Explicit bounds on the coefficients of modular polynomials for the elliptic $j$-invariant

TL;DR

This paper provides explicit upper bounds on the coefficients of elliptic modular polynomials, which are crucial in understanding the relationships between elliptic curves linked by cyclic isogenies, with bounds that are asymptotically optimal.

Contribution

It introduces explicit bounds on the coefficients of modular polynomials for all N, improving understanding of their size and asymptotic behavior.

Findings

Bounds are explicit and applicable for all N

Main term of bounds is asymptotically optimal

Enhances understanding of elliptic curve isogenies

Abstract

We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials $\Phi_N$ for any $N\geq1$. These polynomials vanish at pairs of $j$-invariants of elliptic curves linked by cyclic isogenies of degree $N$. The main term in the bound is asymptotically optimal as $N$ tends to infinity.