Singular moduli of higher level and special cycles

Abstract

We describe the complex multiplication (CM) values of modular functions for Γ0(N ) whose divisor is given by a linear combination of Heegner divisors in terms of special cycles on the stack of CM elliptic curves. In particular, our results apply to Borcherds products of weight 0 for Γ0(N ). By working out explicit formulas for the special cycles, we obtain the prime ideal factorizations of such CM values.

CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1

“…The representation numbers |L(E P , ι P , m, a, µ)| can be determined following [KRY99] (completely explicit in the case of a prime discriminant and up to Galois conjugation in general). We refer to [Ehl15] for details.…”

“…We will relate each of these coefficients to a CM value of a meromorphic modular form of weight zero given by a Borcherds product. Then, we apply the results of [Ehl15] to determine their prime ideal factorization.…”

“…In this case, Ψ(z, g) is a meromorphic modular form of weight zero. By choosing g carefully, we can assure that those special values of Ψ(z, g) lie in the Hilbert class field H k and determine their prime ideal factorization using the results of [Ehl15].…”

“…The second factor on the right-hand side can be shown to be constant for a special choice of g giving the desired relation. We then apply the main result of [Ehl15] which relates the CM values of Borcherds products to the special cycles.…”

“…In [Ehl15], we gave formulas for the multiplicities of the cycles Z(m) which provide a prime ideal decomposition of the algebraic numbers α(a, m) in Theorem 1.2. In the case of a prime discriminant the formulas are particularly simple and explicit.…”