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2017•American Journal of MathematicsOpen access

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

Baskar Balasubramanyam, A. Raghuram

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Abstract

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

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

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Available abstract

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

Key concepts: Mathematics, Automorphic form, Congruence relation, Pure mathematics, Algebraic number field, Automorphic L-function, Field (mathematics), L-function

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