We study how the degrees of irrationality of moduli spaces of polarized K3
surfaces grow with respect to the genus. We prove that the growth is bounded by
a polynomial function of degree 14+ε for any ε>0 and,
for three sets of infinitely many genera, the bounds can be improved to degree
10. The main ingredients in our proof are the modularity of the generating
series of Heegner divisors due to Borcherds and its generalization to higher
codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For
special genera, the proof is also built upon the existence of K3 surfaces
associated with certain cubic fourfolds, Gushel-Mukai fourfolds, and
hyperkaehler fourfolds.Comment: v2: Results substantially improved. We use Kudla's modularity
conjecture to obtain a uniform polynomial bound of the degrees of
irrationality for all moduli spaces of polarized K3s. 20 page