Papers › CM points on Shimura curves via QM-equivariant isogeny volcanoes
CM points on Shimura curves via QM-equivariant isogeny volcanoes
Frederick Saia
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We study CM points on the Shimura curves X₀ᴰ(N)_(/ℚ) and X₁ᴰ(N)_(/ℚ), parametrizing abelian surfaces with quaternionic multiplication and extra level structure. A description of the locus of points with CM by a specified order is obtained for general level, via an isogeny-volcano approach in analogy to work of Clark and Clark--Saia in the D=1 case of modular curves. This allows for a count of all points with CM by a specified order on such a curve, and a determination of all primitive residue fields and primitive degrees of such points on X₀ᴰ(N)_(/ℚ). We leverage computations of least degrees towards the existence of sporadic CM points on X₀ᴰ(N)_(/ℚ).
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