arXiv subjects

Jerome T. Dimabayao

Publications and source records attributed to Jerome T. Dimabayao.

3 recordsLinked to original sources

arXiv2023

A variant of the congruent number problem

A positive integer $n$ is called a $θ$-congruent number if there is a triangle with sides $a,b$ and $c$ for which the angle between $a$ and $b$ is equal to $θ$ and its area is $n\sqrt{r^2 - s^2}$, where $0 < θ< π$, $\cos θ= s/r$ and $0 \leq |s| < r$ are relatively prime integers. The case $θ=π/2$ refers to the classical congruent numbers. It is known that the problem of classifying $θ$-congruent numbers is related to the existence of rational points on the elliptic curve $y^2 = x(x+(r+s)n)(x-(r-s)n)$. In this paper, we deal with a variant of the congruent number problem where the cosine of a fixed angle is $\pm \sqrt{2}/2$.

math.NT↗

arXiv2019

Je{ś}manowicz' conjecture for polynomials

Let $(a,b,c)$ be pairwise relatively prime integers such that $a^2 + b^2 = c^2$. In 1956, Je{ś}manowicz conjectured that the only solution of $a^x + b^y = c^z$ in positive integers is $(x,y,z)=(2,2,2)$. In this note we prove a polynomial analogue of this conjecture.

math.NT↗

arXiv2014

On the vanishing of cohomologies of $p$-adic Galois representations associated with elliptic curves

Let $K$ be a $p$-adic field and $E$ an elliptic curve over $K$ with potential good reduction. For some large Galois extensions $L$ of $K$ containing all $p$-power roots of unity, we show the vanishing of certain Galois cohomology groups of $L$ with values in the $p$-adic representation associated with $E$. This generalizes some results of Coates, Sujatha and Wintenberger.

math.NT↗