Related papers: Semistable abelian varieties with small division f…

We study the arithmetic of division fields of semistable abelian varieties A over the rationals. The Galois group of the 2-division field of A is analyzed when the conductor is odd and squarefree. The irreducible semistable mod 2…

An abelian surface $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal S}_5$-extension with ramification index 5 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be any semistable…

A precise and testable modularity conjecture for rational abelian surfaces A with trivial endomorphisms, End_Q A = Z, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on…

We say that an abelian variety $A_{/\mathbf Q}$ of dimension $g$ is {\em prosaic} if it is semistable, with good reduction at 2 and its points of order $2$ generate a $2$-extension of ${\mathbf Q}$. For $p \equiv 1 \bmod{8}$, let $M_u$ be…

An abelian threefold $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal S}_7$-extension over ${\mathbb Q}$ with ramification index 7 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be…

Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\bar{\rho}_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the…

Let $p$ be a prime, let $K$ be a discretely valued extension of $\mathbb{Q}_p$, and let $A_{K}$ be an abelian $K$-variety with semistable reduction. Extending work by Kim and Marshall from the case where $p>2$ and $K/\mathbb{Q}_p$ is…

Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l=2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction…

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$…

We construct infinitely many abelian surfaces A defined over the rational numbers such that, for a prime ell <= 7, the ell-torsion subgroup of A is not isomorphic as a Galois module to the ell-torsion subgroup of its dual. We do this by…

We characterize decomposable principally polarized abelian varieties of the form $E\times B$, with $E$ an elliptic curve, in two different ways, which are, surprisingly, completely analogous to classical results of curve theory concerning…

The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further, we give…

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an…

Let G be a finite group. For each integral representation $\rho$ of G we consider $\rho-$decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X,H) with $\rho(G)-$action, of dimension equal…

Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we…

Let $A$ be an abelian variety defined over a number field $K$. If $\mathfrak{p}$ is a prime of $K$ of good reduction for $A$, let $A(K)_\mathfrak{p}$ denote the image of the Mordell-Weil group via reduction modulo $\mathfrak{p}$. We prove…

Let $A$ be a 2-dimensional abelian variety defined over a number field $K$. Fix a prime number $\ell$ and suppose $\#A(\mathbb{F}_p) \equiv 0 \pmod{\ell^2}$ for a set of primes $\mathfrak{p} \subset \mathcal{O}_K$ of density 1. When…

In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective representations $\rho$ of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic…

Suppose that $K$ is a field of characteristic 0, $p$ is an odd prime, $r$ a positive integer, $q=p^r$ a prime power. Suppose that $f(x)$ is a polynomial of degree $n > 4$ with coefficients in $K$ and without multiple roots. Let us consider…