Petersson norms of not necessarily cuspidal Jacobi modular forms and applications

Abstract

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on H 2 and those on H 1 × C (here H n denotes the Siegel upper half space of degree n). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms; and as a byproduct we also show that any Siegel modular form of degree 2 is determined by its 'fundamental' Fourier coefficients.

Cones of special cycles of codimension 2 on orthogonal Shimura varieties

“…The main obstacles are the so-called Klingen Eisenstein series, whose coefficient growths behave sometimes as for the Siegel Eisenstein series E k 2 and sometimes as for cusp forms, depending on the chosen sequences of matrices of increasing determinant. In the recent paper [BD18], Böcherer and Das have proposed an extensive study of the growth of these coefficients. The aim of this section is to clarify the previous issue and to recall from [BD18] the necessary results for the purposes of this paper.…”

“…In the recent paper [BD18], Böcherer and Das have proposed an extensive study of the growth of these coefficients. The aim of this section is to clarify the previous issue and to recall from [BD18] the necessary results for the purposes of this paper.…”

“…In fact, we need to know with respect to which sequences (T j ) j∈N in Λ + 2 the coefficients a k 2 (f, T j ) grow with the same order of magnitude of a k 2 (T j ). We illustrate here a solution of this problem following the wording of [BD18].…”

“…The main resource we use to treat this issue is the recent paper [BD18], where the growth of the coefficients of the Klingen Eisenstein series is clarified; we refer to Sections 2 and 3 for the needed background.…”

On the Fourier Coefficients of Siegel modular Forms

“…In fact, these aspects were mentioned as 'difficult open' problems in [33, remark 2.6, 2.7]. While the latter (in degree 2) was addressed to in [7,Prop. 7.7] by the authors, in this paper, we settle both of these questions in the most general case (for full level), in particular including vector-valued modular forms.…”