Related papers: A Prym variety with everywhere good reduction over…

Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic)…

In this article we construct for each integer $n\ge 2$ an abelian variety $A$ of dimension $n$ defined over a number field for which there exists a symmetric slope sequence of length $2n$ that does not appear as the slope sequence of…

Let A be a geometrically simple abelian variety over a number field k, let X be a subgroup of A(k) and let P be an element of A(k). We prove that if P belongs to X modulo almost all primes of k then P already belongs to X.

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share…

Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic…

To any pair of coverings $f_i: X \ra X_i, i = 1,2$ of smooth projective curves one can associate an abelian subvariety of the Jacobian $J_X$, the Prym variety $P(f_1,f_2)$ of the pair $(f_1,f_2)$. In some cases we can compute the type of…

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the…

In this paper we generalize the Deuring theorem on a reduction of elliptic curve with complex multiplication. More precisely, for an Abelian variety $A$, arising after reduction of an Abelian variety with complex multiplication by a CM…

We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational…

We give two generalizations of the Clifford theorem to algebraic surfaces. As an application, we obtain some bounds for the number of moduli of surfaces of general type.

Let C be a hyperelliptic curve of good reduction defined over a discrete valuation field K with algebraically closed residue field k. Assume moreover that char k \ne 2. Given d \in K^*\K^*2, we introduce an explicit description of the…

We show that a principally polarized abelian variety over a field $k$ is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a $k$-point if and only if the polarization and the minimal class are both…

Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel…

Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for…

In this note we prove analogues of the main theorems of complex multiplication for abelian varieties for K3 surfaces. This is done by studying the field of definition of the period morphism for complex K3 surfaces. More precisely we relate…

There is a well known theorem by Deuring which gives a criterion for when the reduction of an elliptic curve with complex multiplication (CM) by the ring of integers of an imaginary quadratic field has ordinary or supersingular reduction.…

Jean-Pierre Serre has conjectured Conj. 3.2.1, in the context of abelian varieties, that there are infinitely primes of good ordinary reduction for a smooth, projective variety over a number field. We prove this conjecture for K3 surfaces…

We consider a family of abelian surfaces over $\mathbb{Q}$ arising as Prym varieties of double covers of genus-$1$ curves by genus-$3$ curves. These abelian surfaces carry a polarization of type $(1,2)$ and we show that the average size of…

Let $K$ be an imaginary quadratic field. Modular forms for GL(2) over $K$ are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over…

Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(\mu_\ell)$ be the…