On two geometric theta lifts

Abstract

takes values in Z q (X). Then for η ∈ Z (p−1)q c (X), the compactly supported closed (p − 1)qforms on X, the Kudla-Millson lift is defined byIt turns out that Λ KM (τ, η) is actually a holomorphic modular form of weight 2 − k, so that we have a mapwhich also factors through cohomology. Moreover, the Fourier coefficients of Λ KM are given by periods of η over the special cycles.Theorem 1.2. Assume D be Hermitian, i.e., q = 2, and let f ∈ H + k with constant coefficient a + (0). We then have the following identity of closed 2-forms on X:Therefore the maps Λ B and Λ KM are naturally adjoint via the standard pairing ( ,Furthermore, this duality factors through cohomology, and H + k /M ! k , respectively. (See also Theorem 6.3.) This is based on the fundamental relationship between the two theta series involved: Theorem 1.3. Let L 2−k be the lowering Maass operator of weight 2 − k on H. Then L 2−k Θ(τ, z, ϕ KM ) = −dd c Θ(τ, z, ϕ 0 ).We show this by switching to the Fock model of the Weil representation. Then the idea for the proof of Theorem 1.2 is given by the following formal (!) calculation:

K3 string theory, lattices and moonshine

“…In [42] Duncan and Mack-Crane proposed two (possibly coinciding) weight 0 index 1 weak Jacobi forms for a certain G g ⊆ SL 2 (Z), denoted φ g,+ (τ , z) and φ g,− (τ , z), to each of the four-plane preserving conjugacy classes of Co 0 . 9 The construction of φ g,+ and φ g,− is based on an N = 1 super VOA of central charge c = 12, which has symmetry group Co 0 [40]. Concretely, one has…”

Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

“…3 of [7], the restriction of ξ 2−k to H 2−k,∞ ( 0 (N ), χ ) defines a surjective map to S k ( 0 (N ), χ). One now argues as in the proof of Lemma 7.3 of N ), χ ) and g ∈ S k ( 0 (N ), χ).…”

“…2 for definitions), more general automorphic forms which have been a source of recent interest due to their connection to Ramanujan's mock theta functions, Borcherds products, derivatives of modular L-functions, and traces of singular moduli (see [2][3][4][5][6][7][8]20,21]). …”

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

“…To justify this vanishing, recall that there is an anti-linear differential operator ξ k that takes weak Maass forms of weight k to weakly holomorphic modular forms of weight 2 − k (see Proposition 3.2 of [5]). It is defined by…”