The Yau-Zaslow conjecture determines the reduced genus 0 Gromov-Witten invariants of K3 surfaces in terms of the Dedekind eta function. Classical intersections of curves in the moduli of K3 surfaces with Noether-Lefschetz divisors are related to 3-fold Gromov-Witten theory via the K3 invariants. Results by Borcherds and Kudla-Millson determine the 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 K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on 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.

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Noether-Lefschetz theory and the Yau-Zaslow conjecture

This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.

Lines, conics, and all that

/pdf/noether-lefschetz-theory-and-the-yau-zaslow-conjecture-2cm92i29ow.pdf

https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2008/0013/0001/SDG-2008-0013-0001-a009.pdf

A survey of Calabi-Yau manifolds

https://hal.archives-ouvertes.fr/hal-01104612/document

The geometry of flex tangents to a cubic curve and its parameterizations

Let X be a K3 surface with a primitive ample divisor H, and let $\beta=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_{\beta}$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it verifies Yau-Zaslow formula in the non primitive class $\beta$.

/pdf/the-number-of-rational-curves-on-k3-surfaces-1phkpabh2d.pdf

Baosen Wu

Papers

The Number of Rational Curves on K3 Surfaces

The number of rational curves on K3 surfaces