Space Of Cusp Forms Research Articles

Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms

Let Goo be a semisimple real Lie group with unitary dual Goo- We produce new upper bounds for the multiplicities with which representations ^ e of cohomological type appear in certain spaces of cusp forms on Goo* The main new idea is to apply noncommutative Iwasawa theory to certain p-adic completions of the cohomology of locally symmetric spaces.

Hypergeometric functions over ${\mathbb {F}_p}$ and relations to elliptic curves and modular forms

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.

Determining modular forms on $${SL_2(\mathbb{Z})}$$ by central values of convolution L-functions

We show that a cuspidal normalized Hecke eigenform g of level one and even weight is uniquely determined by the central values of the family of Rankin– Selberg L-functions \({L(s, f\otimes g)}\) , where f runs over the Hecke basis of the space of cusp forms of level one and weight k with k varying over an infinite set of even integers.

Period polynomials and explicit formulas for Hecke operators on Γ0(2)

AbstractLet Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

Trace formula and trace identity of twisted Hecke operators on the spaces of cusp forms of weight $k+1/2$ and level $32M$

Let M be an odd positive integer, an even quadratic character defined modulo 32M, and ψ a quadratic primitive character of conductor divisible by 8. Then, we can define twisted Hecke operators R ψ T(n 2 ) on the space of cusp forms of weight k + 1/2, level 32M, and character Χ, under certain conditions on the conductors of Χ and ψ. This is a specific feature of the case of half-integral weight. We give explicit trace formulas of the twisted Hecke operators and their trace identities.

Summation methods and distribution of eigenvalues of Hecke operators

Let $p$ be a fixed prime number. Let $S_k(N)$ be the space of cusp forms of weight $k$ and level $N$. We prove a weighted equidistribution theorem for the eigenvalues of the $p$-th Hecke operator $T_p$ acting on $S_k(N)$. This is a variant of a celebrated theorem of Serre.

On the irreducibility of Hecke polynomials

Let T n,k (X) be the characteristic polynomial of the nth Hecke operator acting on the space of cusp forms of weight k for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials T n,k (X). Using this criterion with some machine computation, we show that if there exists n ≥ 2 such that T n,k (X) is irreducible and has the full symmetric group as Galois group, then the same is true of T p,k (X) for each prime p < 4,000,000.

On the central value of symmetric square L-functions

Let S k (N, χ) be the space of cusp forms of weight k, level N and character χ. For $$f \in S_k(N, \chi)$$ let L(s, sym2 f) be the symmetric square L-function and $$L(s, f \otimes f)$$ be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym2 f) and $$L(s, f \otimes f)$$ over a basis of S k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym2 f) does not vanish.

The dimension of the space of Siegel–Eisenstein series of weight one

In general, it is difficult to determine the dimension of the space of Siegel modular forms of low weights. In particular, the dimensions of the spaces of cusp forms are known in only a few cases. In this paper, we calculate the dimension of the space of Siegel–Eisenstein series of weight 1, which is a certain subspace of a complement of the space of cusp forms.

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over \({\mathbb{Q}}\) whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms.

Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials

Let be the vector space of cusp forms of weight w + 2 on the full modular group, and let denote its dual space. Periods of cusp forms can be regarded as elements of . The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span . However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural question: which periods would form a basis of .

Weyl’s law for the cuspidal spectrum of SLn

Let r be a principal congruence subgroup of SL n (Z) and let σ be an irreducible unitary representation of SO(n). Let N Γ cus (λ,σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to a. In this paper we prove that the counting function N Γ cus (λ,σ) satisfies Weyl's law. Especially, this implies that there exist infinitely many cusp forms for the full modular group SL n (Z).

On the Basis of the Space of Cusp Forms of the Level 12

Abstract The basis of the space of cusp forms is constructed for any integer 𝑘 and character mod 12.

The Distribution of the Eigenvalues of Hecke Operators

Results of the papers by Serre and by Conrey, Duke, and Farmer on the distribution of the eigenvalues of the Hecke operators Tp on the space of cusp forms of weight k for a fixed p as k increases are refined. Bibliography: 8 titles.

A relative trace formula proof of the Petersson trace formula

The Petersson trace formula relates spectral data coming from cusp forms to Kloosterman sums and Bessel functions. It was discovered in 1932 [Pe] long before Selberg’s trace formula and can be regarded as the first type of trace formula for automorphic forms. It has proven to be an indispensable tool for estimating the size of the Fourier coefficients of modular forms in many situations. See for example [Se], [IK], and Section 5 of [Iw]. In this paper we will use the relative trace formula to prove a variant of the Petersson trace formula. The resulting generalized formula relates Hecke eigenvalues, Fourier coefficients and Petersson norms of cusp forms (on the spectral side) to Bessel functions and Kloosterman sums (on the geometric side). To state this result, let Sk(N,ω ′) be the space of cusp forms of level N , weight k > 2, and nebentypus ω′ (see Section 3). For an integer n which is prime to N , let F be an orthogonal basis consisting of eigenfunctions for the Hecke operator Tn. Then (see Theorem 3.9)

Large sieve inequalities for [formula omitted]-forms in the conductor aspect

Duke and Kowalski in [A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)] derive a large sieve inequality for automorphic forms on GL ( n ) via the Rankin–Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [Duke and Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)]. As an application, we give a generalization to GL ( n ) of Duke's multiplicity theorem from [Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices (2) (1995) 99–109 (electronic)]; we also establish basic estimates on Fourier coefficients of GL ( n ) forms by computing the ramified factors for GL ( n ) × GL ( n ) Rankin–Selberg integrals.

Defining equations of modular curves

We obtain defining equations of modular curves X0(N), X1(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X0(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.

Unramified Hilbert modular forms, with examples relating to elliptic curves

We give a method to explicitly determine the space of unramified Hilbert cusp forms of weight two, together with the action of Hecke, over a totally real number field of even degree and narrow class number one. In particular, one can determine the eigenforms in this space and compute their Hecke eigenvalues to any reasonable degree. As an application we compute this space of cusp forms for Q(root 509), and determine each eigenform in this space which has rational Hecke eigenvalues. We find that not all of these forms arise via base change from classical forms. To each such eigenform f we attach an elliptic curve with good reduction everywhere whose L-function agrees with that of f at every place.

Dimensions of the spaces of cusp forms and newforms on [formula omitted] and [formula omitted

A formula for the dimension of the space of cuspidal modular forms on Γ 0 ( N ) of weight k ( k ⩾ 2 even) has been known for several decades. More recent but still well-known is the Atkin–Lehner decomposition of this space of cusp forms into subspaces corresponding to newforms on Γ 0 ( d ) of weight k, as d runs over the divisors of N. A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it would be desirable to have a formula in closed form for these dimensions. In this paper we establish such a closed-form formula, not only for these dimensions, but also for the corresponding dimensions of spaces of newforms on Γ 1 ( N ) of weight k ( k ⩾ 2 ). This formula is much more amenable to analysis and to computation. For example, we derive asymptotically sharp upper and lower bounds for these dimensions, and we compute their average orders; even for the dimensions of spaces of cusp forms, these results are new. We also establish sharp inequalities for the special case of weight-2 newforms on Γ 0 ( N ) , and we report on extensive computations of these dimensions: we find the complete list of all N such that the dimension of the space of weight-2 newforms on Γ 0 ( N ) is less than or equal to 100 (previous such results had only gone up to 3).

Trace identities of twisted Hecke operators on the spaces of cusp forms of half-integral weight

Let Rψ be a twisting operator for a quadratic primitive character ψ and ˜ T (n 2 ) the n 2 -th Hecke operator of half-integral weight. When ψ has an odd conductor, we already found trace identities between twisted Hecke operators Rψ ˜ T (n 2 ) of half-integral weight and certain Hecke operators of integral weight for almost all cases (cf. (U1-3)). In this paper, the restriction is removed and we give similar trace identities for every quadratic primitive character ψ, including the case that ψ has an even conductor.