Integral Fourier Coefficients Research Articles

Determination of normalized extremal quasimodular forms of depth 1 with integral Fourier coefficients

The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from [Formula: see text] to the reciprocal of the elliptic modular [Formula: see text]-function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by this change of parameters. Furthermore, by transforming these hypergeometric series expressions into a certain manageable form related to the Atkin(-like) polynomials and using the lemmas that appeared in the study of [Formula: see text]-adic hypergeometric series by Dwork and Zudilin, the integrality problem can be reduced to the fact that a polynomial vanishes modulo a prime power, which we prove. We also prove that all extremal quasimodular forms of depth 1 with appropriate weight-dependent leading coefficients have integral Fourier coefficients by focusing on the hypergeometric expression of them.

Distinguishing newforms by the prime divisors of their Fourier coefficients

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis.

Proportion of modular forms with transcendental zeros for general levels

Let $\Gamma$ be a congruence subgroup such that $\Gamma_{1}(N)\subset\Gamma\subset\Gamma_{0}(N)$ for some positive integer $N$. For a positive integer $k$, let $M_{k,\mathbf{Z}}(\Gamma)$ be the set of modular forms of weight $k$ on $\Gamma$ with integral Fourier coefficients. Let $R_{k}(\Gamma)$ be the set of common zeros in the upper half plane $\mathbf{H}$ of all the modular forms of weight $k$ on $\Gamma$. In this note, we prove that the density of modular forms in $M_{k,\mathbf{Z}}(\Gamma)$ with an algebraic zero $z \notin R_{k}(\Gamma)$ is zero.

Mock modular forms with integral Fourier coefficients

In this paper, we explicitly construct mock modular forms with integral Fourier coefficients by evaluating regularized Petersson inner products involving their shadows, which are unary theta functions of weights 12 and 32. In addition, we also improve the known bounds for the denominators of the coefficients of mock modular forms whose shadows are holomorphic weight one cusp forms constructed by Hecke.

Density of modular forms with transcendental zeros

For an even positive integer k, let Mk,Z(SL2(Z)) be the set of modular forms of weight k on SL2(Z) with integral Fourier coefficients. Let Mk,Ztran(SL2(Z)) be the subset of Mk,Z(SL2(Z)) consisting of modular forms with only transcendental zeros on the upper half plane H except all elliptic points of SL2(Z). For a modular form f(z)=∑n=0∞af(n)e2πinz of weight k(f), letϖ(f):=∑n=0rk(f)|af(n)|, where rk(f)=dimC⁡Mk(f),Z(SL2(Z))⊗C−1. In this paper, we prove that if k=12 or k≥16, then#{f∈Mk,Ztran(SL2(Z)):ϖ(f)≤X}#{f∈Mk,Z(SL2(Z)):ϖ(f)≤X}=1−αkX+O(1X2) as X→∞, where αk denotes the sum of the volumes of certain polytopes. Moreover, if we let MZ=∪k=0∞Mk,Z(SL2(Z)) (resp. MZtran=∪k=0∞Mk,Ztran(SL2(Z))) and φ is a monotone increasing function on R+ such that φ(x+1)−φ(x)≥Cx2 for some positive number C, then we provelimX→∞⁡#{f∈MZtran:ϖ(f)+φ(k(f))≤X}#{f∈MZ:ϖ(f)+φ(k(f))≤X}=1.

Graded rings of integral Jacobi forms

We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of (0,2)-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients. These forms are the data for Borcherds products for the Siegel paramodular groups.

A ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients

We determine the structure over $$\mathbb {Z}$$ of a ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients whose weights are multiples of 4 when the base field is the Gaussian number field $$\mathbb {Q}(\sqrt{-1})$$. Namely, we give a set of generators consisting of 24 modular forms. As an application of our structure theorem, we give the Sturm bounds for such Hermitian modular forms of weight k with $$4\mid k$$, for $$p=2$$, 3. We remark that the bounds for $$p\ge 5$$ are already known.

Petersson inner products of weight-one modular forms

Abstract In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

On mod p singular modular forms

Abstract We show that a Siegel modular form with integral Fourier coefficients in a number field K, for which all but finitely many coefficients (up to equivalence) are divisible by a prime ideal 𝔭 ${\mathfrak{p}}$ of K, is a constant modulo 𝔭 ${\mathfrak{p}}$ . Moreover, we define a notion of mod 𝔭 ${\mathfrak{p}}$ singular modular form and discuss a relation between its weight and the corresponding prime p. We discuss some examples of mod 𝔭 ${\mathfrak{p}}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod 𝔭 ${\mathfrak{p}}$ of Klingen–Eisenstein series.

On the canonical decomposition of generalized modular functions

The authors have conjectured (\cite{KoM}) that if a normalized generalized modular function (GMF) $f$, defined on a congruence subgroup $\Gamma$, has integral Fourier coefficients, then $f$ is classical in the sense that some power $f^m$ is a modular function on $\Gamma$. A strengthened form of this conjecture was proved (loc cit) in case the divisor of $f$ is \emph{empty}. In the present paper we study the canonical decomposition of a normalized parabolic GMF $f = f_1f_0$ into a product of normalized parabolic GMFs $f_1, f_0$ such that $f_1$ has \emph{unitary character} and $f_0$ has \emph{empty divisor}. We show that the strengthened form of the conjecture holds if the first few Fourier coefficients of $f_1$ are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either $f_0=1$ or if the divisor of $f$ is concentrated at the cusps of $\Gamma$.

TWISTED BORCHERDS PRODUCTS ON HILBERT MODULAR SURFACES AND THE REGULARIZED THETA LIFT

We construct a lifting from weakly holomorphic modular forms of weight 0 for SL 2(ℤ) with integral Fourier coefficients to meromorphic Hilbert modular forms of weight 0 for the full Hilbert modular group of a real quadratic number field with an infinite product expansion and a divisor given by a linear combination of twisted Hirzebruch–Zagier divisors. The construction uses the singular theta lifting by considering a suitable twist of a Siegel theta function. We generalize the work by Bruinier and Yang who showed the existence of the lifting for prime discriminants using a different approach.

P-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight

Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL 2(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 Γ0(4N) for N = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications to our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.

Coefficients of half-integral weight modular forms modulo ?j

Suppose that l≥5 is prime, that j≥0 is an integer, and that F(z) is a half-integral weight modular form with integral Fourier coefficients. We give some general conditions under which the coefficients of F are “well-distributed” modulo lj. As a consequence, we settle many cases of a classical conjecture of Newman by proving, for each prime power lj with l≥5, that the ordinary partition function p(n) takes each value modulo lj infinitely often.

Arithmetic properties of mirror map and quantum coupling

We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of the $J$-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function field ${\bf Q}(J)$. This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced $mod\ p$. Under the mirror hypothesis and an integrality assumption, we derive $mod~p$ congruences for the Fourier coefficients. For the quintics, we deduce (at least for $5\not{|}d$) that the degree $d$ instanton numbers $n_d$ are divisible by $5^3$ -- a fact first conjectured by Clemens.

On Hilbert modular forms with integral Fourier coefficients

On Hilbert modular forms with integral Fourier coefficients