The Yau-Zaslow conjecture predicts the genus 0 curve counts of
K
3
K3
surfaces in terms of the Dedekind
η
\eta
function. The classical intersection theory of curves in the moduli of
K
3
K3
surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the
K
3
K3
curve counts. Results by Borcherds and Kudla-Millson determine these classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise.
Via a detailed study of the STU model (determining special curves in the moduli of
K
3
K3
surfaces), we prove the Yau-Zaslow conjecture for all curve classes on
K
3
K3
surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.