International Journal of Number Theory

Dans cet article, nous nous intéressons à un q-analogue aux entiers positifs de la fonction zêta de Riemann, que l'on peut écrire pour s ∈ ℕ* sous la forme ζ_q(s) = ∑_k≥1q^k∑_d|kd^s-1. Nous donnons une nouvelle minoration de la dimension de l'espace vectoriel sur ℚ engendré, pour 1/q ∈ ℤ\{-1; 1} et A entier pair, par 1, ζ_q(3), ζ_q(5), …, ζ_q(A - 1). Ceci améliore un résultat récent de Krattenthaler, Rivoal et Zudilin ([13]). En particulier notre résultat a pour conséquence le fait que pour 1/q ∈ ℤ\{-1; 1}, au moins l'un des nombres ζ_q(3), ζ_q(5), ζ_q(7), ζ_q(9) est irrationnel.

In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s ∈ ℕ* by ζ_q(s) = ∑_k≥1q^k∑_d|kd^s-1. We give a new lower bound for the dimension of the vector space over ℚ spanned, for 1/q ∈ ℤ\{-1; 1} and an even integer A, by 1, ζ_q(3), ζ_q(5), …, ζ_q(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin ([13]). In particular, a consequence of our result is that for 1/q ∈ ℤ\{-1; 1}, at least one of the numbers ζ_q(3), ζ_q(5), ζ_q(7), ζ_q(9) is irrational.

We define two asymptotically optimal towers over quadratic fields, and give the explicit descriptions of the ramification loci and the sets of places splitting completely, which relate to the elliptic modular curves X₀(4ⁿ) and X₀(3ⁿ), respectively. Moreover, in the last section, we determine completely the modularity of a tower given by Maharaj and Wulftange in [18].

Let K be a number field with algebraic closure , let S be a finite set of places of K containing the Archimedean places, and let φ be a Chebyshev polynomial. We prove that if is not preperiodic, then there are only finitely many preperiodic points which are S-integral with respect to α.

For f(x) ∈ ℤ[x] and a ∈ ℤ, we let fⁿ(x) be the nth iterate of f(x), P(f, a) = {p prime: p|fⁿ(a) for some n}, and D(P(f, a)) denote the natural density of P(f, a) within the set of primes. A conjecture of Jones [5] indicates that D(P(f, a)) = 0 for most quadratic f. In this paper, we find an exceptional family of (f, a) such that D(P(f, a)) > 0 by considering f_t(x) = (x + t)² - 2 - t and a_t = f_t(0) for t ∈ ℤ. We prove that if t is not of the form ±M² ± 2 or ±2M² ± 2, then D(P(f_t, a_t)) = ⅓. We also determine D(P(f_t, a_t)) in some cases when the density is not equal to ⅓. Our results suggest a connection between the arithmetic dynamics of the conjugates of x² and the conjugates of x² - 2.

We study a general method of the field intersection problem of generic polynomials over an arbitrary field k via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using multi-resolvent polynomials.

We derive further results on Eichler cohomology of generalized modular forms of arbitrary real weight.

Let N be a prime and let A be a quotient of J₀(N) over Q associated to a newform such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of . Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the special L-value of A and of the order of the Shafarevich–Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special L-values of twists of A, we show that the q-adic valuations of the algebraic part of the special L-value of A and of the Birch and Swinnerton-Dyer conjectural order of the Shafarevich–Tate group of A are both positive even numbers. We also give a formula for the algebraic part of the special L-value of A over quadratic imaginary fields K in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently, over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and ∞) which shows that this algebraic part is a perfect square up to powers of the prime two and of primes dividing the discriminant of K. Finally, for an optimal elliptic curve of arbitrary conductor E, we give a formula for the special L-value of the twist E_-D of E by a negative fundamental discriminant -D, which shows that this special L-value is an integer up to a power of 2, under some hypotheses. In view of the second part of the Birch and Swinnerton-Dyer conjecture, this leads us to the surprising conjecture that the square of the order of the torsion subgroup of E_-D divides the product of the order of the Shafarevich–Tate group of E_-D and the orders of the arithmetic component groups of E_-D, under certain mild hypotheses.

We prove that, for any ∊ > 0, there exists a constant C > 0 such that the interval contains numbers whose all prime factors are smaller than .

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

The goal of this paper is to generalize the main results of [21] and subsequent papers on metric with dependent quantities to the set-up of systems of linear forms. In particular, we establish "joint strong extremality" of arbitrary finite collection of smooth non-degenerate submanifolds of ℝⁿ. The proofs are based on generalized quantitative non-divergence estimates for translates of measures on the space of lattices.

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups.

Let be a real quadratic number field with a prime P lying above p in ℚ. If p splits in K/ℚ and p does not divide the big class number of K, then any pro-p extension of K ramified only at P is finite cyclic. If p is inert in K/ℚ, then there exist infinite extensions of K ramified only at P. Furthermore, for big enough integer k, the ray class field (mod P^k+1) is obtained from the ray class field (mod P^k) by adjoining ζ_p^k+1.

In the case of a regular cyclotomic number field K = ℚ(ζ_p), the explicit structure of ray class groups (mod P^k) is given for any positive integer k, where P is the unique prime in K above p.

In this note we show that for any pair of elliptic curves E₁, E₂ over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v, w, t] such that the sextic twists of the curves E₁, E₂ by D(u, v, w, t) have rank ≥ 2 over the field ℚ(u, v, w, t). A similar result is proved for simultaneous quartic twists of pairs of elliptic curves with j-invariant 1728.

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pⁿ-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.