We explicitly construct the canonical rational models of Shimuracurves, both analytically in terms of modular forms andalgebraically in terms of coefficients of genus 2 curves, in thecases of quaternion algebras of discriminant 6 and 10. This emulatesthe classical construction in the elliptic curve case. We also givefamilies of genus 2 QM curves, whose Jacobians are the correspondingabelian surfaces on the Shimura curve, and with coefficients thatare modular forms of weight 12. We apply these results to showthat our $j$-functions are supported exactly at those primes wherethe genus 2 curve does not admit potentially good reduction, andconstruct fields where this potentially good reduction is attained.Finally, using $j$, we construct the fields of moduli and definitionfor some moduli problems associated to the Atkin--Lehner groupactions.
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