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Register a new userZagier duality and integrality for Fourier coefficients for weakly holomorphic modular forms
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Authors
Yichao Zhang
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In this note, we generalize the isomorphisms to the case when the discriminant form is not necessarily induced from real quadratic fields. In particular, this general setting includes all the subspaces with epsilon-conditions, only two spacial cases of which were treated before. With this established, we shall prove the Zagier duality for canonical bases. Finally, we raise a question on the integrality of the Fourier coefficients of these bases elements, or equivalently we concern the existence of a Miller-like basis for vector valued modular forms.
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Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds theorem on the infinite product expansions of integer weight modular forms on $SL_2(ZZ)$ with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called the Zagier duality. After the result of Zagier, this type duality was studied broadly in various view points including the theory of a mock modular form. In this paper, we consider this problem with the Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using holomorphic Poincare series and their supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights $-k$ and $k+2$ respectively, $k>0$, for a $H$-group. We also investigate the structures of them such as the images under the differential operators $D^{k+1}$ and $xi_{-k}$ and quadric relations of the critical values of their $L$-functions.
In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which is related to the weight of Borcherds lifts when the weight is zero. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, and obtain divisibility results in an orthogonal direction on reduced modular forms.
Let $lambda$ be an integer, and $f(z)=sum_{ngg-infty} a(n)q^n$ be a weakly holomorphic modular form of weight $lambda+frac 12$ on $Gamma_0(4)$ with integral coefficients. Let $ellgeq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{ell^m})(z) = sum_{n=0}^infty b(n)q^n$ has the form [ (f|U_{ell^m})(z) equiv b(0) + sum_{i=1}^{t}sum_{n=1}^{infty} b(d_i n^2) q^{d_i n^2} pmod{ell}, ] where $d_1, ldots, d_t$ are square-free positive integers, and the operator $U_ell$ on formal power series is defined by [ left( sum_{n=0}^infty a(n)q^n right) bigg| U_ell = sum_{n=0}^infty a(ell n)q^n. ] Then, $lambda equiv 0 pmod{frac{ell-1}{2}}$. Moreover, if $tilde{f}$ denotes the coefficient-wise reduction of $f$ modulo $ell$, then we have [ biggl{ lim_{m rightarrow infty} tilde{f}|U_{ell^{2m}}, lim_{m rightarrow infty} tilde{f}|U_{ell^{2m+1}} biggr} = biggl{ a(0)theta(z), a(0)theta^ell(z) in mathbb{F}_{ell}[[q]] biggr}, ] where $theta(z)$ is the Jacobi theta function defined by $theta(z) = sum_{ninmathbb{Z}} q^{n^2}$. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.
Let $M_k^!(Gamma_0^+(3))$ be the space of weakly holomorphic modular forms of weight $k$ for the Fricke group of level $3$. We introduce a natural basis for $M_k^!(Gamma_0^+(3))$ and prove that for almost all basis elements, all of their zeros in a fundamental domain lie on the circle centered at 0 with radius $frac{1}{sqrt{3}}$.
Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Gamma_{0}(4N)$ for
$N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coefficients of Siegel modular forms on Maass Space are obtained using Ikeda lifting.
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