Meromorphic analogues of modular forms generating the kernel of Shimura’s lift

Abstract

Abstract. We study the meromorphic modular forms defined as sums of −k (k ≥ 2) powers of integral quadratic polynomials with negative discriminant. These functions can be viewed as meromorphic analogues of the holomorphic modular forms defined in the same way with positive discriminant, first investigated by Zagier in connection with the Doi-Naganuma map and then by Kohnen and Zagier in connection with Shimura-Shintani lifts. We compute the Fourier coefficients of these meromorphic modular forms and we show that they split into the sum of a meromorphic modular form with computable algebraic Fourier coefficients and a holomorphic cusp form.

Cycle integrals of meromorphic modular forms and Siegel theta functions

“…For with , we consider the functions which transform like modular forms of weight 2 k for . These functions were first studied by Zagier [ 12 ] for , in which case they are cusp forms, and by [ 6 ] for , in which case they are meromorphic modular forms. We obtain a refinement of by summing only over quadratic forms in the equivalence class of a fixed form P .…”

Niebur–Poincaré series and traces of singular moduli

“…of the f k,∆ are rational [22]. Bengoechea [2] introduced analogous functions for negative discriminants and showed that their Fourier coefficients are algebraic for small k. These functions are no longer holomorphic, but have poles at the CM-points of discriminant ∆. They were realized as regularized theta lifts by Bringmann, Kane, and von Pippich [7] and Zemel [26].…”

“…Next we examine algebraicity properties of the Fourier coefficients of f * d,D,N . Bengoechea [2] showed that for ∆ < 0 and k ∈ {2, 3, 4, 5, 7} (so that S 2k = {0}), the Fourier coefficients of f k,∆ lie in the Hilbert class field of Q( √ ∆). We have the following extension to the weight 2 case.…”

“…It has been observed in the remark following Theorem 2.1 of[15] that the Selberg-Kloosterman zeta functionS m,n (s) := a>0 K * (m, n, a) a shas an analytic continuation to s = 3 2 for mn < 0. Since S m,n has only finitely many poles in[1,2], there is an ε > 0 such that the function S d, n 2 D r σ has an analytic continuation to σ > −ε. This gives a locally uniform bound for σ > −ε and we obtain the analytic continuation for the sum over the positive n by just plugging in s = 0.…”