Borcherds products on unitary groups

Abstract

In the present paper, we provide a construction of the multiplicative Borcherds lift for unitary groups U(1, m), which takes weakly holomorphic elliptic modular forms as input functions and lifts them to automorphic forms having infinite product expansions and taking their zeros and poles along Heegner divisors. In order to transfer Borcherds' theory to unitary groups, we construct a suitable embedding of U(1, m) into O(2, 2m). We also derive a formula for the values taken by the Borcherds products at cusps of the symmetric domain of the unitary group. Further, as an application of the lifting, we obtain a modularity result for a generating series with Heegner divisors as coefficients, along the lines of Borcherds' generalization of the Gross-Zagier-Kohnen theorem.

Local Borcherds Products for Unitary Groups

“…Theorem 0.2 can be seen a local analog to the global obstruction result showed by the author in [10,Section 5], which in turn is a unitary group version of the obstruction theory developed by Borcherds using Serre-duality [see 1, Theorem 3.1]. We discuss the relationship between the local and the global obstruction theories in section 4.1, and also how the two theorems relate to the quite similar results obtained by Bruinier and Freitag in the setting of orthogonal groups [see 4, Proposition 5.2, Theorem 5.4].…”

“…Since the statement of Theorem 4.1 holds for all sufficiently small ǫ, passing to the direct limit we get the statement for the local Picard group at the cusp ℓ. Now Theorem 4.1 formally resembles a global obstruction statement for unitary groups from [10] in the style of Borcherds [2]. It can be states as follows (by [10, Lemma 5, Theorem 4]): Since by results of Bruinier [3], the local obstruction space S k (ρ * D ) can be embedded into the global obstruction space S k (ρ * L ) [see also 4, Section 5], the global obstruction equation implies the local one.…”

“…For λ ∈ D ′ , H ∞ (λ) is the restriction of a local Heegner divisor attached to λ and, similarly, H ℓ (β, m) is the restriction of a composite local Heegner divisor, in the local Picard group for a generic boundary component of the symmetric domain, defined by kℓ as a two-dimensional isotropic subspace over Q. This follows from the embedding theory developed by the author in [9,10].…”

“…Through the quotient topology, this construction yields a topology on Γ\ (C ∪ {[ℓ]}). The complex structure is defined though the pullback under the canonical projection C ∩ {I R ; I ∈ Iso(V )} → X * Γ,BB , locally for each cusp, see [9] for details. This way, one gets the structure of a normal complex space on X * Γ,BB .…”

Weakly holomorphic modular forms on $$\Gamma _{0}(4)$$ and Borcherds products on unitary group $$\mathrm{U}(2,1)$$

“…We should mention that the analogue of the Borcherds product to unitary Shimura varieties of type (n, 1) has been worked out by Hofmann [15] (see also [12]). The Borcherds product expansion in the unitary case is a little more complicated as it is a Fourier-Jacobi expansion rather than Fourier expansion; the coefficients are theta functions rather than numbers.…”