International Journal of Number Theory

In [A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, in Proc. St. Petersburg Mathematical Society, Vol. 12, American Mathematical Society Translations, Series 2, Vol. 219 (American Mathematical Society, Providence, RI, 2006), pp. 1–85] Fukaya and Kato presented equivariant Tamagawa number conjectures that implied a very general (non-commutative) Iwasawa main conjecture for rather general motives. In this article we apply their methods to the case of one-parameter families of motives to derive a main conjecture for such families. On our way there we get some unconditional results on the variation of the (algebraic) λ- and μ-invariant. We focus on the results dealing with Selmer complexes instead of the more classical notion of Selmer groups. However, where possible we give the connection to the classical notions.

In this paper a convergence theorem for continued fractions with negative parameters is proved. We define a transformation of a certain kind of continued fractions to the same kind of continued fractions. This transformation is obtained by multiple application of the Bauer–Muir transform and then using the limiting process. It is shown that a double application of this transformation is the identity transformation.

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

Let I denote the set of all irrational numbers, θ ∈ I, and simple continued fraction expansion of θ be [a₀, a₁, …, a_n, …]. Then a₀ is an integer and {a_n}_n≥1 is an infinite sequence of positive integers. Let M_n(θ) = [0, a_n, a_n-1, …, a₁] + [a_n+1, a_n+2, …]. Then the set of numbers {lim sup M_n(θ) ∣ θ ∈ I} is called the Lagrange Spectrum 𝔏. Notably 3 is the first cluster point of 𝔏. Essentially lim inf 𝔏 or . Perron [Über die approximation irrationaler Zahlen durch rationale, I, S.-B. Heidelberg Akad. Wiss., Abh. 4 (1921) 17 pp; Über die approximation irrationaler Zahlen durch rationale, II, S.-B. Heidelberg Akad. Wiss., Abh.8 (1921) 12 pp.] has found that lim inf{lim sup M_n(θ) ∣ θ = [a₀, a₁, a₂, …, a_n, …] and . This article forwards the value of lim inf{lim sup M_n(θ) ∣ θ = [a₀, a₁, …, a_n, …] and a_n ≥ 4 frequently}, a long awaited cluster point of Lagrange Spectrum.

In this paper, we will study the dynamical Manin–Mumford problem, focusing on the question of polarizability for endomorphisms of an abelian variety A and on the action of a Frobenius and its Verschiebung on the diagonal subvariety of A × A. We complete the study with different polarizability criteria.

Let {E_n} be the Euler numbers. In the paper we determine E_2^mk+b – E_b modulo 2^m+7, where k and m are positive integers and b ∈ {0, 2, 4, …}.

In a recent paper, we considered integers n for which the polynomial xⁿ – 1 has a divisor in ℤ[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial xⁿ – 1 has a divisor in 𝔽_p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.

For each integer n ≥ 2, let be the index of composition of n, where γ(n) ≔ ∏_p∣np. The index of composition of an integer measures the multiplicity of its prime factors. In this paper, we obtain a new asymptotic formula of the sum ∑_n≤xλ^-k(n). Furthermore, we improve the error term under the Riemann Hypothesis.

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad–Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X₁(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X₁(N), we deduce that, for sufficiently large N, X₁(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

We investigate the solubility of the congruence xy ≡ 1 (mod p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.

Recently, Williams discovered explicit formulas of the coefficients c(n) in the Fourier series expansions of a class of eta quotients. Motivated by the results obtained by Williams, we find that the coefficients c(2n) in the Fourier series expansions of another class of eta quotients can be represented as a linear combination of σ₃(n), σ₃(n/2), σ₃(n/3) and σ₃(n/6). One example is

In this study we use some known convolution sums to find the representation number for each of the three octonary quadratic forms , and .

Dans cet article, on montre que les orbites sous Galois des invariants modulaires associés à des courbes elliptiques complexes sans multiplication complexe variant dans une même classe d'isogénie s'équidistribuent dans la courbe modulaire vers la probabilité hyperbolique. La démonstration repose sur des arguments de théorie ergodique, notamment le théorème de Ratner (cf. [A. Eskin et H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), ainsi que sur le théorème de l'image ouverte de Serre [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] dans le cas où les invariants modulaires considérés sont algébriques sur Q, et des résultats de G. Shimura dans le cas transcendant [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)].

In this article, it is shown that Galois orbits of invariants associated with non-CM and pairwise isogeneous complex elliptic curves equidistribute in the classical modular curve towards the hyperbolic probability measure. The proof is based on arguments from ergodic theory, especially Ratner's theorem on unipotent flows (cf. [A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), as well as on Serre's open image theorem [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] in case of algebraic invariants, and on G. Shimura's work in the transcendant case [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)].