Mock-modular forms of weight one - UCLA Department of Mathematics

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16 W. DUKE AND Y. LI where V p = ( { p 1, p ≡ 1 (mod 4) , ε 1) p = i, p ≡ 3 (mod 4) Since p ≡ 3 (mod 4), we know that ε p = i. From the definition of pr −ɛ , we have (37) (pr −ɛ f)(z) = 1 (−ɛih(z) + f(z)) 2 ( = 1 −ɛi √ (f| 1 W p V p )(z) + ∑ ) (1 − χ p (n)ɛ) a(n, y)q n ∈ H !,ɛ 2 p 1 (p). n∈Z Notice that the action V p produces an expansion with coefficients supported on multiples of p. So if pr −ɛ (f) = 0, then a(n, y) = 0 whenever χ p (n) = −ɛ and f ∈ H !,ɛ 1 (p) by definition. Conversely if f ∈ H !,ɛ 1 (p), then pr −ɛ (f) ∈ H !,+ 1 (p) ∩ H !,− 1 (p), which means the coefficients ξ 1 (pr −ɛ (f)) ∈ M ! 1(p) are supported on multiples of p. A lemma due to Hecke ([33, Lemma 6, p.32] ) implies that (38) M !,+ 1 (p) ∩ M !,− 1 (p) = {0}. So ξ 1 (pr −ɛ (f)) = 0 and pr −ɛ (f) is holomorphic and also contained in M !,+ 1 (p) ∩ M !,− 1 (p). By the same argument, pr −ɛ (f) vanishes. From this, we can deduce that H !,ɛ 1 (p) is the eigenspace of iU p W p with eigenvalue ɛ, and pr ɛ restricted to H !,ɛ 1 (p) is the identity. Using the relationship (f| 1 Wp 2 )(z) = −f(z), we can then deduce equation (34). It is easy to see that we have the following direct sum decomposition H 1(p) ! = H !,+ 1 (p) ⊕ H !,− 1 (p) f = pr + (f) + pr − (f). Applying the same procedure to the harmonic Maass form associated to any mock-modular form in M ! 1(p), we have the same decomposition of space M ! 1(p) = M !,+ 1 (p) ⊕ M !,− 1 (p). The decomposition of the space H 1(p) ! above behaves well under the action of ξ 1 . By Theorem 2.2, we can prove a statement about surjectivity of ξ 1 : H1(p) ɛ → S1 −ɛ (p). Furthermore, there is a nice characterization of the space S1(p) ɛ as a consequence of facts about weight one modular forms. Lemma 3.3. For ɛ = ±1, we have a surjection (39) ξ 1 : H ɛ 1(p) → S −ɛ 1 (p). In other words, for any g(z) ∈ S −ɛ 1 (p), there exists a mock-modular form ˜g(z) ∈ M ɛ 1(p) whose shadow is g(z). In addition, S + 1 (p) contains all dihedral cusp forms. If there are 2d − non-dihedral forms in S 1 (p), then the spaces S + 1 (p) and S − 1 (p) have dimensions 1 2 (H(p)−1)+d − and d − respectively Proof. By Theorem 2.2, the following map is surjective (40) ξ 1 : H 1 (p) → S −ɛ 1 (p). □
MOCK-MODULAR FORMS OF WEIGHT ONE 17 Since ξ 1 commutes with the action of U p and W p and conjugates their coefficients, it commutes with the operator pr ɛ as follows (41) ξ 1 (pr ɛ (f)) = pr −ɛ (ξ 1 (f)). The operator pr −ɛ is the identity when restricted to S1 −ɛ (p). So the preimage of S1 −ɛ (p) under ξ 1 lies in H1(p) ɛ and the map ξ 1 : H1(p) ɛ → S1 −ɛ (p) is surjective for ɛ = ±1. Now if f ∈ S 1 (p) is a dihedral newform, then it is a linear combination of theta series, whose n th coefficient is zero if χ p (n) = −1. So f ∈ S 1 + (p) by definition. If f(z) = ∑ n≥1 c(f, n)qn is a octahedral or icosahedral newform, then f(z) and f(z) := f(z) = ∑ n≥1 c(f, n)qn are linearly independent newforms. When l ≠ p is a prime number, we have the relationship (42) c(f, l) = χ p (l)c(f, l). This is a consequence of the formula of the adjoint of the l th Hecke operator with respect to the Petersson inner product [34, p.21]. By the definition of S1(p), ɛ we have f + f ∈ S 1 + (p) and f − f ∈ S1 − (p). Since the number of octahedral and icosahedral newforms is set to be 2d − , we obtain the formulae for the dimensions. □ Remark: The spaces S ɛ 1(p) should be compared to the spaces M + and M − in (9.1.2) in [36]. If all octahedral and icosahedral newforms f(z) ∈ S 1 (p) satisfy (43) (f| 1 W p )(z) = −if(z), then M + is the span of the weight one Eisenstein series and S + 1 (p) and M − = S − 1 (p). Besides compatibility with ξ 1 , the decomposition also behaves nicely with respect to the regularized inner product. As a consequence of Lemma 3.1, we have the following proposition. Proposition 3.4. Let f(z) ∈ M !,ɛ 1 (p), g(z) ∈ S ɛ′ 1 (p) and ˜g(z) ∈ M −ɛ′ 1 (p) with shadow g(z) and Fourier expansion ∑ n∈Z c+ (n)q n . If ɛ = ɛ ′ , then (44) 〈f, g〉 reg = ∑ n∈Z δ(n)c(f, −n)c + (n). Here δ(n) is 2 if p|n and 1 otherwise. If ɛ ≠ ɛ ′ , then 〈f, g〉 reg = 0. Proof. Let ĝ(z) ∈ H −ɛ′ 1 (p) be the harmonic Maass form associated to ˜g(z). Then both assertions are immediate consequences of Lemma 3.1 and (f| 1 W p )(z) = −iɛ(f| 1 U p )(z), (ĝ| 1 W p )(z) = −iɛ ′ (ĝ| 1 U p )(z). Since the adjoint of W p (resp. U p ) is W −1 p = −W p (resp. W p U p Wp −1 ) with respect to the regularized inner product, it is easy to check that pr ɛ is self-adjoint, which also proves the second claim. □ When g(z) above is zero, i.e. ˜g(z) is modular, equation (44) reduces to an orthogonal relationship between Fourier coefficients of weakly holomorphic modular forms. In particular, we can take ˜g(z) to be cusp forms and obtain relations satisfied by {c(f, −n) : n ≥ 1}. These relations turn out to characterize the space M !,ɛ 1 (p), which gives a nice characterization of the space M ɛ 1(p). Proposition 3.5. Let g(z) ∈ S1(p). ɛ Then there exists a mock-modular form ˜g(z) ∈ M −ɛ 1 (p) with shadow g(z) and prescribed principal part ∑ n
Page 1 and 2: MOCK-MODULAR FORMS OF WEIGHT ONE W.
Page 3 and 4: MOCK-MODULAR FORMS OF WEIGHT ONE 3
Page 15: MOCK-MODULAR FORMS OF WEIGHT ONE 15
Page 31 and 32: Similar to the proof of Theorem 5.1
Page 39 and 40: where T := (T kk ′) 0≤k,k ′

Mock-modular forms of weight one - UCLA Department of Mathematics

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