Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms

Kohnen and Sengupta (Proc. Am. Math. Soc. 137(11) (2009) 3563–3567) showed that if two Hecke eigencusp forms of weight $$k_1$$ and $$k_2$$ respectively, with $$1<k_1<k_2$$ over $$\Gamma _0({N})$$ , have totally real algebraic Fourier coefficients $$\lbrace a(n) \rbrace $$ and $$\lbrace b(n) \rbrace $$ respectively for $$n \ge 1$$ with $$a(1)=1=b(1)$$ , then there exists an element $$\sigma $$ of the absolute Galois group $$\mathrm{Gal}({\bar{\mathbb {Q}}}/{\mathbb {Q}})$$ such that $$a(n)^{\sigma } b(n)^{\sigma } < 0$$ for infinitely many n. Later Gun et al. (Arch. Math. (Basel) 105(5) (2015) 413–424) extended their result by showing that if two Hecke eigen cusp forms, with $$1<k_1<k_2$$ , have real Fourier-coefficients $$\lbrace a(n)\rbrace $$ and $$\lbrace b(n)\rbrace $$ for $$n \ge 1$$ and $$a(1)b(1) \ne 0$$ , then there exists infinitely many n such that $$a(n)b(n) > 0$$ and infinitely many n such that $$a(n)b(n) < 0$$ . When $$k_1=k_2$$ , the simultaneous sign changes of Fourier coefficients of two normalized Hecke eigen cusp forms follow from an earlier work of Ram Murty (Math. Ann. 262 (1983) 431–446). In this note, we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $$\Gamma _0({N})$$ where the coefficients lie in an arithmetic progression. Next, we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigencusp forms for the full modular group.

It is known that if a nonzero cusp form has real Fourier coefficients, then its Fourier coefficients change signs infinitely often. In this paper, we prove that there is a codimension one subspace in the space of holomorphic modular forms of square-free level such that all of its non-zero forms have similar sign change property.

Sign changes of Fourier coefficients of Hilbert modular forms

We prove a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form of degree 2. In addition, we prove a quantitative result for the number of sign changes of the primitive Fourier coefficients. We give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2 over certain imaginary quadratic extensions.

This survey gives an account of background and the recent development concerning sign changes of Fourier coefficients of modular forms, which includes the great contributions of other authors. We are attempting to elucidate interesting viewpoints, ideas and methods which, even unspecified, may not originate from us -- the present authors. Moreover, we formulate some questions for future study.

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

The purpose of this paper is to derive that the square of Fourier coefficients a(n) at a square free positive integer n of modular forms f of half integral weight belonging to Kohnen's spaces of arbitrary odd level and of arbitrary primitive character is essentially equal to the critical value of the zeta function attached to the modular form F of integral weight which is the image of f under the Shimura correspondence. Previously, KohnenZagier had obtained an analogous result in the case of Kohnen's spaces of square free level and of trivial character. Our results give some generalizations of them of KohnenZagier. Our method of the proof is similar to that of Shimura's paper concerning Fourier coefficients of Hilbert modular forms of half integral weight over totally real fields.

We prove a quantitative result for the number of sign changes of the Fourier coefficients of half-integral weight cusp forms in the Kohnen plus space, provided the Fourier coefficients are real numbers.

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $$t\mathbb {Z}^2$$ , we investigate the sign changes in the $$tp^2$$ -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level.

The quantum degeneracies of Bogomolny‐Prasad‐Sommerfield (BPS) black holes of octonionic magical supergravity in five dimensions are studied. Quantum degeneracy is defined purely number theoretically as the number of distinct states in charge space with a given set of invariant labels. Quantum degeneracies of spherically symmetric stationary BPS black holes are given by the Fourier coefficients of modular forms of exceptional group . Their charges take values in the lattice defined by the exceptional Jordan algebra over integral octonions. The quantum degeneracies of rank 1 and rank 2 BPS black holes are given by the Fourier coefficients of singular modular forms and . The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on embeddings of cubic rings A into the exceptional Jordan algebra we show that the quantum degeneracies of rank 1 black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms (HMFs) of . If the discriminant of the cubic ring A is with p a prime number then the isotropic lines in the 24 dimensional orthogonal complement of A define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. The current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity is also reviewed.

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

In this paper we consider the hyperbolic Kac–Moody algebra \(\mathcal {F}\) associated with the generalized Cartan matrix Open image in new window. Its connection to Siegel modular forms of genus 2 was first studied by A. Feingold and I. Frenkel. The denominator function of \(\mathcal{F}\) is not an automorphic form. However, Gritsenko and Nikulin extended \(\mathcal{F}\) to a generalized Kac–Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as an infinite product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac–Moody algebra which contains \(\mathcal {F}\) and whose denominator function is related to a product of Dedekind η-functions. In particular, root multiplicities of the generalized Kac–Moody algebra are determined by Fourier coefficients of a modular form. As the main results of this paper, we compute asymptotic formulas for these Fourier coefficients using the method of Hardy–Ramanujan–Rademacher, and obtain an asymptotic bound for root multiplicities of the algebra \(\mathcal{F}\). Our method can be applied to other hyperbolic Kac–Moody algebras and to other modular forms as demonstrated in the later part of the paper.

In this paper, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus was dealt in an earlier work of Choie, the first author and Kohnen.

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over ℚ associated to a rational quaternion algebra into the Shimura surface associated to the base change of the quaternion algebra to a real quadratic field. After extending the associated moduli problems over ℤ we obtain an arithmetic threefold with a embedded arithmetic surface, which we view as a cycle of codimension one. We then construct a family, indexed by totally positive algebraic integers in the real quadratic field, of codimension two cycles (complex multiplication points) on the arithmetic threefold. The intersection multiplicities of the codimension two cycles with the fixed codimension one cycle are shown to agree with the Fourier coefficients of a (very particular) Hilbert modular form of weight 3/2. The results are higher dimensional variants of results of Kudla-Rapoport-Yang, which relate intersection multiplicities of special cycles on the integral model of a Shimura curve to Fourier coefficients of a modular form in two variables.