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There are several formulas for the number of orbits of the projective line under the action of subgroups of $$\mathrm{GL}_2$$ GL 2 . We give an interpretation of two such formulas in terms of the geometry of elliptic curves, and prove a more general formula for a large class of congruence subgroups of Bianchi groups. Our formula involves the number of walks on a certain graph called an isogeny...
Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in $${\mathbb P}^3$$ P3 . Let $$E'$$ E′ be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover $$C'$$ C′ of $$E'$$ E′ with the property that C and $$C'$$ C′ are represented by the same cohomology class on the 4-torsion. In fact we give equations for $$C'$$ C′ as a curve...
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