Abstract
This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer \(n\) and any positive integer \(d\), \(\mathrm {PSp}_n(\mathbb {F}_{\ell ^d})\) or \(\mathrm {PGSp}_n(\mathbb {F}_{\ell ^d})\) occurs as a Galois group over the rational numbers for a positive density set of primes \(\ell \). The result depends on some work of Arthur’s, which is conditional, but expected to become unconditional soon. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of \(\hbox {GL}_n({\mathbb {A}}_\mathbb {Q})\) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series apply.
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Notes
Here we are excluding the possibility that the global parameter for \(\tau \) is non-generic at the same time as \(\tau _q\) belongs to a non-generic \(A\)-packet, in which case the parameter for \(\pi _q\) would be the transfer of the non-generic parameter for \(\tau _q\) (so not supercuspidal). We thank Gordan Savin for asking to clarify.
More precisely there are only finitely many irreducible subrepresentations isomorphic to \(\pi '_F\) in the space of cuspforms on \(\hbox {GL}_n({\mathbb {A}}_F)\) (with trivial central character).
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Acknowledgments
The authors would like to thank the referee for a careful reading. S. A.-d.-R. was partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain. She thanks the University of Barcelona for its hospitality during several short visits. S. A.-d.-R. would like to thank X. Caruso for his explanations concerning tame inertia weights and the Hodge–Tate weights. L. V. D. was supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain and by an ICREA Academia Research Prize. S. W. S. was partially supported by NSF grant DMS-1162250 and Sloan Fellowship. G. W. acknowledges partial support by the Priority Program 1489 of the Deutsche Forschungsgemeinschaft (DFG) and by the Fonds National de la Recherche Luxembourg (INTER/DFG/12/10).
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Arias-de-Reyna, S., Dieulefait, L.V., Shin, S.W. et al. Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties. Math. Ann. 361, 909–925 (2015). https://doi.org/10.1007/s00208-014-1091-x
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DOI: https://doi.org/10.1007/s00208-014-1091-x