Abstract
In this article, we study deformations of conjugate self-dual Galois representations. The study is twofold. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.
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Acknowledgements
This article is a byproduct of the AIM SQuaREs project Geometry of Shimura varieties and arithmetic application to L-functions conducted by the five authors from 2017 to 2019. We would like to express our sincere gratitude and appreciation to the American Institute of Mathematics for their constant and generous support of the project, and to the staff members at the AIM facility in San Jose, California for their excellent coordination and hospitality. We thank Toby Gee for providing us with an improvement on Proposition 4.5(2), hence Theorem 4.8. Finally, we would like to thank the anonymous referees for careful reading and helpful comments.
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Y. L. supported by NSF (Grant No. DMS–1702019) and a Sloan Research Fellowship; Y. T. supported by NSFC (Grant No. 12225112/12231001) and CAS Project for Young Scientists in Basic Research (Grant No. YSBR-033); L. X. supported by NSF (Grant No. DMS–1502147/DMS–1752703), NSFC (Grant No. 12071004) and the Chinese Ministry of Education; W. Z. supported by NSF (Grant No. DMS–1838118/DMS–1901642); X. Z. supported by NSF (Grant No. DMS–1902239) and a Simons Fellowship
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Liu, Y.F., Tian, Y.C., Xiao, L. et al. Deformation of Rigid Conjugate Self-dual Galois Representations. Acta. Math. Sin.-English Ser. 40, 1599–1644 (2024). https://doi.org/10.1007/s10114-024-1409-x
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DOI: https://doi.org/10.1007/s10114-024-1409-x