Related papers: Arithmetic E_8 lattices with maximal Galois action

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We…

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points…

We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil…

We study relatively minimal surfaces equipped with a strongly isotrivial elliptic fibration in positive characteristic by means of the notion of equivariantly normal curves introduced and developed recently by Brion. Such surfaces are…

We study the biregular and birational geometry of degree 6 del Pezzo surfaces with Picard number 1, defined over an arbitrary perfect field. Using Galois cohomology techniques, we obtain an explicit description of cocycles for such surfaces…

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) \'etale fundamental group. A key result in our investigation is that these…

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first…

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…

We give a classification of all possible $2$-adic images of Galois representations associated to elliptic curves over $\mathbb{Q}$. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop…

Following the philosophy of arithmetic topology, we describe a point of view which helps look at surfaces and $p$-adic fields in a "uniform way", and show that results on mapping class groups can be extended to this point of view, and thus…

Monodromy groups, i.e. the groups of isometries of the intersection lattice L_X:=H_2/torsion generated by the monodromy action of all deformation families of a given surface, have been computed in math.AG/0006231 for any minimal elliptic…

We classify pairs $(S, \gamma)$, consisting of a rational elliptic surface $S$ and a Galois cover $\gamma$ of the base, which satisfy a condition we call $\mathcal{L}$-stability. We explain how to use the theory of Mordell-Weil lattices to…

It follows from an observation of A. Coble in 1919 that the automorphism group of an unnodal Enriques surface contains the $2$-congruence subgroup of the Weyl group of the $E_{10}$-lattice. In this article, we determine how much bigger the…

We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the…

Over the past two years we have improved several of the (Mordell-Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P_i…

We study the automorphism groups of Mori Del Pezzo fibrations over a smooth projective curve $C$ of positive genus. From that, we obtain a classification of maximal connected algebraic subgroups of $\mathrm{Bir}(C\times \mathbb{P}^2)$. Our…

We introduce one of the most beautiful algebraic varieties known, a quintic hypersurface in projective five-space, which is invariant under the action of the Weyl group of $E_6$. This variety is intricately related with many other moduli…

We classify equivariantly Gorenstein log del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite…

Given a self-similar groupoid action $(G,E)$ on the path space of a finite graph, we study the associated Exel-Pardo \'etale groupoid ${\mathcal G}(G,E)$ and its $C^*$-algebra $C^*(G,E)$. We review some facts about groupoid actions, skew…

Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at $m$ ($m\geq n+2$) points in the projective space $\mpp^n(\mc)$. In this paper we study the case where all the points of blow-ups lie on a certain…