Sieving Zagier’s Harmonic Xxxxx Form Sample Clauses

Sieving Zagier’s Harmonic Xxxxx Form. ‌ We require the following result, which shows that we can define holomorphic modular forms whose coefficients are supported on fundamental discriminants satisfying local conditions and are given by class numbers. Given sets S+, S−, S0 as in Theorem 1.2.1, we let AΣ be defined as the set of positive integers n such that the following hold: 1. For p ∈ S+ ∪ S− ∪ S−, p2 ‡ n. 2. Q(√−n) splits at the primes in S+, ramifies at the primes in S0, and is inert at the primes in S−. 1.1. Let S+, S−, S0 be sets as in Theorem 1.2.1, and assume that S− is nonempty. Then there is a weight 3 modular form HΣ(z) = Σ∞ a(n)qn on Γ0(NΣ), where Nσ is as in equation (1.5), such that a(n) = H(n) n ∈ AΣ 0 otherwise The idea is to take combinations of twists of Zagier’s function H(z) to obtain holomorphic modular form. The key properties of H(z) that allow us to do this are 1. The Fourier expansion of the non-holomorphic part is supported on terms of the form q−n2 , which allows us to use twisting to annihilate the non-holomorphic part of H(z), and 2. H(z) has moderate growth at poles, which ensures that any linear combina- tion of twists of H(z) will not have any exponential singularities, as a weakly holomorphic modular form would. Σ For χ, ψ Dirichlet characters modulo m and N , the twist of G(z) := n∞=0 a(n, y)qn ∈ Hk(Γ0(N ), ψ) by χ is given by Σ Gχ(z) = χ(n)a(n, y)qn. n∈Z If d is a positive integer, the operators U (d), V (d) are defined, as one does when working with holomorphic modular forms, by Σ (G|U (d))(z) := a(dn, y)qn n∈Z and Σ (G|V (d))(z) := a(n, y)qdn. n∈Z