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

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28 W. DUKE AND Y. LI The proof of Proposition 6.1 follows the same steps as in the proof of Theorem 5.1 and Corollary 5.2. First, we will express 〈f lift,N,B , g ψ 〉 reg in terms of other regularized inner products and the limit of an infinite sum. Then, via Green’s function, we will relate the limit of the infinite sum to the height pairings between Heegner divisors on J 0 (N), the Jacobian of X 0 (N). Since the Heegner divisor is trivial on J 0 (N), the height pairing is the same as the value of some modular function evaluated at a Heegner point. From there, we can deduce equation (68). Before presenting the proof, we need some notations and facts about level N modular forms. For this part, we only require N to be a prime number. Let M 2(N), ! resp. M 0(N), ! be the space of weakly holomorphic weight 2 modular forms, resp. modular functions, of level N, and trivial nebentypus. Denote the subspaces of M 2(N) ! of cusp forms and holomorphic modular forms by M 2 (N), S 2 (N) respectively. Let M !,new 0 (N) be the subspace of M 0(N) ! containing weakly holomorphic modular forms f(z) satisfying (69) (f|W N )(z) = −N(f|U N )(z). The following lemma shows that M ! 0(N) can be be decomposed into the direct sum of M !,new 0 (N) and level one modular functions. Lemma 6.2. Let f(z) be a modular function of prime level N. Then it can be written uniquely as f(z) = f 1 (Nz) + f 2 (z), such that f 1 (z) is a modular function of level 1 and f 2 (z) ∈ M !,new 0 (N). Proof. Let a, b ∈ Z. With some matrix calculations, we have ( ( )) f| UN W N + a N (z) = a−1 f(z) + N TrN 1 (f)(Nz) ( ( ) ( )) f| UN W N + a UN W N N + b N (z) = (a−1)(b−1) f(z) + a+b+N−1 Tr N N 2 N 1 (f)(Nz) Set f j (z) as follows and we are done f 1 (z) = N N+1 TrN 1 (f)(z), f 2 (z) = − N N+1 NW N ) (z) − f(z)) . Alternatively, one can characterize the space M ! 0(N) via the space S 2 (N). Definition 6.3. A set of numbers {λ m : m ≥ 1} is called a relation for S 2 (N) if (i) λ m ∈ C is non-zero for all but finitely many m. (ii) For any cusp form h(z) = ∑ m≥1 c(h, m)qm ∈ S 2 (N), the numbers {λ m } satisfy ∑ (70) δ N (m)λ m c(h, m) = 0. m≥1 where δ N (m) = N + 1 if N|m and 1 otherwise. A relation {λ m } m≥1 is called integral if λ m ∈ Z for all m ≥ 1. Denote Λ N the set of all relations for S 2 (N). For any f(z) ∈ M !,new 0 (N), one knows that the principal part coefficients, {c(f, −m) : m ≥ 1}, is a relation for S 2 (N) from works by Petersson. By considering the Fourier expansion of vector-valued Poincaré series, (See [7], [27], [32]), one can also show that the converse is true. So given λ = {λ m } ∈ Λ N , the expression ∑ λ m q −m m≥1 □
MOCK-MODULAR FORMS OF WEIGHT ONE 29 is necessarily the principal part of the Fourier expansion at infinity of a unique function f λ (z) ∈ M !,new 0 (N). Alternatively, this statement follows essentially from an application of Serre duality [5, §3]. Suppose λ ∈ Λ N is integral. Then the n th Fourier coefficient of f λ (z) is integral when n ≠ 0 since f λ is a rational function of j(z) and j(Nz). Let g N be the genus of Γ 0 (N). From [1], we know that for all h(z) ∈ S 2 (N), ord ∞ (h(z)) ≤ dim(S 2 (N)) = g N ≤ (N + 1)/12. Using matrix computations similar to that in the proof of Lemma 6.2, one can show that (71) h| 2 U N = −h| 2 W N for all h(z) ∈ S 2 (N), implying that U N ◦ U N is the identity operator on S 2 (N). Combining these facts together gives us the following lemma about M !,new 0 (N). Lemma 6.4. The space M !,new 0 (N) is spanned by modular functions ∞∑ f m (z) = q −m + c m (n)q n , n=−g N with m ≥ g N + 1. For these m, the coefficient c m (n) are integers when n ≠ 0. Furthermore, for all m ≥ g N + 1, f N 2 m(z) − N+1 δ N (m) f m(z) = q −N 2m − N+1 δ N (m) q−m + O(1). There is a similar duality statement dictating the existence of forms in M ! 2(N). Lemma 6.5. For every n ≥ 0, there exists P n,N (z) = ∑ m∈Z c(P n,N, m)q m ∈ M ! 2(N) such that (i) At the cusp infinity, P n,N (z) = q −n + O(q). (ii) (P n,N |W N )(z) = −(P n,N |U N )(z). Furthermore, if f(z) = ∑ m∈Z c(f, m)qn ∈ M !,new 0 (N), then ∑ δ N (m)c(f, m)c(P n,N , −m) = 0. m∈Z Now, we will recall some facts about Heegner points on the modular curve from [22] and their height pairings on the Jacobian. Let X 0 (N) be the natural compactification of Y 0 (N), the open modular curve over Q classifying pairs of elliptic curves (E, E ′ ) with an order N cyclic isogeny φ : E → E ′ . The complex points of X 0 (N) has a structure of a compact Riemann surface and can be identified with H ∪ P 1 (Q) modulo the action of Γ 0 (N). Heegner points on X 0 (N) corresponds to pair of elliptic curves (E, E ′ ) having complex multiplication by the same ring O in some imaginary quadratic field F . This occurs only when there exists an O-ideal n dividing N such that O/n is cyclic of order N. In this case, E(C) is isomorphic to C/a with a an invertible O-submodule in F . Since this a can be chosen independent of homothety by elements in F × , we only need to consider [a], the class of a in Pic(O). So a Heegner point on X 0 (N) can be expressed as the triplet (O, n, [a]). To find the image of such a Heegner point in H, choose an oriented basis 〈ω 1 , ω 2 〉 of a such that an −1 has basis 〈ω 1 , ω 2 /N〉. Then τ [a],n := ω 1 /ω 2 ∈ Γ 0 (N)\H is the image of this Heegner point. For our purpose, F = Q( √ −p) and O = O F . Then we require χ p (N) = 1 for Heegner points to exist. In that case, N splits into nn in F with n = ZN + Z bn+√ −p 2 .
Page 1 and 2: MOCK-MODULAR FORMS OF WEIGHT ONE W.
Page 3 and 4: MOCK-MODULAR FORMS OF WEIGHT ONE 3
Page 27: MOCK-MODULAR FORMS OF WEIGHT ONE 27
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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