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On the embedding problem for $ 2^+S_4$ representations


Author: Ariel Pacetti
Journal: Math. Comp. 76 (2007), 2063-2075
MSC (2000): Primary 11F80; Secondary 11F37
DOI: https://doi.org/10.1090/S0025-5718-07-01940-0
Published electronically: April 24, 2007
MathSciNet review: 2336282
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Abstract: Let $ 2^+S_4$ denote the double cover of $ S_4$ corresponding to the element in $ \operatorname{H}^2(S_4,\mathbb{Z}/2\mathbb{Z})$ where transpositions lift to elements of order $ 2$ and the product of two disjoint transpositions to elements of order $ 4$. Given an elliptic curve $ E$, let $ E[2]$ denote its $ 2$-torsion points. Under some conditions on $ E$ elements in $ \operatorname{H}^1(\operatorname{Gal}_{\mathbb{Q}},E[2])\backslash \{ 0 \}$ correspond to Galois extensions $ N$ of $ \mathbb{Q}$ with Galois group (isomorphic to) $ S_4$. In this work we give an interpretation of the addition law on such fields, and prove that the obstruction for $ N$ having a Galois extension $ \tilde N$ with $ \operatorname{Gal}(\tilde N/ \Q) \simeq 2^+S_4$ gives a homomorphism $ s_4^+:\operatorname{H}^1(\operatorname{Gal}_{\mathbb{Q}},E[2]) \rightarrow \operatorname{H}^2(\operatorname{Gal}_\mathbb{Q}, \mathbb{Z}/2\mathbb{Z})$. As a corollary we can prove (if $ E$ has conductor divisible by few primes and high rank) the existence of $ 2$-dimensional representations of the absolute Galois group of $ \mathbb{Q}$ attached to $ E$ and use them in some examples to construct $ 3/2$ modular forms mapping via the Shimura map to (the modular form of weight $ 2$ attached to) $ E$.


References [Enhancements On Off] (What's this?)

  • 1. Pilar Bayer and Gerhard Frey, Galois representations of octahedral type and $ 2$-coverings of elliptic curves, Math. Z. 207 (1991), no. 3, 395-408. MR 1115172 (92d:11058)
  • 2. Michael Bungert, Construction of a cuspform of weight $ 3/2$, Arch. Math. (Basel) 60 (1993), no. 6, 530-534. MR 1216696 (94f:11035)
  • 3. H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1977, pp. 69-78. Lecture Notes in Math., Vol. 627. MR 0472703 (57:12396)
  • 4. Teresa Crespo, Explicit construction of $ \tilde A\sb n$ type fields, J. Algebra 127 (1989), no. 2, 452-461. MR 1028464 (91a:12006)
  • 5. -, Explicit construction of $ 2S\sb n$ Galois extensions, J. Algebra 129 (1990), no. 2, 312-319. MR 1040941 (91d:11135)
  • 6. Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $ 1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507-530 (1975). MR 0379379 (52:284)
  • 7. Larry Dornhoff, Group representation theory. Part A: Ordinary representation theory, Marcel Dekker Inc., New York, 1971. MR 0347959 (50:458a)
  • 8. Benedict H. Gross, Heights and the special values of $ L$-series, Number theory (Montreal, Que., 1985), CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115-187. MR 894322 (89c:11082)
  • 9. A. Jehanne, Realization over $ \mathbb{Q}$ of the groups $ \tilde A\sb 5$ and $ \hat {A}\sb 5$, J. Number Theory 89 (2001), no. 2, 340-368. MR 1845242 (2002f:12005)
  • 10. Arnaud Jehanne, Sur les extensions de $ {\bf Q}$ à groupe de Galois $ S\sb 4$ et $ \widetilde S\sb 4$, Acta Arith. 69 (1995), no. 3, 259-276. MR 1316479 (95m:11127)
  • 11. Winfried Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32-72. MR 660784 (84b:10038)
  • 12. J. Larry Lehman, Levels of positive definite ternary quadratic forms, Math. Comp. 58 (1992), no. 197, 399-417, S17-S22. MR 1106974 (92f:11057)
  • 13. PARI/GP, version 2.2.8, http://pari.math.u-bordeaux.fr/, 2004.
  • 14. Anna Rio, Dyadic exercises for octahedral extensions ii, Submitted (2005).
  • 15. Bruno Schoeneberg, Elliptic modular functions: an introduction, Springer-Verlag, New York, 1974. MR 0412107 (54:236)
  • 16. Jean-Pierre Serre, L'invariant de Witt de la forme $ {\rm Tr}(x\sp 2)$, Comment. Math. Helv. 59 (1984), no. 4, 651-676. MR 780081 (86k:11067)
  • 17. Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440-481. MR 0332663 (48:10989)
  • 18. -, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. MR 1291394 (95e:11048)
  • 19. Willaim Stein, The modular forms explorer, http://modular.ucsd.edu/mfd/mfe/.
  • 20. Gonzalo Tornaría, Tables of ternary quadratic forms (part of computational number theory), http://www.ma.utexas.edu/users/tornaria/cnt/, 2004.
  • 21. Masaru Ueda, The decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators, J. Math. Kyoto Univ. 28 (1988), no. 3, 505-555. MR 965416 (90a:11054)

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Additional Information

Ariel Pacetti
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria. C.P:1428, Buenos Aires, Argentina
Email: apacetti@dm.uba.ar

DOI: https://doi.org/10.1090/S0025-5718-07-01940-0
Keywords: Galois representations, Shimura correspondence
Received by editor(s): July 14, 2005
Received by editor(s) in revised form: March 11, 2006
Published electronically: April 24, 2007
Additional Notes: The author was supported by a CONICET grant
The author would like to thank the “Universitat de Barcelona” where this work was done
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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