Related papers: Collision of Orbits on an Elliptic Surface

Let $P : \Sigma \rightarrow S$ be a finite degree covering map between surfaces. Rafi and Schleimer show that there is an induced quasi-isometric embedding $\Pi : \mathcal{C}(S) \rightarrow \mathcal{C}(\Sigma)$ between the associated curve…

We determine and list all possible configurations of singular fibres on rational elliptic surfaces in characteristic three. In total, we find that 267 distinct configurations exist. This result complements Miranda and Persson's…

An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $\phi:E\rightarrow E^\prime$ comes equipped with an invariant $\alpha_{\phi/K}$ that measures the valuation of the leading term of the formal group homomorphism…

We investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces. For the projectivization of the Atiyah bundle over an elliptic curve with a…

Surfaces of amplitude 1 in ordinary projective space are of general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in $\mathbf{P}(1,2,a,b)$ of amplitude…

The congruent number elliptic curves are defined by $E_d: y^2=x^3-d^2x$, where $d\in \mathbb{N}.$ We give a simple proof of a formula for $L(\mathrm{Sym}^2(E_d),3)$ in terms of the determinant of the elliptic trilogarithm evaluated at some…

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion…

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin \mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing a rational…

Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where…

Let $\mathrm{Hilb}^gS$ be the Hilbert scheme of $g$ points on a K3 surface $S$. Suppose that $\mathrm{Pic}S\cong\Z C$ where $C$ is a smooth curve with $C^2=2(g-1)n^2$. We prove that $\mathrm{Hilb}^gS$ is a Lagrangian fibration.

Let $P$ and $Q$ be two points on an elliptic curve defined over a number field $K$. For $\alpha\in \text{End}(E)$, define $B_\alpha$ to be the $\mathcal{O}_K$-integral ideal generated by the denominator of $x(\alpha(P)+Q)$. Let…

Motivated by Miranda and Ascher--Bejleri's works on compactifications of the moduli space of rational elliptic surfaces with a section, we study constructions and boundaries of compact moduli spaces of elliptic surfaces with a multiple…

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is…

Let $X$ be a smooth projective surface over a number field $K$. Assume that $X$ has an elliptic fibration over $\mathbb{P}^1_K$ with at least one singular fibre and a section. Let $\mathcal{X}/U$ be a smooth projective model of $X$ over…

We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by…

We describe all the elliptic models with section on the Shioda supersingular K3 surface X of Artin invariant 1 over an algebraically closed field of characteristic 3. In particular, we compute elliptic parameters and Weierstrass equations…

We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a…

The complete classification of the orbits on subspaces under the action of the projective stabiliser of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of…

The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular…

Let $E_{m,n}$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^2x+n^2$, where $m$ and $n$ are positive integers. Brown and Myers showed that the curve $E_{1,n}$ has rank at least two for all $n$. In the present paper, we…