Papers › Generic models for genus 2 curves with real multiplication
Generic models for genus 2 curves with real multiplication
Alex Cowan, Sam Frengley, Kimball Martin
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Explicit models of families of genus 2 curves with multiplication by √(D) are known for D= 2, 3, 5. We obtain generic models for genus 2 curves over ℚ with real multiplication in 12 new cases, including all fundamental discriminants D < 40. A key step in our proof is to develop an algorithm for minimisation of conic bundles fibred over ℙ². We apply this algorithm to simplify the equations for the Mestre conic associated to the generic point on the Hilbert modular surface of fundamental discriminant D < 100 computed by Elkies--Kumar.
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