Related papers: Algebraic-geometric codes with many automorphisms …
We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension of the dyadic field generated by the three-torsion points of the elliptic curve. As an application, we give a…
We develop an explicit geometric construction of automorphisms of finite fields arising from isogeny cycles. Let $k$ be a finite field, $E/k$ an elliptic curve, and $\ell$ an integer coprime to $\mathrm{char}(k)$. Let $\mathfrak{h}$ be an…
We present four new examples of plane rational curves with two Galois points in positive characteristic, and determine the number of Galois points for three of them. Our results are related to a problem on projective linear groups.
Algebraic geometry codes on the Hermitian curve have been the subject of several papers, since they happen to have good performances and large automorphism groups. Here, those arising from the Singer cycle of the Hermitian curve are…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
We define Convolutional Goppa Codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in particular some Maximum-Distance Separable (MDS)…
We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also develop some tools for…
Expository paper discussing AG or Goppa codes arising from curves, first from an abstract general perspective then turning to concrete examples associated to modular curves. We will try to explain these extremely technical ideas using a…
I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…
Two mappings in a finite field, the Frobenius mapping and the cyclic shift mapping, are applied on lines in PG($n,p$) or codes in the Grassmannian, to form automorphisms groups in the Grassmanian and in its codes. These automorphisms are…
In this paper, we focus on some characterizations for curves in the Galilean and Pseudo-Galilean space.
The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a…
This article computes the Galois groups of congruence covers arising in the context of certain hyperbolic triangle groups. As a consequence of this computation, the genera of the respective curves are deduced.
The arrangement of all Galois lines for the Artin-Schreier-Mumford curve in the projective 3-space is described. It may be surprising that there exist infinitely many Galois lines intersecting this curve.
A fundamental problem in arithmetic geometry is to determine the image of the mod $N$ Galois representation for all elliptic curves over $\mathbb{Q}$ and integers $N \geq 1$. For a given subgroup $G \le…
We construct special conics configurations from some points configurations which are the singularities of the dual of a quartic curve.
We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an…
This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible…