Related papers: Detecting linear dependence on a simple abelian va…

We consider a local to global principle for detecting linear dependence of nontorsion points, by reduction maps, in the Mordell-Weil group of an abelian variety over a number field.

We define a certain class of simple varieties over a field $k$ by a constructive recipe and show how to control their (equivariant) truncating invariants. Consequently, we prove that on simple varieties: (i) if $k=\overline{k}$ and…

Let $\mathcal{A}$ be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number $p$. Denote by ${\rm A}$ an abelian variety over a finite field of characteristic $p$, obtained by the reduction…

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties. We study the problem of counting the number of principal polarizations modulo the natural action of the automorphism group of the abelian variety on a very…

We prove that if $B$ is a $p$-block with non-trivial defect group $D$ of a finite $p$-solvable group $G$, then $\ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian…

This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper…

For a quartic primitive CM field $K$, we say that a rational prime $p$ is {\it evil} if at least one of the abelian varieties with CM by $K$ reduces modulo a prime ideal $\gerp| p$ to a product of supersingular elliptic curves with the…

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many…

By adapting the classical proof of Jordan's theorem on finite subgroups of linear groups, we show that every approximate subgroup of the unitary group U_n(C) is almost abelian.

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

We partially answer, in terms of monodromy, Murty and Patankar's question: Given an absolutely simple abelian variety over a number field, does it have simple specializations at a set of places of positive Dirichlet density? The answer is…

Let $p$ be a prime number, $K$ be the henselization of the rational functions over the finite field $\mathbb{F}_p$ and $R$ be the ring of additive polynomials over K. We show that the field of Laurent series over $\mathbb{F}_p$ is decidable…

Let w be an elliptic element of the Weyl group of a connected reductive group G. Let X be the set of pairs (g,B) where g is an element of G, B is a Borel subgroup of G and B,gBg^{-1} are in relative position w. Then G acts naturally on X.…

We give an efficient, deterministic algorithm to decide if two abelian varieties over a number field are isogenous. From this, we derive an algorithm to compute the endomorphism ring of an elliptic curve over a number field.

Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $\rho_A$ encodes the Galois action on the torsion points…

Let K be a global field, let S be a finite set of primes of K containing the archimedean primes and let A be an abelian variety over K. We generalize the duality theorem established in our paper "On Neron class groups of abelian varieties"…

Let $G$ be a $(2,m,n)$-group and let $x$ be the number of distinct primes dividing $\chi$, the Euler characteristic of $G$. We prove, first, that, apart from a finite number of known exceptions, a non-abelian simple composition factor $T$…

I give a short proof of the following algebraic statement: if a vertex algebra is simple, then its underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra.

In this paper we consider ideal sheaves associated to the singular loci of a divisor in a linear system $|L|$ of an ample line bundle on a complex abelian variety. We prove an effective result on their (continuous) global generation, after…

Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties…