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A quaternionic Saito–Kurokawa lift and cusp forms on G2
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-06-30, DOI: 10.2140/ant.2021.15.1213
Aaron Pollack

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.