The main result of this paper implies that if an abelian variety over a field $F$ has a maximal isotropic subgroup of $n$-torsion points all of which are defined over $F$, and $n\ge 5$, then the abelian variety has semistable reduction away from $n$. This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its $n$-torsion points are defined over a field $F$ and $n\ge 3$, then the abelian variety has semistable reduction away from $n$. We also give information about the Néron models...

We propose a definition of sign of imaginary quadratic fields. We give an example of such functions, and use it to define new invariants that are roots of the classical Ramachandra invariants. Also we introduce signed ordinary distributions and compute their signed cohomology by using Anderson's theory of double complex.

We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)dx+Q(x,y)dy$ with polynomial coefficients of degree $\le d$, the number of ovals of the foliation $H=\mathrm{const}$, which yield the zero value of the complete Abelian integral $I\left(t\right)={\oint }_{H=t}\omega$, grows at most as $exp{O}_H\left(d\right)$ as $d\to \infty$, where $O_H\left(d\right)$ depends only on $H$. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let $f_1\left(t\right),\cdots ,f_n\left(t\right)$, $t\in K⋐ℝ$, be a fundamental system of real solutions...

We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.

We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using the third exterior...

We study the codimension two locus $H$ in ${𝒜}_g$ consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class $\left[H\right]\in C{H}^2\left({𝒜}_g\right)$ for every $g$. For $g=4$, this turns out to be the locus of Jacobians with a vanishing theta-null. For $g=5$, via the Prym map we show that $H\subset {𝒜}_5$ has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of $\overline{{𝒜}_5}$ and show that the component $\overline{N_0^{\text{'}}}$ of the Andreotti-Mayer...