Fourier Coefficients Of Modular Forms Research Articles

L${L}$‐functions of Kloosterman sheaves

AbstractIn this article, we study a family of motives associated with the symmetric power of Kloosterman sheaves constructed by Fresán, Sabbah, and Yu. They demonstrated that for , the ‐functions of extend meromorphically to and satisfy the functional equations conjectured by Broadhurst and Roberts. Our work aims to extend these results to the ‐functions of some of the motives , with , as well as other related two‐dimensional motives. In particular, we prove several conjectures of Evans type, which relate moments of Kloosterman sheaves and Fourier coefficients of modular forms.

Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional & Hilbert Modular Forms

AbstractThe 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.

Fourier coefficients of automorphic L-functions

We prove two unconditional results on Fourier coefficients of modular forms: Let [Formula: see text] be a Hecke eigen cusp form of weight [Formula: see text] and level [Formula: see text] ([Formula: see text] square free), and [Formula: see text] be the normalized Hecke eigenvalues. Let [Formula: see text] be the least prime [Formula: see text] such that [Formula: see text]. Then we show that in a family of modular forms of weight [Formula: see text] and level [Formula: see text], except for a density zero set, [Formula: see text] for some [Formula: see text]. Second, we show that outside a density zero set, a modular form is determined by Fourier coefficients up to [Formula: see text] for some [Formula: see text].

Toric Periods and 𝑝-adic Families of Modular Forms of Half-Integral Weight

The primary goal of this work is to construct p p -adic families of modular forms of half-integral weight, by using Waldspurger’s automorphic framework to make the results as comprehensive and precise as possible. A secondary goal is to clarify the role of test vectors as defined by Gross-Prasad in the elucidation of general formulae for the Fourier coefficients of modular forms of half-integral weight in terms of toric periods of the corresponding modular forms of integral weight. As a consequence of our work, we develop a generalization of a classical formula due to Shintani, and make precise the conditions under which Shintani’s lift vanishes. We also give a number of results on test vectors for ramified representations which are of independent interest.

On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Abstract In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 \epsilon>0 , there exist infinitely many natural numbers 𝑛 such that τ ⁢ ( p n ) \tau(p^{n}) has at least 2 ( 1 - ϵ ) ⁢ log ⁡ n log ⁡ log ⁡ n 2^{(1-\epsilon)\frac{\log n}{\log\log n}} distinct prime factors for almost all primes 𝑝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above.

Exponential sums in prime fields for modular forms

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski’s bound for exponential sums attached to certain linear recurrence sequences over finite fields.

A supplement to “On the Fourier coefficients of Hilbert modular forms of half-integral weight over arbitrary algebraic number fields, Tsukuba J. Math. 37(2013), 1-11”

The purpose of this note is to prove that the Shintani lift of Hilbert modular forms over algebraic number fields commutes with the action of Hecke operators. We show our assertion using the commutativity of the Shimura lift with Hecke operators and some properties of adjoint mappings. This commutativity of the Shintani lift plays an essential role for the proof of the Waldspurger-type theorem concerning the Fourier coefficients of modular forms of half-integral weight over arbitrary algebraic number fields.

C<sub>2</sub> Invariants of Hourglass Chains via Quadratic Denominator Reduction

We introduce families of four-regular graphs consisting of chains of hourglasses which are attached to a finite kernel. We prove a formula for the $c_2$ invariant of these hourglass chains which only depends on the kernel. For different kernels these hourglass chains typically give rise to different $c_2$ invariants. An exhaustive search for the $c_2$ invariants of hourglass chains with kernels that have a maximum of ten vertices provides Calabi-Yau manifolds with point-counts which match the Fourier coefficients of modular forms whose weights and levels are [4,8], [4,16], [6,4], and [9,4]. Assuming the completion conjecture, we show that no modular form of weight 2 and level $\leq1000$ corresponds to the $c_2$ of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in $c_2$ invariants of $\phi^4$ quantum field theory.

Generalized Paley graphs and their complete subgraphs of orders three and four

Let \(k \ge 2\) be an integer. Let q be a prime power such that \(q \equiv 1 ({\mathrm{mod}}\,\,k)\) if q is even, or, \(q \equiv 1 ({\mathrm{mod}}\,\,2k)\) if q is odd. The generalized Paley graph of order q, \(G_k(q)\), is the graph with vertex set \(\mathbb {F}_q\) where ab is an edge if and only if \({a-b}\) is a kth power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in \(G_k(q)\), \(\mathcal {K}_4(G_k(q))\), which holds for all k. This generalizes the results of Evans, Pulham and Sheehan on the original (\(k=2\)) Paley graph. We also provide a formula, in terms of Jacobi sums, for the number of complete subgraphs of order three contained in \(G_k(q)\), \(\mathcal {K}_3(G_k(q))\). In both cases, we give explicit determinations of these formulae for small k. We show that zero values of \(\mathcal {K}_4(G_k(q))\) (resp. \(\mathcal {K}_3(G_k(q))\)) yield lower bounds for the multicolor diagonal Ramsey numbers \(R_k(4)=R(4,4,\ldots ,4)\) (resp. \(R_k(3)\)). We state explicitly these lower bounds for small k and compare to known bounds. We also examine the relationship between both \(\mathcal {K}_4(G_k(q))\) and \(\mathcal {K}_3(G_k(q))\), when q is prime, and Fourier coefficients of modular forms.

Rademacher expansions and the spectrum of 2d CFT

A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and c > 1. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin j ≠ 0. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

Periodicities for Taylor coefficients of half-integral weight modular forms

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $\tau=i$ of the classical Jacobi theta function. Here, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular forms of half-integral weight.

Ramanujan graphs and exponential sums over function fields

We prove that q+1-regular Morgenstern Ramanujan graphs Xq,g (depending on g∈Fq[t]) have diameter at most (43+ε)logq⁡|Xq,g|+Oε(1) (at least for odd q and irreducible g) provided that a twisted Linnik–Selberg conjecture over Fq(t) is true. This would break the 30 year-old upper bound of 2logq⁡|Xq,g|+O(1), a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips, and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms. We also unconditionally construct infinite families of Ramanujan graphs that prove that 43 cannot be improved.

Fourier expansions at cusps

In this article, we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. We also show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is a result of Shimura on the interplay between the actions of $$\mathrm {GL}_2^+(\mathbb {Q})$$ and $${{\,\mathrm{Aut}\,}}(\mathbb {C})$$ on modular forms.

Metric number theory of Fourier coefficients of modular forms

We discuss the approximation of real numbers by Fourier coefficients of newforms, following recent work of Alkan, Ford and Zaharescu. The main tools used here, besides the (now proved) Sato-Tate Conjecture, come from metric number theory.

On a q-deformation of modular forms

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work, such partial sums are related to the radial asymptotics of infinite q-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric q-deformation of two CM modular forms of weight 3 and discuss the corresponding q-congruences.

Sequences, modular forms and cellular integrals

AbstractIt is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

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

In this paper, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms [Formula: see text] and [Formula: see text] with Fourier coefficients [Formula: see text] and [Formula: see text] respectively, we consider the following questions: existence of infinitely many primes [Formula: see text] such that [Formula: see text]; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.

On unbounded denominators and hypergeometric series

We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first result is a necessary and sufficient criterion, applicable to all but finitely many primes, for determining when the coefficients of a hypergeometric series with rational parameters are p-adically unbounded (equivalent but different conditions were found earlier by Dwork in [12] and Christol in [9]). We then show that the set of unbounded primes for a given series is, up to a finite discrepancy, a finite union of the set of primes in certain arithmetic progressions and we explain how this set can be computed. We characterize when the density of the set of unbounded primes is 0 (a similar result is found in [9]), and when it is 1. Finally, we discuss the connection between this work and the unbounded denominators conjecture for Fourier coefficients of modular forms.

Non-vanishing of products of Fourier coefficients of modular forms of half-integral weight

We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform of half-integral weight.

On the density of the odd values of the partition function, II: An infinite conjectural framework

We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that p(n) is odd exactly 50% of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality.A striking consequence is that, under suitable existence conditions, if any t-multipartition function is odd with positive density and t≢0(mod3), then p(n) is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today.Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms.