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

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20 W. DUKE AND Y. LI we arrive at (52) 〈g ψ , g ψ 〉 = −H(p) ∑ A∈Cl(F ) Since ψ 2 is non-trivial, we have equation (50). ψ 2 (A) log( √ y A |η(τ A )| 2 ). Remarks. i) By the same procedure, one can analyze the inner product between any weight one newforms. In particular, if g(z), h(z) ∈ S 1 (p) arises from different types of Galois representations, then 〈g, h〉 = 0. ii) This proposition is really a proven case of Stark’s conjecture in the abelian, rank one case. Next we prove the existence of the special pre-image. Proposition ∑ 4.2. Let ψ be a non-trivial character of Cl(F ). Then there exists ˜g ψ (z) = n≥n 0 r + ψ (n)qn ∈ M − 1 (p) with shadow g ψ (z) such that (i) When χ p (n) = 1 or n < − p+1 , the coefficient 24 r+ ψ (n) equals to zero. (ii) For n ≤ 0, the coefficients r + ψ (n) are of the form r + ψ (n) = −β ∑ A∈Cl(F ) ψ 2 (A) log |u(n, A)|, where β ∈ Q depends only on p, and u(n, A) ∈ O × H depends only on n and A. (iii) Let σ C ∈ Gal(H/F ) be the element associated to the class C ∈ Cl(F ) via Artin’s isomorphism. Then it acts on the units u(n, A) by σ C (u(n, A)) = u(n, AC −1 ). Proof. From Lemma 3.3, we know that the dimension of S + 1 (p), denoted by d + , is 1 2 (H(p) − 1)+d − . Let {g 1 , g 2 , . . . , g d+ } be a basis of S + 1 (p). Suppose we have chosen n 1 , n 2 , . . . , n d+ > 0 such that χ p (n k ) ≠ −1 and the matrix P := [c(g j , n k )] 1≤j,k≤d+ is invertible. Then for any ψ, there exists ˜g ψ (z) ∈ M − 1 (p) with shadow g ψ (z) and principal part coefficients r + ψ (−n) = 0 for all positive integer n not in the set {n 1, n 2 , . . . , n d+ } by Proposition 3.5. Furthermore, these r + ψ (−n) can be uniquely determined from solving a d + × d + system of equations. We will show that such ˜g ψ (z) can be made to satisfy the proposition. First, we can choose {n j : 1 ≤ j ≤ d + } such that n j ≤ (p + 1)/24. Let {g 1 , g 2 , . . . , g d+ } to be a q-echelon basis. Then these n j ’s can be chosen to be bounded by the supremum of the set {ord ∞ h(z) : h(z) ∈ S 1 + (p)}. Now, the square of a cusp form in S 1 (p) is in S 2 (p), the space of weight 2 cusp form of trivial nebentypus on Γ 0 (p). This gives rise to a holomorphic differential form on the modular curve X 0 (p). Atkin showed in [1] that ∞ is not a Weiestrass point of the modular curve X 0 (p), i.e. ord ∞ f(z) is no more than the genus of X 0 (p). Using the Riemann-Hurwitz formula (see [37]), we know that the genus of X 0 (p) is bounded by (p + 1)/12. So the n j ’s can be chosen to be bounded by (p + 1)/24. □
MOCK-MODULAR FORMS OF WEIGHT ONE 21 Let {g j (z) : 1 ≤ j ≤ (H(p) − 1)/2} be the dihedral newforms, and label the class group characters as {ψ j : 1 ≤ j ≤ H(p) − 1} such that g ψj (z) = g j (z) and ψ j = ψ H(p)−j for 1 ≤ j ≤ (H(p) − 1)/2. When (H(p) + 1)/2 ≤ j ≤ d + , let g j (z) be linear combinations of non-dihedral newforms such that g j (z) has integral coefficients and the set {g j (z) : (H(p) + 1)/2 ≤ j ≤ d + } is linearly independent. Then for each non-trivial ψ, the values {r + ψ (−n k) : 1 ≤ k ≤ d + } satisfies the equation P · [δ(n k )r + ψ (−n k)] 1≤k≤d+ = R, where R is the d + × 1 matrix (〈g j , g ψ 〉) 1≤j≤d+ . By Proposition 4.1, the inner product 〈g j , g ψ 〉 ∑ is either 0 or 1 12 A∈Cl(F ) ψ2 (A) log |u A |. So we have ⎛ ⎞ 0. ∑ 0 1 R = 12 A ψ2 (A) log |u A | ⎜ ⎟ ⎝ 0. ⎠ 0 Let A 0 be the principal class in Cl(F ) and label the non-principal ( classes as A i for 1 ≤ i ≤ H(p) − 1 such that A −1 = A H(p)−i . Let M ′ M1 0 be the matrix 0 Id d− ), where M 1 is an i 1 (H(p) − 1) × 1 (H(p) − 1) matrix defined by 2 2 [ψ j (A i ) + ψ j (A H(p)−i ) − 2] 1≤i,j≤(H(p)−1)/2 and Id d− is the d − × d − identity matrix. The product M ′ P is the matrix ⎛ ⎞ H(p)(r A1 (n 1 ) − r A0 (n 1 )) · · · H(p)(r A1 (n d+ ) − r A0 (n d+ )) . . .. . M ′ H(p)(r P = A(H(p)−1)/2 (n 1 ) − r A0 (n 1 )) · · · H(p)(r A(H(p)−1)/2 (n d+ ) − r A0 (n d+ )) , c(g ⎜ (H(p)+1)/2 , n 1 ) · · · c(g (H(p)+1)/2 , n d+ ) ⎝ . ⎟ . .. . ⎠ c(g d+ , n 1 ) · · · c(g d+ , n d+ ) which has integer coefficients. It is also non-singular since {ϑ Aj − ϑ A0 : 1 ≤ j ≤ (H(p) − 1)/2} ∪ {g j : (H(p) + 1)/2 ≤ j ≤ d + } is a basis of S 1 + (p). The product M ′ R is the d + × 1 matrix ⎛ ∑ A ψ2 (A) log |u A (A 1 )| where Also by (47), we have M ′ R = ⎜ ⎝ 1 12 1 12 ⎞ ∑ . A ψ2 (A) log |u A (A (H(p)−1)/2 )| , ⎟ 0. ⎠ 0 u A (B) := u A √ B u A √ B −1 u 2 A σ C (u A (B)) = u AC −1(B) ∈ O × H .
Page 1 and 2: MOCK-MODULAR FORMS OF WEIGHT ONE W.
Page 3 and 4: MOCK-MODULAR FORMS OF WEIGHT ONE 3
Page 19: MOCK-MODULAR FORMS OF WEIGHT ONE 19
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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