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Galois Groups Via Atkin-Lehner Twists

Pete L. Clark
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Abstract

Using Serre's proposed complement to Shih's Theorem, we obtain PSL_2(F_p) as a Galois group over Q for at least 614 new primes p. Under the assumption that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3/8 of the primes not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture concerning class numbers of quadratic fields. The key issue is to understand rational points on Atkin-Lehner twists of X_0(N). In an appendix, we explore the existence of local points on these curves.

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Pete L. Clark. 2005-09-09. Galois Groups Via Atkin-Lehner Twists. https://arxiv.org/abs/math/0506490

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