Prof. Dr. Florian Breuer (University of Newcastle)

For every positive integer $N$, the modular polynomial $\Phi_N(X,Y)$ has integer coefficients and vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of order $N$. These polynomials have applications in cryptography and define integral (but singular) models for the modular curves $X_0(N)$. Their coefficients grow rapidly with $N$. In this talk, I will explain recent joint work with Fabien Pazuki and Desir\’ee Gij\’on G\’omez obtaining explicit upper and lower bounds on the size of these coefficients. Our methods also lead to explicit bounds on the heights of Hecke images. If time allows, I can also outline analogous results for Drinfeld modular polynomials.