Modular Forms Of Weight Research Articles

On Shimura lifting of Hilbert modular forms

Let $${\mathfrak {N}}_{0}$$ be a integral ideal divisible by 4, of a totally real field K. We show that there is the Shimura lifting map of a space of Hilbert modular forms with character modulo $${\mathfrak {N}}_{0}$$ of half-integral weight, to the space of Hilbert modular forms of integral weight under some condition. In particular it is shown that if $$16|{\mathfrak {N}}_{0}$$ , then any Hilbert modular forms of weight at least 5 / 2 has the Shimura lift. As an application, we compute the Shimura lifts of the third powers of theta series for $$K={\mathbf {Q}}(\sqrt{2})$$ and $$K={\mathbf {Q}}(\sqrt{5})$$ , and obtain the formulas for the numbers of representations of totally positive integers in K as sums of three integral squares.

Sums of quadratic functions with two discriminants

In a 1999 paper, Zagier discusses a construction of a function Fk,D(x) defined for an even integer k≥2, and a positive discriminant D. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the D-th Fourier coefficient of weight k+1/2 Eisenstein series constructed by H. Cohen.In this note we consider a construction which works both for even and odd positive integers k. Our function Fk,D,d(x) depends on two discriminants d and D with signs sgn(d)=sgn(D)=(−1)k, degenerates to Zagier's function when d=1, namely,Fk,D,1(x)=Fk,D(x), and has very similar properties. In particular, we prove that the average value of Fk,D,d(x) is again a Fourier coefficient of H. Cohen's Eisenstein series of weight k+1/2, while now the integer k≥2 is allowed to be both even and odd.

Representations by some quinary quadratic forms of level 8

Let rQ(n) be the representation number of a nonnegative integer n by a certain quinary quadratic form Q of level 8. Then the associated theta function is a modular form of weight 5/2 for Γ0(8) associated with the character (8⋅). We express this theta function as a linear combination of Hecke eigenforms and find the general formula of the representation number rQ(n). As a consequence, we show that rQ(n) satisfies some partially multiplicative relations by applying Fricke involution and Hecke operators on the associated theta functions.

Nonzero coefficients of half-integral weight modular forms mod $$\ell $$ ℓ

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $$\ell $$ l . Among the applications of this we show that there are $$\gg \sqrt{X}/\log \log X$$ ≫ X / log log X integers $$n \le X$$ n ≤ X for which the partition function p(n) is not divisible by $$\ell $$ l , and that there are $$\gg \sqrt{X}/\log \log X$$ ≫ X / log log X values of $$n \le X$$ n ≤ X for which c(n), the nth Fourier coefficient of the j-invariant, is odd.

On Hecke L-functions attached to half-integral weight modular forms

We investigate non-vanishing properties of L(f,s) on the real line, when f is a Hecke eigenform of half-integral weight k+12 on Γ0(4).

On Vector-Valued Siegel Modular Forms of Degree 2 and Weight $(j,2)$

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight \mathrm{Sym}^j\otimes\mathrm{det}^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.

Conjugacy growth series for wreath product finitary symmetric groups

In recent work, Bacher and de la Harpe define and study conjugacy growth series for finitary permutation groups. In two subsequent papers, Cotron, Dicks, and Fleming study the congruence properties of some of these series. We define a new family of conjugacy growth series for the finitary alternating wreath product that are related to sums of modular forms of integer and half-integral weights, the so-called mixed weight modular forms. The previous works motivate the study of congruences for these series. We prove that congruences exist modulo powers of all primes p ge 5. Furthermore, we lay out a method for studying congruence properties for sums of mixed weight modular forms in general.

CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups

Abstract Given a modular form f {{f}} of even weight larger than two and an imaginary quadratic field K {{K}} satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f {{f}} enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s method [21], as adapted by Nekovář [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to f {{f}} and K {{K}} and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.

Local and global Maass relations

We characterize the irreducible, admissible, spherical representations of mathrm{GSp}_4(F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito–Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito–Kurokawa lifts in terms of Fourier coefficients of half-integral weight modular forms or Jacobi forms. As an additional application of our methods, we give a new characterization of Saito–Kurokawa lifts involving a certain average of Fourier coefficients.

Zeros of weakly holomorphic modular forms of level 5

Let [Formula: see text] be the space of weakly holomorphic modular forms of weight [Formula: see text] and level [Formula: see text] that are holomorphic away from the cusp at [Formula: see text]. We study a canonical basis for [Formula: see text] and the locations of zeros of this basis in a fundamental domain. We give a lower bound for the number of zeros lying on a lower boundary arc of the fundamental domain.

Fourier–Jacobi periods of classical Saito–Kurokawa lifts

Kohnen–Skoruppa (Invent Math 95(3): 541–558, 1989) proved a formula for the ratio of the Petersson inner products of the half integral weight modular form and its Saito–Kurokawa lifting. We give an interpretation of this formula in the framework of the refined Gan–Gross–Prasad conjecture. This provides us with an example of the refined Gan–Gross–Prasad conjecture for the nontempered representations.

P-Adic properties of coefficients of certain half-integral weight modular forms

In this paper, we study the parallel cases of Zagier's and Folsom–Ono's grids of weakly holomorphic (resp. weakly holomorphic and mock modular) forms of weights 3/2 and 1/2, investigating their p-adic properties under the action of Hecke operators.

Coefficients of half-integral weight modular forms and indivisibility of class numbers of imaginary quadratic fields

In this paper, we investigate the nonvanishing coefficients of half-integral weight modular forms modulo a prime. We also consider the indivisibility of class numbers of imaginary quadratic fields.

A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form

Let f be a modular form of weight k and Nebentypus $\psi$. By generalizing a construction of Dabrowski and Delbourgo, we construct a p-adic L-function interpolating the special values of the L-function $L(s,\mathrm{Sym}^2(f)\otimes \xi)$, where $\xi$ is a Dirichlet character. When s=k-1 and $\xi=\psi^{-1}$, this p-adic L-function vanishes due to the presence of a so-called trivial zero. We give a formula for the derivative at s=k-1 of this p-adic L-function when the form f is Steinberg at p. If the weight of f is even, the conductor is even and squarefree, and the Nebentypus is trivial this formula implies a conjecture of Benois.

ON SIGN CHANGES FOR ALMOST PRIME COEFFICIENTS OF HALF‐INTEGRAL WEIGHT MODULAR FORMS

For a half-integral weight modular form of weight on such that () are real, we prove for a fixed suitable natural number that changes sign infinitely often as varies over numbers having at most prime factors, assuming the analog of the Ramanujan conjecture for Fourier coefficients of half-integral weight forms.

Shimura lifting of modular forms of weight 3/2

We show that there is the Shimura lifting map of \(\mathbf {M}_{3/2}(N,\chi _{0})\) to \(\mathbf {M}_{2}(N/2,\chi _{0}^{2})\) for any natural number N divisible by 4 and for any odd Dirichlet character \(\chi _{0}\), which was already proved in the case of higher weight (Tsuyumine, Tsukuba J Math 23:465–483, 1999). As the application, we show that the lift of the weighted average of theta series of quadratic forms of an odd number of variables in a class, is an Eisenstein series of integral weight.

Subconvexity for half integral weight L-functions

We prove a subconvexity bound in the conductor aspect for $L(s,f,\chi)$ where $f$ is a half integer weight modular form. This $L$-function has analytic continuation and functional equation, but no Euler product. Due to the lack of an Euler product, one does not expect a Riemann hypothesis for half integer weight modular forms. Nevertheless one may speculate a Lindelof-type hypothesis, and this current subconvexity result is an indication towards its truth.

Theta series associated to symmetric matrices

Given positive integers and , a vector of and a symmetric positive definite integral matrix whose diagonal entries are even, we consider an associated series similar to theta series and prove that this series is a modular form of weight for a congruence subgroup with character.

Some non-congruence subgroups and the associated modular curves

In this paper, we study two families of normal subgroups of Γ0(2) with abelian quotients and their associated modular curves. They are similar to Fermat groups and Fermat curves in some aspects but very different in other aspects. Almost all of them are non-congruence subgroups. These modular curves are either projective lines or hyperelliptic curves. There are few modular forms of weight 1 for these groups. We also determine their cuspidal divisor class groups and show that these groups are finite.

Byte weight enumerators and modular forms of genus r

For a positive integer m, let R be either the ring ℤ2m of integers modulo 2m or the quaternionic ring Σ2m = ℤ2m + αℤ2m + βℤ2m + γℤ2m with α = 1 + î, β = 1 + ĵ and [Formula: see text], where [Formula: see text] are elements of the ring ℍ of real quaternions satisfying [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.