arXiv subjects

Dong Quan Ngoc Nguyen

Publications and source records attributed to Dong Quan Ngoc Nguyen.

21 records · Page 2Linked to original sources

arXiv2014

Carlitz module analogues of Mersenne primes, Wieferich primes, and certain prime elements in cyclotomic function fields

In this paper, we introduce a Carlitz module analogue of Mersenne primes, and prove Carlitz module analogues of several classical results concerning Mersenne primes. In contrast to the classical case, we can show that there are infinitely many composite Mersenne numbers. We also study the acquaintances of Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz module context that were first introduced by Dinesh Thakur.

math.NT↗

arXiv2014

Algebraic families of hyperelliptic curves violating the Hasse principle

In $2000$, Colliot-Thélène and Poonen showed how to construct algebraic families of genus one curves violating the Hasse principle. Poonen explicitly constructed an algebraic family of genus one cubic curves violating the Hasse principle using the general method developed by Colliot-Thélène and himself. The main result in this paper generalizes the result of Colliot-Thélène and Poonen to arbitrarily high genus hyperelliptic curves. More precisely, for $n > 5$ and $n \not\equiv 0 \pmod{4}$, we show that there is an algebraic family of hyperelliptic curves of genus $n$ that is counterexamples to the Hasse principle explained by the Brauer-Manin obstruction.

math.NT↗

arXiv2012

Generalized Mordell curves, generalized Fermat curves, and the Hasse principle

A generalized Mordell curve of degree $n \ge 3$ over $\bQ$ is the smooth projective model of the affine curve of the form $Az^2 = Bx^n + C$, where $A, B, C$ are nonzero integers. A generalized Fermat curve of signature $(n, n, n)$ with $n \ge 3$ over $\bQ$ is the smooth projective curve of the form $Ax^n + By^n + Cz^n = 0$ for some nonzero integers $A, B, C$. In this paper, we show that for each prime $p$ with $p \equiv 1 \pmod{8}$ and $p \equiv 2 \pmod{3}$, there exists a threefold $\cX_p \subseteq \bP^6$ such that certain rational points on $\cX_p$ produce infinite families of non-isomorphic generalized Mordell curves of degree $12n$ and infinite families of generalized Fermat curves of signature $(12n, 12n, 12n)$ for each $n \ge 2$ that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction. We also show that the set of special rational points on $\cX_p$ producing generalized Mordell curves and generalized Fermat curves that are counterexamples to the Hasse principle is infinite, and can be constructed explicitly.

math.NT↗