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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: 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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  • [1] K. Belabas and E. Fouvry, Sur le $ 3$-rang des corps quadratiques de discriminant premier ou presque premier, Duke Math. J. 98 (1999), 217-268. MR 1695199 (2000i:11167)
  • [2] J. Ellenberg, $ \mathbb{Q}$-curves and Galois representations, Progr. Math. 224, 93-103. MR 2058645 (2005g:11088)
  • [3] J. González. Equations of hyperelliptic modular curves, Ann. Inst. Fourier (Grenoble) 41 (1991), 779-795. MR 1150566 (93g:11064)
  • [4] G. Malle, Genus zero translates of three point ramified Galois extensions, Manuscripta Math. 71 (1991), 97-111. MR 1094741 (92f:12009)
  • [5] B. Mazur, Modular elliptic curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1988), 33-186.
  • [6] A. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449-462. MR 0364259 (51:514)
  • [7] K. Ono and C. Skinner, Nonvanishing of quadratic twists of modular L-functions, Invent. Math. 134 (1998), 651-660. MR 1660945 (2000a:11077)
  • [8] J. Quer, $ \mathbb{Q}$-curves and abelian varieties of $ {\rm GL}_2$-type, Proc. London Math. Soc. (3) 81 (2000), 285-317. MR 1770611 (2001j:11040)
  • [9] K.-y. Shih, On the construction of Galois extensions of function fields and number fields, Math. Ann. 207 (1974), 99-120. MR 0332725 (48:11051)
  • [10] K.-y. Shih, $ p$-division points on certain elliptic curves, Comp. Math. 36 (1978), 113-129. MR 0515041 (80d:10042)
  • [11] J.-P. Serre, Topics in Galois Theory, Research Notes in Mathematics 1, Jones and Bartlett, 1992. MR 1162313 (94d:12006)
  • [12] V. Vatsal, Rank-one twists of a certain elliptic curve, Math. Ann. 311 (1998), 791-794. MR 1637976 (99i:11050)

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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

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