Galois groups via Atkin-Lehner twists
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- by Pete L. Clark PDF
- Proc. Amer. Math. Soc. 135 (2007), 617-624
Abstract:
Using Serre’s proposed complement to Shih’s Theorem, we obtain $PSL_2(\mathbb {F}_p)$ as a Galois group over $\mathbb {Q}$ for at least $614$ new primes $p$. Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for $\frac {3}{8}$ of the primes that were 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.References
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Additional Information
- Pete L. Clark
- Affiliation: Department of Mathematics and Statistics, 1126 Burnside Hall, McGill University, 805 Sherbrooke West, Montreal, QC, Canada H3A 2K6
- Address at time of publication: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 767639
- Email: clark@math.mcgill.ca, pete@math.uga.edu
- Received by editor(s): June 30, 2005
- Received by editor(s) in revised form: September 15, 2005
- Published electronically: September 15, 2006
- Communicated by: Ken Ono
- © Copyright 2006 by the author
- Journal: Proc. Amer. Math. Soc. 135 (2007), 617-624
- MSC (2000): Primary 11G18, 12F12
- DOI: https://doi.org/10.1090/S0002-9939-06-08493-0
- MathSciNet review: 2262856