Related papers: The constant of the support problem for abelian va…

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

Let $A$ be a finite subset of $\ffield$, the field of Laurent series in $1/t$ over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that…

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $\epsilon > 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(\epsilon)$,…

We give several generalisations of the Deuring reduction criterion for elliptic curves to abelian varieties of higher dimension. In particular the Newton polygon of the reduction of an abelian variety A with complex multiplication by F at a…

Let $a>1$ be an integer. Denote by $l_a(n)$ the multiplicative order of $a$ modulo integer $n\geq 1$. We prove that there is a positive constant $\delta$ such that if $x^{1-\delta}\log^3 x = o(y)$, then $$ \frac1y \sum_{a<y} \frac1x…

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…

A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly by…

Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a good category of abelian motives over the algebraic closure of a finite field and a reduction functor to it from the category of CM-motives. Consequentely, one…

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it…

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…

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group…

Let $K$ be a number field, and let $A$ be an Abelian variety over $K$ which has no CM isogeny-factors over $\overline{K}$. We prove that $A$ has only finitely many torsion points over the maximal $n$-step-solvable extension of $K$ for any…

Irreducible crystalline representations of dimension 2 of Gal(Qpbar/Qp) depend up to twist on two parameters, the weight k and the trace of frobenius a_p. We show that the reduction modulo p of such a representation is a locally constant…

Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show…

In this paper we improve our previous results on classification of groups of points on abelian varieties over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $k$-isogeny class.

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset…

Let p be an odd prime. Let K_p = Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of K_p lying over p. Let G be the Galois group of K_p. Let v be a primitive root mod p. Let sigma be a Q-isomorphism of K_p. Let P(sigma) =…

Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a…

Let Z be a subvariety of the moduli space of principally polarised abelian varieties of dimension g over the complex numbers. Suppose that Z contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny…

We give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a consequence (of our theorem 9.1) one obtains that for every prescribed odd prime characteristic $p$ every bounded…