Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties
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Author(s)
Arias-de-Reyna, Sara
Dieulefait, Luis V.
Shin, Sug Woo
Wiese, Gabor
Date Issued
September 2014
Journal
Mathematische Annalen
Publisher
Springer-Verlag
Citation
Arias-de-Reyna, Sara, Luis V. Dieulefait, Sug Woo Shin, and Gabor Wiese. “Compatible Systems of Symplectic Galois Representations and the Inverse Galois Problem III. Automorphic Construction of Compatible Systems with Suitable Local Properties.” Math. Ann. 361, no. 3–4 (September 7, 2014): 909–925. © 2104 Springer-Verlag
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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, PSp[subscript n](F[subscript ℓ[superscript d]]) or PGSp[subscript n](F[subscript ℓ[superscript d]]) occurs as a Galois group over the rational numbers for a positive density set of primes ℓ. 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 GL[subscript n](A[subscript 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00208-014-1091-x
