TY - JOUR

T1 - Arithmetic distribution of tempered components of cuspidal representations of GL(3)

AU - Yang, Liyang

N1 - Funding Information:
I am very grateful to Dinakar Ramakrishnan for pleasant discussions. I would like to thank Kimball Martin for his helpful suggestions. Sincere thanks are also due to the anonymous referee for his/her careful reading and valuable comments.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/3

Y1 - 2023/3

N2 - Let π=⊗v′πv be an arbitrary unitary cuspidal representation of GL (3) over a number field F. We show, for certain ray classes C, that ∑p2∈C|aπ(p)|<1logN(p)N(p)1/2=+∞,which implies that there are infinitely many unramified places v in C such that πv’s are tempered with Hecke eigenvalues lying inside the open unit disk. Furthermore, when F= Q, we consider the problem on the least prime p in an arithmetic progression such that πp satisfies the Ramanujan conjecture. An effective upper bound of Linnik type for such a prime p is proved.

AB - Let π=⊗v′πv be an arbitrary unitary cuspidal representation of GL (3) over a number field F. We show, for certain ray classes C, that ∑p2∈C|aπ(p)|<1logN(p)N(p)1/2=+∞,which implies that there are infinitely many unramified places v in C such that πv’s are tempered with Hecke eigenvalues lying inside the open unit disk. Furthermore, when F= Q, we consider the problem on the least prime p in an arithmetic progression such that πp satisfies the Ramanujan conjecture. An effective upper bound of Linnik type for such a prime p is proved.

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U2 - 10.1007/s00209-023-03213-w

DO - 10.1007/s00209-023-03213-w

M3 - Article

AN - SCOPUS:85148347100

SN - 0025-5874

VL - 303

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3

M1 - 66

ER -