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Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic,…

A rational number $r$ is a \textbf{realizable exponent} for a graph $H$ if there exists a finite family of graphs $\mathcal{F}$ such that $\mathrm{ex}(n,H,\mathcal{F})=\Theta(n^r)$, where $\mathrm{ex}(n,H,\mathcal{F})$ denotes the maximum…

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n$. We investigate positivity properties of the spaces $R_e(X)$ of degree $e$ rational curves in $X$. We show that for small $e$, $R_e(X)$ has no rational curves meeting…

A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of $\pi$. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this…

We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over projective…

This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify…

It is easy to find a right-angled triangle with integer sides whose area is 6. There is no such triangle with area 5, but there is one with rational sides (a `\emph{Pythagorean triangle}'). For historical reasons, integers such as 6 or 5…

Let $\ell$ be an odd prime and $d$ a positive integer. We determine when there exists a degree-$d$ number field $K$ and an elliptic curve $E/K$ with $j(E)\in\mathbb{Q}\setminus\{0,1728\}$ for which $E(K)_{\mathrm{tors}}$ contains a point of…

For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $\rho_E : G \to GL_2(\hat{\mathbb{Z}})$, where $G$ is the absolute Galois group of the…

We study an interplay between operator algebras and geometry of rational elliptic curves. Namely, let $\mathcal{O}_B$ be the Cuntz-Krieger algebra given by square matrix $B=(b-1, ~1, ~b-2, ~1)$, where $b$ is an integer greater or equal to…

Let $E$ be an elliptic curve, defined over a quartic extension $K$ of $\mathbb{Q}$, with $j(E) \in \mathbb{Q}$. In this paper, we classify the possible torsion subgroup structures $E(K)_{\text{tors}}$.

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its…

Given a lower-triangular matrix of real numbers, one can ask the following four total-positivity questions: total positivity of the triangle itself; total positivity of its row-reversal; Toeplitz-total positivity of its row sequences…

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability…

A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be…

For two positive integers $r$ and $s$ with $r\geq 2s-2$, if $G$ is a graph of order $3r+4s$ such that $d(x)+d(y)\geq 4r+4s$ for every $xy\not\in E(G)$, then $G$ independently contains $r$ triangles and $s$ quadrilaterals, which partially…

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we…

We are interested in the following semilinear elliptic problem: \begin{equation*} \begin{cases} -\Delta u + \lambda u = u^{p-1} \ \text{in} \ T,\\ u > 0, u = 0 \ \text{on} \ \partial T,\\ \int_{T}u^{2} \, dx= c \end{cases} \end{equation*}…

In Descartes' five circle problem integer curvatures (inverse radii) are considered. The positive integer curvature triple [c_1, c_2, c_3] (dimensionless), with non-decreasing entries for three given mutually touching circles, leading to…

A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after…