Binary Quadratic Forms, Shintani Lifts, and Twisted L-Values
Abstract
Let F∈ S2knew(M,χ2) be a Hecke newform, and D a fundamental discriminant with (−1)χ(−1)D>0. Via the Shimura Correspondence, we get associated modular forms of half-integral weight, and Waldspurger showed that for such g=∑ cnqn ∈ Sk+1/2(4M',χ;F), if c|D|(g)≠ 0, then |c|D|(g)|2 ∝ |D|k−1/2L(F⊗ χ̅⊗ χD,k). Kohnen refined this in the case that f∈ S2knew(N) for N odd and squarefree, making explicit |c|D|(g)|2 for g ∈ S+k+1/2(4N;f). However, c|D|(g)=0 unless (D/p)=wp(f) for all p|N. In this paper, we produce more generally g ∈ S+k+1/2(4 lcm(N,N12),χ;f⊗ χ), where χ is a character of odd squarefree conductor N1, using a family of generalised Shimura/Shintani maps ΨχD,ΨχD* due to Kojima and Tokuno. Extending their work (and Kohnen's), we compute its Petersson norm, and give an explicit formula for all the |cm(g)|2. The key technical ingredient is an explicit set of representatives for L0M1,M(M12D2)/Γ0(M) of binary quadratic forms. We also use a new multiplicity-one result based on work of Kohnen, Ueda, and Waldspurger. In the case N1|N, we reconstruct forms previously considered by Baruch-Mao and Prasanna, which change Kohnen's condition to (D/p)=−wp(f) for all p|N1, thus relating twisted L-values to squares of Fourier coefficients for all classes of discriminants.
Related
- S-squareful values of binary quadratic forms
- On the polynomial values represented by binary quadratic forms
- Positive Values of Inhomogeneous Quadratic Forms of Type (1, 4): The Constant Γ_{1,4} = 8
- On the average of class numbers of indefinite binary quadratic forms
- Integral Forms of Twisted Quantum Affine Algebras and Quantum Frobenius
BibTeX
@misc{chambers2026binary,
title = {{Binary Quadratic Forms, Shintani Lifts, and Twisted L-Values}},
author = {Mark Chambers},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32470},
eprint = {2610.32470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32470},
}
RIS
TY - UNPB
TI - Binary Quadratic Forms, Shintani Lifts, and Twisted L-Values
AU - Mark Chambers
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32470
UR - https://arxiv.org/abs/2610.32470
AB - Let F∈ S_2k^new(M,χ^2) be a Hecke newform, and D a fundamental discriminant with (−1)χ(−1)D>0. Via the Shimura Correspondence, we get associated modular forms of half-integral weight, and Waldspurger showed that for such g=∑ c_nq^n ∈ S_k+1/2(4M',χ;F), if c_|D|(g)≠ 0, then |c_|D|(g)|^2 ∝ |D|^k−1/2L(F⊗ χ̅⊗ χ_D,k). Kohnen refined this in the case that f∈ S_2k^new(N) for N odd and squarefree, making explicit |c_|D|(g)|^2 for g ∈ S^+_k+1/2(4N;f). However, c_|D|(g)=0 unless (D/p)=w_p(f) for all p|N. In this paper, we produce more generally g ∈ S^+_k+1/2(4 lcm(N,N_1^2),χ;f⊗ χ), where χ is a character of odd squarefree conductor N_1, using a family of generalised Shimura/Shintani maps Ψ_χ^D,Ψ_χ^D^* due to Kojima and Tokuno. Extending their work (and Kohnen's), we compute its Petersson norm, and give an explicit formula for all the |c_m(g)|^2. The key technical ingredient is an explicit set of representatives for L^0_M_1,M(M_1^2D^2)/Γ_0(M) of binary quadratic forms. We also use a new multiplicity-one result based on work of Kohnen, Ueda, and Waldspurger. In the case N_1|N, we reconstruct forms previously considered by Baruch-Mao and Prasanna, which change Kohnen's condition to (D/p)=−w_p(f) for all p|N_1, thus relating twisted L-values to squares of Fourier coefficients for all classes of discriminants.
ER -