We prove two “large images” results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p > 13 not dividing d, provided that d is divisible by q (but d 6= q) with q = 2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing dD, where D = disc(...

A rational (resp. integral) distance set is a subset S of the plane R such that for all s, t ∈ S, the distance between s and t is a rational number (resp. is an integer). Huff [4] considered rational distance sets S of the following form: given distinct a, b ∈ Q∗, S contains the four points (0,±a) and (0,±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q∗. Such a point (x, 0) m...

J. Stix proved that a curve of positive genus over $\mathbb{Q}$ which maps to non-trivial Brauer-Severi variety satisfies the section conjecture. We prove that, if $X$ is number field $k$ and Weil restriction $R_{k/\mathbb{Q}}X$ admits rational map variety, then As consequence, $P$ such corestriction $\operatorname{cor}_{k/\mathbb{Q}}([P])\in\operatorname{Br}(\mathbb{Q})$ non-trivial,

We prove that if the moduli $\mathbb{Q}$-b-divisor of a basic slc-trivial fibration is b-numerically trivial then it $\mathbb{Q}$-b-linearly trivial. As consequence, we part semi-ample when base space curve.

This paper compares the coherent groups of load buses based on Reactive Reserve Margin (RRB) obtained from the V-Q curve analysis and a novel generator sensitivity based method for optimal placement of STATCOM. The V-Q curve is one of the popular methods to study voltage instability problems in power systems during the post transient period or critical peak loads. The reactive reserve margin ob...

We give a formula for divisors of modular units on $$X_1(N)$$ and use it to prove that the $$\mathbb {Q}$$ -gonality curve is bounded above by $$[{11N^2}/{840}]$$ , where $$[\cdot ]$$ denotes nearest integer.

Given a family of quadratic twists fixed elliptic curve defined over Q $\mathbb {Q}$ , we investigate the average rank in subfamily having non-torsion rational point almost minimal height. We show particular that analytic is greater than one.

A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all the rational points on certain curves of genus 2. 0 Introduction The following classical result of Chabauty [3] is a curiosity of the literature in that there has been a 50 year period during which applications have been ...

A conjecture is formulated for an upper bound on the number of points in PG(2, q) of a plane curve without linear components, defined over GF(q). We prove a new bound which is half way from the known bound to the conjectured one. The conjecture is true for curves of low or high degree, or with rational singularity. Let C be a plane curve of degree n, defined over GF(q), without (rational) linea...

We give a geometric interpretation of the entropy of the supertubes with fixed conserved charges and angular momenta in two different approaches using the DBI action and the supermembrane theory. By counting the geometrically allowed microstates, it is shown that both the methods give consistent result on the entropy. In doing so, we make the connection to the gravity microstates clear. 1 Intro...