EVIDENCE FOR MONOTHEISM BETWEEN RELIGION AND PHILOSOPHY

The research is about the Islamic faith and it is interested in proving the monotheism issue from an Islamic and rational point of view. This will be fulfilled through a survey of evidences from Sunnah and a reading into philosophic approaches of the most well-known Arab-Islamic philosophers andthinlres such as ELFARABI, Ibn-SINA, el kendy, Ibn Roched and Cheik Mohamed Abdou ….This research aims mainly at: - Firmly establishing Religious thinking attempting to reconcile religion and philosophy. - Drawing a path where Ideology and philosophy could huddle together in harmony.

Rational points and derived equivalence

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

Mazur’s rational torsion result for pointless genus one curves: examples

This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

Dual Complex of Log Fano Pairs and Its Application to Witt Vector Cohomology

We prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro–Fujino type and a rational point formula in Dimension 3.

Rational Points of Some Elliptic Curves Related to the Tilings of the Equilateral Triangle

Let n be a positive and squarefree integer. We show that the equilateral triangle can be dissected into $$n\cdot k^2$$ n · k 2 congruent triangles for some k if and only if $$n\le 3$$ n ≤ 3 , or at least one of the curves $$C_n :y^2 =x(x-n)(x+3n)$$ C n : y 2 = x ( x - n ) ( x + 3 n ) and $$C_{-n} : y^2 =x(x+n)(x-3n)$$ C - n : y 2 = x ( x + n ) ( x - 3 n ) has a rational point with $$y\ne 0$$ y ≠ 0 . We prove that if p is a positive prime such that $$p\equiv 7$$ p ≡ 7 (mod 24), then $$C_p$$ C p and $$C_{-p}$$ C - p do not have such points. Consequently, for these primes the equilateral triangle cannot be dissected into $$p\cdot k^2$$ p · k 2 congruent triangles for any k.