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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Galois groups via Atkin-Lehner twists


Author: Pete L. Clark
Journal: Proc. Amer. Math. Soc. 135 (2007), 617-624
MSC (2000): Primary 11G18, 12F12
Published electronically: September 15, 2006
MathSciNet review: 2262856
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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.


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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
Email: clark@math.mcgill.ca, pete@math.uga.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08493-0
PII: S 0002-9939(06)08493-0
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
Article copyright: © Copyright 2006 by the author