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Research in the Mathematical Sciences

, 2:24

First Online: 24 November 2015Received: 09 June 2015Accepted: 30 September 2015

Abstract

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the moduli space of pairs C, E of a genus 2 curve C and elliptic curve E with a map of degree n from C to E, as well as a tautological family over the base, for \2 \le n \le 11\. We also analyze the resulting models from the point of view of arithmetic geometry, and produce several interesting curves on them.

KeywordsK3 surfaces Moduli spaces Hilbert modular surfaces Genus-2 curves Jacobians Elliptic curves Mathematics Subject ClassificationPrimary 11F41 Secondary 14J28 14H40  Download to read the full article text



Autor: Abhinav Kumar

Fuente: https://link.springer.com/



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