O}_D\) then \(p < |D|N\) is clear. Otherwise, by [14], the orders \(\mathcal{O}_D\) and \(\mathcal{O}_{Df^2}\) embed optimally into \(\mathcal{O}_{J,p}\). The result now follows from [15]. ◻
The author thanks Fabien Pazuki for interesting discussions.