L-Functions and Rational Points on Galois Twists of Modular Curves
Studies elliptic curves with a prescribed mod-p torsion group scheme by realizing their classifying space as a Galois twist of the modular curve X(p), then investigates the L-functions and rational points of these twisted curves. Learners see how moduli of elliptic curves, two-dimensional Galois representations, Euler systems, and the Bloch-Kato conjecture combine to control the arithmetic of elliptic curves whose Galois image lies in the normalizer of a nonsplit Cartan subgroup.
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This PhD thesis studies Galois twists of the modular curve X(p) and the arithmetic of elliptic curves whose mod-p torsion is a prescribed finite group scheme. It first shows, in the sense of Katz and…