Galois Groups Via Atkin-Lehner Twists

Details Galois Groups Via Atkin-Lehner Twists Galois Groups Via Atkin-Lehner Twists

2005-06-23

arxiv.org/abs/math/0506490v3

Mathematics

Number Theory

An "appendix" has been added containing additional results. There are many minor expository changes. 8 pages

Scientific paper

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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