Space Of Cusp Forms Research Articles

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

Paramodular groups and theta series

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling or pullback of Eisenstein series method. For this we give a new geometric proof of Garrett’s double coset decomposition which works in our more general situation.

Linear independence of even periods of modular forms

We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three even periods are linearly independent. We also prove an asymptotic result for an arbitrary number of even periods. These results are achieved by studying the Rankin-Cohen brackets of Eisenstein series.

Estimates for the Fourier coefficients of the Duke–Imamoglu–Ikeda lift

Abstract Let k and n be positive even integers. For a Hecke eigenform h in the Kohnen plus subspace of weight k - n 2 + 1 2 {k-\frac{n}{2}+\frac{1}{2}} for Γ 0 ⁢ ( 4 ) {\Gamma_{0}(4)} , let I n ⁢ ( h ) {I_{n}(h)} be the Duke–Imamoglu–Ikeda lift of h to the space of cusp forms of weight k for Sp n ⁡ ( ℤ ) {\operatorname{Sp}_{n}({\mathbb{Z}})} . We then give an estimate of the Fourier coefficients of I n ⁢ ( h ) {I_{n}(h)} . It is better than the usual Hecke bound for the Fourier coefficients of a Siegel cusp form.

Rankin–Cohen brackets of Eisenstein series

We prove that Rankin–Cohen brackets of Eisenstein series generate the whole space of cuspforms in several cases. This generalizes the classical result that products of two Eisenstein series generate the whole space of modular forms.

On the Miller basis for the space of cusp forms

In this paper, we study the existence of the Miller basis for [Formula: see text], the space of cusp forms of weight [Formula: see text] and level [Formula: see text]. We seek an easy criterion to determine whether or not the space [Formula: see text] admits the Miller basis. In particular, we prove that if the level [Formula: see text] is sufficiently composite, then [Formula: see text] does not have the Miller basis. We also prove that there exists a constant [Formula: see text] such that [Formula: see text] admits the Miller basis if and only if [Formula: see text] admits the Miller basis when [Formula: see text].

Ramanujan-style congruences for prime level

We establish Ramanujan-style congruences modulo certain primes \(\ell \) between an Eisenstein series of weight k, prime level p and a cuspidal newform in the \(\varepsilon \)-eigenspace of the Atkin–Lehner operator inside the space of cusp forms of weight k for \(\Gamma _0(p)\). Under a mild assumption, this refines a result of Gaba–Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler–Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by \(\ell .\) The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.

Sato-Tate distribution of p-adic hypergeometric functions

Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the p-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the p-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight kge 4 and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1, 2, 6].

ON THE NATURE OF ADDITIONAL SPACE AT CUTTING OF SPACES OF CUSP FORMS

In the article we study a space of cusp forms by the method of cutting. This space is a direct sum of the subspace of forms divided by the fixed cusp form named the cutting function and the additional space. If the additional space is zero we have the situation of exact cutting. In common case the cutting is not exact and it is important to research the nature of the additional space. We prove that the basis of the additional space can be described by the space of cusp forms of small weight. This weight is not more than 14 and often is equal to 4. We give examples of all cutting functions for all levels. We prove the theorem about the basis of the additional space to the space of cusp forms in the space of modular forms of the same level, weight and character. We use properties of eta-products, Biagioli formula for orders in cusps and Cohen Oesterle formula for dimensions.

Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms

Abstract Extending our previous work we construct weakly holomorphic Hecke eigenforms whose period polynomials correspond to elements in a basis consisting of odd and even Hecke eigenpolynomials induced by only cusp forms. As an application of our results, we give an explicit construction of the holomorphic parts of harmonic weak Maass forms that are good for Hecke eigenforms. Moreover, we give an explicit construction of the Hecke-equivariant map between the space of weakly holomorphic cusp forms and two copies of the spaces of cusp forms, and show that the map is compatible with the corresponding map on the spaces of period polynomials.

The trace of [formula omitted] takes no repeated values

We prove that the trace of the Hecke operator T2 acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.

Some remarks on the critical values of Rankin–Selberg L-series

We denote by Sl(N,ψ) (resp. Sk(N,χ)) the space of cusp forms of weight l (resp. k) for Γ0(N) with nebentypus ψ (resp. χ). We give an arithmetic expression for the critical values of the Rankin–Selberg L-series D(s,f⊗g) associated with a pair (f,g) of modular forms for SL2(Z). It is also shown that a cusp form g∈Sl(N,ψ) is uniquely determined by certain critical values of the family of Rankin–Selberg L-series D(s,f⊗g), where f runs over a fixed orthogonal basis of Sk(N,χ).

Computing models for quotients of modular curves

We describe an algorithm for computing a {mathbb {Q}}-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining q-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves.

Arithmetic of weakly holomorphic Hecke eigenforms

In [10], Guerzhoy found the dual space for the space of cusp forms of even integral weight and studied that space with respect to the Hecke operators. In this paper, we extend that result for higher level cases. Especially, we find a basis for the dual space consisting of eigenforms for the Hecke operators, show how to calculate those eigenforms explicitly, and examine algebraicity of the Fourier coefficients of eigenforms.

Fourier coefficients of level 1 Hecke eigenforms

Lehmer's 1947 conjecture on whether $\tau(n)$ vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of $\tau(n)$. For example, among the odd numbers $\alpha$ such that $|\alpha|<99$, we determine that $$\tau(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have that $\tau(n) \neq -|\alpha|$ and that $\tau(n) \notin \{9,25,27,39,$ $75,81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $\Delta E_4 = \sum_{n = 1}^{\infty} \tau_{16}(n)q^n$, we show that \begin{align*}\tau_{16}(n) \notin &\{\pm \ell: 1\leq \ell \leq 99, \ell \text{ is odd}, \ell \neq 33,55,59,67,73,83,89,91\} & \quad \quad \quad \quad \quad \cup \{-33,-55,-59,-67,-89,-91\}.\end{align*} Furthermore, we implement congruences given by Swinnerton-Dyer to rule out additional large primes which divide numerators of specific Bernoulli numbers. To obtain these results, we make use of the theory of Lucas sequences, methods for solving high degree Thue equations, Barros' algorithm for solving hyperelliptic equations, and the theory of continued fractions.

Compactification of Drinfeld moduli spaces as moduli spaces of 𝐴-reciprocal maps and consequences for Drinfeld modular forms

We construct a compactification of the moduli space of Drinfeld modules of rank r r and level N N as a moduli space of A A -reciprocal maps. This is closely related to the Satake compactification but not exactly the same. The construction involves some technical assumptions on N N that are satisfied for a cofinal set of ideals N N . In the special case where A = F q [ t ] A=\mathbb {F}_q[t] and N = ( t n ) N=(t^n) , we obtain a presentation for the graded ideal of Drinfeld cusp forms of level N N and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect similar results in general, but the proof will require more ideas.

Newton polygons of Hecke operators

In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator $$T_2$$ for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at $$n\le 15$$ , then it agrees with the Newton polygon of $$T_2$$ up to n.

The number of representations of integers by generalized Bell ternary quadratic forms

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

Bases of spaces of harmonic weak Maass forms and Shintani lifts of harmonic weak Maass forms

We construct bases of the space of harmonic weak Maass forms of weight $$\kappa \in \frac{1}{2}{\mathbb {Z}}$$ . Using these bases, we obtain a Shintani lift from a positive integral weight harmonic weak Maass form to a half-integral weight harmonic weak Maass form, which reduces to the classical Shintani lift on the space of cusp forms.

Drinfeld cusp forms: oldforms and newforms

Let p=(P) be any prime of Fq[t], let m be any ideal of Fq[t] not divisible by p and consider the space of Drinfeld cusp forms of levelmp, i.e. for the modular group Γ0(mp). Using degeneracy maps, traces and Fricke involutions we offer definitions for p-oldforms and p-newforms which turn out to be subspaces stable with respect to the action of the Atkin operator UP. We provide eigenvalues and/or slopes for p-oldforms and p-newforms and a condition to get the whole space of cusp forms as the direct sum between them.