Galois groups via Atkin-Lehner twists

Author:
Pete L. Clark

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

Published electronically:
September 15, 2006

MathSciNet review:
2262856

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Abstract | References | Similar Articles | Additional Information

Abstract: Using Serre's proposed complement to Shih's Theorem, we obtain as a Galois group over for at least new primes . Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 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 . 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:
https://doi.org/10.1090/S0002-9939-06-08493-0

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