Abstract

For every positive integer , there is a modular polynomial which vanishes exactly at pairs of -invariants of elliptic curves linked by a cyclic isogeny of degree . The coefficients of these polynomials are notoriously large; their asymptotic growth (with implicit constants) has been known since the 1980s. Since these polynomials have applications in cryptography it is important to obtain explicit bounds on their coefficients. In this talk, I report on joint work with Fabien Pazuki and Desirée Gijón Gómez proving asymptotically optimal upper and lower bounds on these coefficients.