Related papers: Explicit constructions of cyclic N-isogenies

Let X be a projective complex K3 surface. Beauville and Voisin singled out a 0-cycle c_X on X of degree 1: it is represented by any point lying on a rational curve in X. Huybrechts proved that the second Chern class of a rigid simple…

Let $p$ be a prime and let $\mathbf{E}$ be an elliptic curve defined over the finite field $\mathbb{F}_p$ of $p$ elements. For a point $G\in\mathbf{E}(\mathbb{F}_p)$ the elliptic curve congruential generator (with respect to the first…

The modular decomposition of a graph is a canonical representation of its modules. Algorithms for computing the modular decomposition of directed and undirected graphs differ significantly, with the undirected case being simpler, and…

Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree $\ell$, their structure has been investigated graph-theoretically. We generalise the notion of $\ell$-isogeny graphs to…

For each open subgroup $H\leq \operatorname{GL}_2(\widehat{\mathbb{Z}})$, there is a modular curve $X_H$, defined as a quotient of the full modular curve $X(N)$, where $N$ is the level of $H$. The genus formula of a modular curve is well…

Given an integer $N \geq 3$, we prove that for any ring $R$ and any finite locally free $R$-group scheme $G$ which is fppf-locally (over $R$) isomorphic the $N$-torsion subscheme of some elliptic curve $E/R$, there is a smooth affine curve…

Basepoint free cycles on the moduli space $\overline{M}_{0,n}$ of stable n-pointed rational curves, defined using Gromov-Witten invariants of smooth projective homogeneous spaces X are studied. Intersection formulas to find classes are…

This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along…

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on…

Let $\L_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $\L_g^G$ for $G \iso \Z_n, \Z_2{\o}\Z_n, \Z_2{\o}A_4$, or $SL_2(3)$. We show that for $G \iso…

Categorical resolutions of singularities are a replacement of resolution of singularities within the realm of triangulated categories. They allow the study of the derived category of a singular variety $X$ via a triangulated category that…

The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are…

We consider smooth plane curves $\mathcal{X}$ of degree $d\geq4$, defined over an algebraically closed field of characteristic $0$, that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering…

We give a general formula for generators of the NL-cone, the cone of effective linear combinations of irreducible components of Noether-Lefschetz divisors, on an orthogonal modular variety. We then fully describe the NL-cone and its…

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…

We split the program of explicit descent of elliptic curves into two parts. For $n=3$ and $n=5,$ we first display a model for the universal elliptic curve $E$ with full level $n$ structure and describe the map of rational points of $E$ to…

We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that…

For an integer $n\geq 7$, we investigate the matroid realization space of a specific deformation of the regular $n$-gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic…

Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $\mathrm{End}_{\bar{\mathbb{Q}}}(E)=\mathbb{Z}$ and which admits a non-trivial cyclic $\mathbb{Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation…

Let $X$ be a smooth projective algebraic curve of genus $g\geq 2$ defined over a field $K$. We show that $X$ can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering $X\to X/\Aut(X)$ is of type…