Arithmeticity of holomorphic cuspforms on Hermitian symmetric domains

Abstract

We prove Galois equivariance of ratios of Petersson inner products of holomorphic cuspforms on symplectic, unitary, or Hermitian orthogonal groups. As a consequence, we show that the ratios of Petersson norms of such cuspforms with the same Hecke eigenvalues are algebraic. We also show that spaces of such cuspforms of sufficiently high fixed weight and level are spanned by theta series.

Automorphic forms for some even unimodular lattices

“…In the latter case, say u = −v 2 , then a/v = −a/v, which implies that a/v is an integer multiple unique up to scaling [Min,§2.3]. We may also call on [LU,§4] for the non-vanishing of Z ∞ (s 0 , F, Φ). We may now proceed as in the proof of [LU,Theorem 3].…”

“…We may also call on [LU,§4] for the non-vanishing of Z ∞ (s 0 , F, Φ). We may now proceed as in the proof of [LU,Theorem 3]. This is close to Böcherer's idea of using the Siegel-Weil formula to substitute for the Eisenstein series in a pull-back formula/doubling integral [Bo], and our condition N > 2m is in order to apply Theorem 9.1, with 2m substituted for m because of the doubling.…”

“…We need to know that Z ∞ (s 0 , F, Φ) p|D Z p (s 0 , F, Φ) = 0. For p | D an argument of Lanphier and Urtis [LU,§4,case v ∤ n] shows that Z p (s 0 , F, Φ) = 0. To justify this, note that even though U m,m is ramified at such p, the maximal compact subgroup U m,m (Z p ) is special (if not hyperspecial), as noted in [AK, §2.1], so spherical vectors are still…”