Skew-holomorphic Jacobi forms of index 1 and Siegel modular forms of half-integral weight

Abstract

The isomorphism between Kohnen's plus space and Jacobi forms of index 1 was given by Eichler-Zagier. In this article, we generalize this isomorphism for higher degree in the case of skew-holomorphic Jacobi forms. r 2004 Elsevier Inc. All rights reserved.

A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences

“…This notion of plus space was generalized to general degree and used in the comparison with holomorphic and skew holomorphic Jacobi forms of general degree (cf. [9], [5], [7]). We review this "plus" subspace for our case.…”

“…We remark that when p = 2, we can also define an Euler 2-factor for F ∈ S + k−1/2, j ( 0 (4), ψ) in the same way as in [7]. Indeed, we can similarly define T * i (2) as in [9] and [5] by the pull back of the Hecke operators on holomorphic or skew holomorphic Jacobi forms. Denoting by λ * (2) and ω * (2) = ω(2) the eigenvalues of these operators, we can then define an Euler 2-factor as above.…”

“…For simplicity, we assume here that n = 2 though the results can be easily generalized. Most of the materials in this section are in [3], [17], [2], [9], [7], [5], [13], and [23], [6]. A definition of vector valued Klingen Eisenstein series has not been treated in the above papers and we sketch it here but it is almost the same as in the known cases.…”

“…for holomorphic or skew holomorphic Jacobi forms is given in [9] or [5] for the scalar valued case. The vector valued case is obtained by replacing the automorphy factor det k by ρ.…”

Classification of the space spanned by theta series and applications

“…For more details, see also [14], [3], [11] and [12]. Denote the symplectic group over the integers of degree n by Sp n (Z).…”