Narrow Results

We obtain new upper bounds for the mean squared modulus of sums ∑_n∈ℕA_nχ(n), where the sequence (A_n) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (c_m) satisfying c_m = 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has A_n = ∑_{uvm = n} α(u)β(v)c_m (n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as log_q(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.

Fix a prime N, and consider the action of the Hecke operator T_N on the space of modular forms of full level and varying weight κ. The coefficients of the matrix of T_N with respect to the basis {E₄ⁱ E₆^j | 4i + 6j = κ} for can be combined for varying κ into a generating function F_N. We show that this generating function is a rational function for all N, and present a systematic method for computing F_N. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.

In this paper, we prove the following theorem: Let $𝔽$ be an algebraic closure of a finite field of characteristic $p>3$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\mathbb{Q}$ to $\mathrm{\text{GL}}(3,𝔽$) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of $\rho$ is contained in the intersection of the stabilizers of the line spanned by $e_2$ and the plane spanned by $e_1,e_3$, where $\{e_i\}$ denotes the standard basis. Such $\rho$ will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of $\rho$ is squarefree, that the predicted weight $(a,b,c)$ lies in the lowest alcove, and that $c\not\equiv b+1(modp-1)$, we prove that $\rho$ is attached to a Hecke eigenclass in $H_2(\mathrm{\Gamma },M)$, where $\mathrm{\Gamma }$ is a subgroup of finite index in $\mathrm{\text{SL}}(3,\mathbb{Z})$ and $M$ is an $𝔽\mathrm{\Gamma }$-module. The particular $\mathrm{\Gamma }$ and $M$ are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms with sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight $k\ge 0$, which is related to the weight of Borcherds lifts when $k=0$. In particular, we see that such divisibility of the weight of Borcherds lifts only exists for $\mathbb{Q}(\sqrt{5})$. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, we obtain divisibility results in an “orthogonal” direction on reduced modular forms.

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels.

In Mathematische Werke, Hecke defines the operator $T_p$ and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL$(\mathrm{\Delta })$ is again a theta series associated to a collection of forms in CL$(\mathrm{\Delta })$. We state and prove an explicit formula for the action of $T_p$ on a binary quadratic form of negative discriminant.

We generalize Merel’s work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols. Among the key inputs is a computation of the action of Hecke operators on Manin symbols, building upon the Heilbronn–Merel matrices constructed by Mohamed.

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels.