International Journal of Number Theory

We show that a positive proportion of the Poincaré series do not vanish identically when either the index or the weight varies over an interval of suitable length, the other one being fixed.

We prove explicit lower bounds for the density of the sets of primes p such that eigenvalues λ_p of a Siegel cusp form of degree 2 satisfy c₂ > λ_p > c₁, for c₁, c₂ real. A similar result is also proved for the set of primes such that ∣λ_p∣ > c.

A number is cubefull if the exponent of every prime in its prime factorization is at least three. In this paper, we give, for a fixed l, the number of pairs of cubefull numbers n, n + l such that n is less than a given quantity.

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants on arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the remainder terms are sharper.

Let F be the cubic field of discriminant –23 and let be its ring of integers. By explicitly computing cohomology of congruence subgroups of , we computationally investigate modularity of elliptic curves over F.

In this paper the metric theory of Diophantine approximation of linear forms that are of mixed type is investigated. Khintchine–Groshev theorems are established together with the Hausdorff measure generalizations. The latter includes the original dimension results obtained in [H. Dickinson, The Hausdorff dimension of sets arising in metric Diophantine approximation, Acta Arith.68(2) (1994) 133–140] as special cases.

We study modular Galois representations mod p^m. We show that there are three progressively weaker notions of modularity for a Galois representation mod p^m: We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a level-lowering result ("stripping-of-powers of p away from the level"): A mod p^m strongly modular representation of some level N_p^r is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod p^m corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod p^m to any "dc-weak" eigenform, and hence to any eigenform mod p^m in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general.

We study the distribution of solutions n to the congruence σ(n) ≡ a (mod n). After excluding obvious families of solutions, we show that the number of these n ≤ x is at most x^½+o(1), as x → ∞, uniformly for integers a with ∣a∣ ≤ x^¼. As a concrete example, the number of composite solutions n ≤ x to the congruence σ(n) ≡ 1 (mod n) is at most x^½+o(1). These results are analogues of theorems established for the Euler ϕ-function by the third-named author.

We show that a character sum attached to a family of 3-isogenies defined on the fibers of a certain elliptic surface over 𝔽_p relates to the class number of the quadratic imaginary number field . In this sense, this provides a higher-dimensional analog of some recent class number formulas associated to 2-isogenies of elliptic curves.

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with Petersen [On ideal well-rounded lattices, Int. J. Number Theory8(1) (2002) 189–206]. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.

It is known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that its shifts ζ(s + iτ, α), τ ∈ ℝ, approximate with a given accuracy any analytic function uniformly on compact subsets of the strip D = {s ∈ ℂ : ½ < σ < 1}. Let H(D) denote the space of analytic functions on D equipped with the topology of uniform convergence on compacta. In the paper, the classes of functions F : H(D) → H(D) such that F(ζ(s, α)) is universal in the above sense are considered. For example, if F is continuous and, for each polynomial p = p(s), the set F^-1{p} is non-empty, then F(ζ(s, α)) with transcendental α is universal.

We show that one simple zero of a Maass L-function implies infinitely many simple zeros of the L-function. Since Strömbergsson found simple zeros of three different even Maass L-functions, we know that there are at least three Maass L-functions having infinitely many simple zeros.

An exact formula for distribution of sequences of fractional parts is proved. This provides us with a possibility for computational study of their discrepancies by the means of integer-number programming.

In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. Explicit formulas are obtained for the number of representations of a positive integer n by the quadratic forms

For a pair (π, σ) of cuspidal automorphic representations of GL_n and GL_n-1, both of non-vanishing cohomology, we show algebraicity properties of critical values of the associated Rankin–Selberg L-function twisted by finite-order characters. A certain non-vanishing assumption about the associated archimedean Rankin–Selberg pairings on the cohomology is established for n = 3.