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A constraint for twist equivalence of cusp forms on \(\mathrm{GL}(n)\). (English) Zbl 1515.11037

Summary: This note answers, and generalizes, a question of Kaisa Matomäki. We show that given two cuspidal automorphic representations \(\pi_1\) and \(\pi_2\) of \(\mathrm{GL}(n)\) over a number field \(F\) of respective conductors \(N_1, N_2\), every character \(\chi\) such that \(\pi_1\otimes\chi\simeq\pi_2\) of conductor \(Q\), satisfies the bound: \(Q^n\mid N_1N_2\). If at every finite place \(v, \pi_{1,v}\) is a discrete series whenever it is ramified, then \(Q^n\) divides the least common multiple \([N_1,N_2]\).

MSC:

11F11 Holomorphic modular forms of integral weight
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F55 Other groups and their modular and automorphic forms (several variables)
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