A conditional construction of Artin representations for real analytic Siegel cusp forms of weight (2,1)

Abstract

Let F be a vector-valued real analytic Siegel cusp eigenform of weight (2, 1) with the eigenvalues − 5 12 and 0 for the two generators of the center of the algebra consisting of all Sp4(R)-invariant differential operators on the Siegel upper half plane of degree 2. Under natural assumptions in analogy of holomorphic Siegel cusp forms, we construct a unique symplectically odd Artin representation ρF : G Q −→ GSp4(C) associated to F . For this, we develop the arithmetic theory of vector-valued real analytic Siegel modular forms. Several examples which satisfy these assumptions are given by using various transfers and automorphic induction.

Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional & Hilbert Modular Forms

“…[ 66 ] The Fourier coefficients of non‐singular higher weight forms were obtained by Kim and Yamauchi in ref. [68, 69] based on work by Karel. [ 70 ] Using the notation of refs.…”

“…To our knowledge the relationship between higher powers of $$E_4(Z)$$ and the higher weight k modular forms or cusp forms of ref. [68–70] have not yet been studied by mathematicians.…”

An Equidistribution Theorem for Holomorphic Siegel Modular Forms for and Its Applications

“…The algebra of all Sp 4 (R)-invariant differential operators on H 2 is isomorphic to C[Ω, ∆], the commutative polynomial ring of two variables, where Ω is the degree 2 Casimir element, and ∆ is the degree 4 element. (see Section 5 of [33] for the details and Ω = ∆ 1 , ∆ = ∆ 2 in the notation there.) It is easy to see that (3,1), or (0, 0), the two eigenvalues are 0 and 4.…”