Tohoku Mathematical Journal

Commutation relations of Hecke operators for Arakawa lifting

Atsushi Murase and Hiro-aki Narita

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T. Arakawa, in his unpublished note, constructed and studied a theta lifting from elliptic cusp forms to automorphic forms on the quaternion unitary group of signature $(1, q)$. The second named author proved that such a lifting provides bounded (or cuspidal) automorphic forms generating quaternionic discrete series. In this paper, restricting ourselves to the case of $q=1$, we reformulate Arakawa's theta lifting as a theta correspondence in the adelic setting and determine a commutation relation of Hecke operators satisfied by the lifting. As an application, we show that the theta lift of an elliptic Hecke eigenform is also a Hecke eigenform on the quaternion unitary group. We furthermore study the spinor $L$-function attached to the theta lift.

Article information

Tohoku Math. J. (2), Volume 60, Number 2 (2008), 227-251.

First available in Project Euclid: 7 July 2008

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Zentralblatt MATH identifier

Primary: 11F55: Other groups and their modular and automorphic forms (several variables)

Theta lifting Hecke operators Spinor $L$-functions


Murase, Atsushi; Narita, Hiro-aki. Commutation relations of Hecke operators for Arakawa lifting. Tohoku Math. J. (2) 60 (2008), no. 2, 227--251. doi:10.2748/tmj/1215442873.

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