A quaternionic Saito–Kurokawa lift and cusp forms on G2

Pollack, Aaron

doi:10.2140/ant.2021.15.1213

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

Aaron Pollack

Vol. 15 (2021), No. 5, 1213–1244

DOI: 10.2140/ant.2021.15.1213

Abstract

We consider a special theta lift $𝜃 (f)$ from cuspidal Siegel modular forms $f$ on ${Sp}_{4}$ 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 $G_{2} \subseteq SO (4, 4)$ , we obtain cuspidal modular forms on $G_{2}$ 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 $G_{2}$ 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

Vol. 15, No. 5, 2021