Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field Article Swipe

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Baskar Balasubramanyam , A. Raghuram ·

YOU? · · 2017 · Open Access · · DOI: https://doi.org/10.1353/ajm.2017.0017 · OA: W2964176149

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 ${\rm GL}(n)\times{\rm GL}(n)$.