Related papers: A parametrized set of explicit elements of $\Sha(E…

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these…

Ideal class pairings map the rational points of rank $r\geq 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D) \geq \frac{1}{2}(c(E)-\varepsilon)(\log…

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least…

We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the…

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in…

A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and…

In this article, we introduce a new object, a virtual quadratic space, and its group of isometries. They are presented as natural generalizations of quadratic spaces and orthogonal groups. It is then shown that by replacing quadratic spaces…

We establish a classification of the values of \( N \) for which an elliptic curve defined over \( \mathbb{Q} \) with square discriminant admits an \( N \)-isogeny. Furthermore, we determine the values of \( N \) for which two elliptic…

We revisit the classification of Lorentz homogeneous spaces of dimension $3$, and relax usual completeness assumptions. In particular, non-unimodular elliptic plane waves, and only them, are neither locally symmetric nor locally isometric…

We provide an explicit classification of supersingular elliptic curves in characteristic~3 into isomorphism classes, and give explicit formulae for their point counts. This report was written specifically to support implementation of point…

We generalise the notion of the Tate-Shafarevich group of an elliptic K3 surface with a section to the Tate-Shafarevich group of a K3 surface endowed with a linear system. The construction, which uses Grothendieck's special Brauer group,…

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Any finite set $S$ of closed points of $C$ gives rise to an integral domain…

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension…

Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the Tate-Shafarevich group of the abelian surface B has…

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight…

Given a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The…

The main aim of the paper is to study in greater detail absolutely homogeneous structures (that is, objects with the property that each partial isomorphism extends to a global automorphism), with special emphasis on metric spaces and…

Let $C$ be a hyperelliptic curve defined over $\mathbb{Q}$, whose Weierstrass points are defined over extensions of $\mathbb{Q}$ of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case…

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over the complex numbers. Then the primitive cohomology $H^{1,1}_{prim}(X)$ has, up to a sign, a natural orthonormal basis $(\eta_i)_{i\in [1, N]}$ given by certain meromorphic…

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the…