A constraint for twist equivalence of cusp forms on GL$(n)$

Dinakar Ramakrishnan; Liyang Yang

A constraint for twist equivalence of cusp forms on GL$(n)$

Number Theory 2020-01-07 v3

Authors: Dinakar Ramakrishnan , Liyang Yang

Abstract

This Note answers, and generalizes, a question of Kaisa Matom\"aki. We show that give two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL_n$ 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 $[N_1, N_2].$

Keywords

cuspidal automorphic representations gl(n) twist equivalence conductors discrete series number fields

Cite

@article{arxiv.1906.01047,
  title  = {A constraint for twist equivalence of cusp forms on GL$(n)$},
  author = {Dinakar Ramakrishnan and Liyang Yang},
  journal= {arXiv preprint arXiv:1906.01047},
  year   = {2020}
}

Comments

The main result of the earlier version remains unchanged in this version. The main change is that the proof has been modified to "not use" anything about Galois representations or the local Langlands conjecture. Instead this new proof works totally on the "automorphic side", using only the representation theory of GL(n) and division algebras

A constraint for twist equivalence of cusp forms on GL$(n)$

Abstract

Keywords

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