International Journal of Number Theory

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL₂(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(p^jn))_n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

We evaluate the convolution sums ∑_{l,m∈ℕ,l+2m=n} σ₃(l)σ₃(m), ∑_{l,m∈ℕ,l+3m=n} σ₃(l) × σ₃(m), ∑_{l,m∈ℕ,2l+3m=n} σ₃(l)σ₃(m) and ∑_{l,m∈ℕ,l+6m=n} σ₃(l)σ₃(m) for all n ∈ ℕ using the theory of modular forms and use these convolution sums to determine the number of representations of a positive integer n by the quadratic forms Q₈ ⊕ Q₈ and Q₈ ⊕ 2Q₈, where the quadratic form Q₈ is given by

Let K and L be two number fields. Previously, Iwasawa proved a theorem about Hecke characters of K with values in L under the assumptions that K is Galois over ℚ, K is a CM-field and K ⊆ L. In this paper, we remove these hypotheses.

Let ω be a point in the upper half plane, and let Γ be a discrete, finite covolume subgroup of PSL₂(ℝ). We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice Γω, intersected with increasingly large balls centered at ω. We prove this conjecture for Γ = PSL₂(ℤ) and ω an elliptic point.

Let G be a finite cyclic group of order n. The Erdős–Ginzburg–Ziv theorem states that each sequence of length 2n - 1 over G has a zero-sum subsequence of length n. A sequence without a zero-sum subsequence of length n is called n-zero-sum free. Savchev and Chen characterized all the n-zero-sum free sequences of length n + k - 1 over G, where . In the present paper, we determine all the n-zero-sum free sequences of length over G.

Bringmann, Mahlburg and Rhoades proved asymptotic formulas for all the even moments of the ranks and cranks of partitions with polynomial error terms. In this paper, motivated by their work, we apply the same method and obtain asymptotics for the two rank moments of overpartitions.

Let k ≥ 1 be an integer and ped_k(n) be the number of k-tuple partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). In this paper, we examine the behavior of ped₄(n) modulo 8 in detail. As a result, we obtain infinite families of congruences for ped₄(n) modulo 8.

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚ_p, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

Let Y_K → X_K be a ramified cyclic covering of curves, where K is a cyclotomic field. In this work we study the p-rank of the reduction mod p of a model of the Jacobian of Y_K. In this way, we obtain counterparts of the Deuring polynomial, defined for elliptic curves, for genus greater than one. We provide a new point of view of this subject in terms of L-functions. To carry out this study we use the relationship between Jacobi sums and L-functions. This is established in [A. Weil, Jacobi sums as "Grössencharaktere", Trans. Amer. Math. Soc. 73 (1952) 487–495] for the case of Fermat curves. We also give a new proof of a result of Deligne concerning the constant terms of these L-functions and Jacobi sums.

In recent work, Andrews, Chan, and Kim extend a result of Garvan about even rank and crank moments of partitions to positive moments. In a similar fashion we extend a result of Mao about even rank moments of overpartitions. We investigate positive Dyson-rank, M₂-rank, first residual crank, and second residual crank moments of overpartitions. In particular, we prove a conjecture of Mao which states that the positive Dyson-rank moments are larger than the positive M₂-rank moments. We also prove some additional inequalities involving rank and crank moments of overpartitions, including an interlacing property.

In this paper, we construct some families of polynomials defining Salem numbers which approach integer numbers and we give some results on their beta expansion. As a consequence we obtain in a simple way, for any integer number q ≥ 2, many sequences of Salem numbers which are beta numbers and converge to q from both sides.

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Over the function field of a smooth projective curve over an algebraically closed field, we investigate the set of S-integral elements in a forward orbit under a rational function by establishing some analogues of the classical Siegel theorem.

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field , p-ring spaces V_p(c) modulo c are introduced by defining a morphism ψ : f ↦ V_p(f) from the divisor lattice ℕ of positive integers to the lattice 𝒮 of subspaces of the direct product V_p of the p-elementary class group 𝒞/𝒞^p and unit group U/U^p of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group Gal(N | ℚ) and sharing a common discriminant d_N and conductor c over K. The number m_p(d, c) of these extensions is given by a formula in terms of positions of p-ring spaces in 𝒮, whose complexity increases with the dimension of the vector space V_p over the finite field 𝔽_p, called the modified p-class rank σ_p of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σ_p ≤ 1 only. Here, the results are extended to σ_p = 2, underpinned by concrete numerical examples.

We define Atkin's inner product for the Fricke groups of levels 5 and 7, and show that under certain condition the Atkin orthogonal polynomials in these cases exist. We also show the connection between extremal quasimodular forms of depth 1 and the Atkin orthogonal polynomials for these groups. Finally we propose a conjecture that relates our Atkin orthogonal polynomials to supersingular elliptic curves.

In this paper, we characterize whether the pure sextic fields with square-free integers m ≢ ±1(Mod 9) have power integral bases or do not; if m ≡ 2, 3 (Mod 4), then have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic subfields. Based on the works of Kovács and Pethő, several examples on application of monogenic fields to CNS (Canonical Number System) are shown.