Related papers: Essential dimension of isogenies

We consider the question of when a Jacobian of a curve of genus $2g$ admits a $(2,2)$-isogeny to two polarized dimension $g$ abelian varieties. We find that one of them must be a Jacobian itself and, if the associated curve is…

The main goal of this paper is to present an algorithm bounding the dimension of a linear system of curves of given degree (or monomial basis) with multiple points in general position. As a result we prove the Hirschowitz--Harbourne…

In this note, we establish the hardness of approximation of the problem of computing the minimal size of a $\delta$-sufficient reason for decision trees.

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$.…

We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or…

Bounded infinite graphs are defined on the basis of natural physical requirements. When specialized to trees this definition leads to a natural conjecture that the average connectivity dimension of bounded trees cannot exceed two. We verify…

The set of badly approximable $m \times n $ matrices is known to have Hausdorff dimension $mn $. Each such matrix comes with its own approximation constant $c$, and one can ask for the dimension of the set of badly approximable matrices…

We give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura variety of A_{g-1}, contained in the Prym locus. First we give such a bound for a germ passing through a Prym variety of a k-gonal…

This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of…

Fix a number field $k$ and a rational prime $\ell$. We consider abelian varieties whose $\ell$-power torsion generates a pro-$\ell$ extension of $k(\mu_{\ell^\infty})$ which is unramified away from $\ell$. It is a necessary, but not…

We prove that the expected area of the amoeba of a complex plane curve of degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ and once rescaled by $\ln(d)^2$, is asymptotically bounded from below by $3/4$. In order to get this…

We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh…

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of…

Let $[X,\lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce…

We prove that the irreducible components of primitive class Severi varieties of general abelian surfaces are completely determined by the maximal factorization through an isogeny of the maps from the normalized curves.

The Narasimhan-Nori conjecture asks for a closed formula for the number of non-isomorphic principal polarizations of any given abelian variety. In this paper, we introduce a new algorithm that gives a lower bound on the number of…

Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness…

In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection.…

A conjecture of De Concini Kac and Procesi provides a bound on the minimal possible dimension of an irreducible module for quantized enveloping algebras at an odd root of unity. We pose the problem of the existence of modules whose…

We prove that for each dimension not less than five there exists a contraction between solvable Lie algebras that can be realized only with matrices whose Euclidean norms necessarily approach infinity at the limit value of contraction…