Divisibility of the coefficients of modular polynomials

TL;DR

This work analyzes divisibility properties of modular polynomials Φ_N by studying coefficients after shifting by a singular modulus J. It derives explicit $p$-adic lower bounds on the coefficients $a_{i,j}$ of Φ_N(X+J,Y+J) in terms of the quantity $C_J(N,π)$ and local data, with sharp behavior at small primes and for primes $p$ with supersingular reduction. The proofs combine a general valuation framework, a Vandermonde-based interpolation lemma, and deformation theory of elliptic curves to construct families with prescribed $p$-adic distance to $J$, together with explicit isogeny computations (e.g., Velu's formula) for small primes. The results have implications for understanding reduction types, improving CRT-based computations of Φ_N, and connecting to the theory of differences of singular moduli in the spirit of Gross–Zagier and related work on modular curves and CM theory.

Abstract

Let $N>1$ and let $Φ_N(X,Y)\in\mathbb{Z}[X,Y]$ 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 of $Φ_N(X+J, Y+J)$ for certain algebraic numbers $J$, in particular $J=0$ and other singular moduli. It turns out that these coefficients are highly divisible by small primes at which $J$ is supersingular.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Remark 2.4
• Theorem 3.1