Vol. 15, No. 5, 2021

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A quaternionic Saito–Kurokawa lift and cusp forms on $G_2$

Aaron Pollack

Vol. 15 (2021), No. 5, 1213–1244
Abstract

We consider a special theta lift 𝜃(f) from cuspidal Siegel modular forms f on Sp4 to “modular forms” 𝜃(f) on SO(4,4) in the sense of our prior work (Pollack 2020a). This lift can be considered an analogue of the Saito–Kurokawa lift, where now the image of the lift is representations of SO(4,4) that are quaternionic at infinity. We relate the Fourier coefficients of 𝜃(f) to those of f, and in particular prove that 𝜃(f) is nonzero and has algebraic Fourier coefficients if f does. Restricting the 𝜃(f) to G2 SO(4,4), we obtain cuspidal modular forms on G2 of arbitrarily large weight with all algebraic Fourier coefficients. In the case of level one, we obtain precise formulas for the Fourier coefficients of 𝜃(f) in terms of those of f. In particular, we construct nonzero cuspidal modular forms on G2 of level one with all integer Fourier coefficients.

Keywords
$G_2$ modular forms, Saito–Kurokawa, cusp forms, theta correspondence, Fourier coefficients
Mathematical Subject Classification 2010
Primary: 11F03
Secondary: 11F30, 20G41
Milestones
Received: 29 February 2020
Revised: 16 September 2020
Accepted: 21 October 2020
Published: 30 June 2021
Authors
Aaron Pollack
Department of Mathematics
University of California, San Diego
La Jolla, CA
United States