Abstract
We show that, for each \(n>0\), there is a family of elliptic surfaces which are covered by the square of a curve of genus \(2n+1\), and whose Hodge structures have an action by \({{\mathbb {Q}}}(\sqrt{-n})\). By considering the case \(n=3\), we show that one particular family of K3 surfaces are covered by the squares of curves of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in \({\mathbb {P}}^4\) with 15 ordinary double points up to a map of finite degree of K3 surfaces. This gives an explicit proof of the Kuga–Satake–Deligne correspondence for these K3 surfaces and any K3 surfaces related to them by maps of finite degree, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.
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Acknowledgements
C. Ingalls was partially supported by an NSERC Discovery Grant. A. Logan thanks the Tutte Institute for Mathematics and Computation for its support of his external research. A. Logan and O. Patashnick thank the Institut Henri Poincaré and the organizers of the Reinventing Rational Points program for their support and hospitality during their spring 2019 stay. We thank the referee for detailed and helpful reports, Frank Calegari and Cecília Salgado for interesting discussions, and a preliminary reviewer for a very helpful comment vis-à-vis the Hodge conjecture.
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Ingalls, C., Logan, A. & Patashnick, O. Explicit coverings of families of elliptic surfaces by squares of curves. Math. Z. 302, 1191–1238 (2022). https://doi.org/10.1007/s00209-022-03090-9
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DOI: https://doi.org/10.1007/s00209-022-03090-9