Modular Curves of Prime-power Level with Infinitely Many Rational Points
نویسنده
چکیده
For each open subgroup G of GL2(Ẑ) containing −I with full determinant, let XG/Q denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in G. Up to conjugacy, we determine a complete list of the 248 such groups G of prime power level for which XG(Q) is infinite. For each G, we also construct explicit maps from each XG to the j-line. This list consists of 220 modular curves of genus 0 and 28 modular curves of genus 1. For each prime `, these results provide an explicit classification of the possible images of `-adic Galois representations arising from elliptic curves over Q that is complete except for a finite set of exceptional j-invariants.
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