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Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 139, Number 3, June 2017
- pp. 641-679
- 10.1353/ajm.2017.0017
- Article
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We prove an integrality result for the value at $s=1$ of the adjoint $L$-function associated to a cohomological cuspidal automorphic representation on ${\rm GL}(n)$ over any number field. We then show that primes (outside an exceptional set) dividing this special value give rise to congruences between automorphic forms. We also prove a non-vanishing property at infinity for the relevant Rankin-Selberg $L$-functions on ${\rm GL}(n)\times{\rm GL}(n)$.