Rational products of singular moduli

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

8 - THE SYSTEM OF RATIONAL NUMBERS

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

Let p≥3 be a prime number. A Fermat curve over Q of exponent p is defined by an equation of the form axp+byp+czp=0, where a, b, c are non-zero rational numbers. We prove in this article that there exist infinitely many Fermat curves defined over Q, of exponent p, pairwise non Q-isomorphic, contradicting the Hasse principle.

We prove the generalized Hyers-Ulam stability of the generalized quadratic functional equation <TEX>$$f(rx\;+\;sy)\;=\;r^2f(x)\;+\;s^2f(y)\;+\;\frac{rs}{2}[f(x\;+\;y)\;-\;f(x\;-\;y)]$$</TEX> in fuzzy Banach spaces, where r, s are non-zero rational numbers with <TEX>$r^2\;+\;s^2\;{\neq}\;1$</TEX>.

The degrees-of-freedom of a K-user Gaussian interference channel (GIFC) has been defined to be the multiple of (1/2) log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> P at which the maximum sum of achievable rates grows with increasing P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GIFCs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are non-zero rational numbers. More specifically, for all K > 2, we find a class of K-user GIFCs that is dense in the GIFC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIFC with non-zero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.

It is known that there exists a multiple-unicast network which has a rate 1 linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes. In this paper, we present a generalization of this result for a linear network coding solution of any rate. Specifically, we show that for any non-zero positive rational number k/n, there exists a multiple-unicast network which has a rate k/n fractional linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes.

A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We present also a general construction of Heegner points and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to the cube sum problem.

Given a multiplicative group of nonzero rational numbers and a positive integer m m , we consider the problem of determining the density of the set of primes p p for which the order of the reduction modulo p p of the group is divisible by m m . In the case when the group is finitely generated the density is explicitly computed. Some examples of groups with infinite rank are considered.

The problem of expressing numbers as the sum of two cubes of non-zero rational numbers has a long history, some results being given by Diophantus. Several mathematicians have since contributed to our knowledge of the problem, in particular in the discovery of numbers which cannot be expressed as the sum of two cubes. Fermat discovered that 1, 18 and 36 belong to this class of numbers, Euler and Legendre demonstrated that so do 3, 4 and 5 and some infinite sets of such numbers were discovered by Sylvester. However, much of interest still remains to be discovered in this area. In this article I hope to both summarise the existing results on this problem and to produce a new class of numbers which cannot be expressed as the sum of two cubes. Most importantly, a general method of tackling the problem will be described which should stimulate further investigation.

We consider some lacunary power series with rational coefficients in 𝕈p. We show that under certain conditions these series take transcendental values at non-zero rational number arguments, and we determine the classes of these transcendental values with respect to Mahler’s classification of p -adic numbers.

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(\alpha)$. For fixed $\alpha\in \bar{\mathbb Q}$, the map $t\mapsto m_t(\alpha)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $\alpha$. It remains open to determine whether there is a sequence of elements $\alpha_n\in \bar{\mathbb Q}$ such that the number of exceptional points for $\alpha_n$ tends to $\infty$ as $n\to \infty$. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the $t$-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.

Les fondements des mathématiques de la géométrie d'euclide à la relativité générale et à l'intuitionisme: Par F. Gonseth, Proffeseur à l'Université de Berne. Préface de M. Jacquwa Hadamard, Membre de l'Institut, Professeur au Collège de France et à l'École POlytechnique, xiv–243 pages, 25.1 × 16.5 cm., paper. Paris, Librairie Scientifique, Albert Blanchard, 1926. Price, 25 francs

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1, . . . , eαm,m⩾ 2, whereα0= 0,α1, . . . ,αmarem+ 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies onmare improved.