An explicit height bound for the classical modular polynomial

TL;DR

This paper establishes an explicit upper bound on the logarithmic height of classical modular polynomials Phi_m for prime m, improving understanding of their size and complexity.

Contribution

The paper provides a new explicit height bound for Phi_m, derived from Cohen's theorem, and supplies a comprehensive table of values for m up to 3607.

Findings

Proved that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m

Derived a simplified bound h(Phi_m) <= 6 m log m + 18 m

Compiled a table of h(Phi_m) for m <= 3607

Abstract

For a prime m, let Phi_m be the classical modular polynomial, and let h(Phi_m) denote its logarithmic height. By specializing a theorem of Cohen, we prove that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m. As a corollary, we find that h(Phi_m) <= 6 m log m + 18 m also holds. A table of h(Phi_m) values is provided for m <= 3607.