arXiv subjects
Publications and source records attributed to Yuri G. Zarhin.
We study Prym varieties of ramified (at precisely two points) double covers of smooth irreducible complex projectives curves that admit an automorphism of prime order $p>2$. Using Galois theory, we give an explicit constructions of Prym varieties that are not isomorphic to jacobians (even if one ignores the polarizations).
Let $d>1$ be an integer and $K_0$ a perfect field such that $char(K_0)$ does not divide $d$. Let $n>d$ be an integer that is prime to $d$. Let $f(x)\in K_0[x]$ be a degree $n$ monic polynomial without repeated roots, and $\mathcal{C}_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(\mathcal{C}_{f,d})$ be the Jacobian of the $K_0$-curve $\mathcal{C}_{f,d} $. As usual, we identify $\mathcal{C}_{f,d}$ with its canonical image in $J(\mathcal{C}_{f,d})$ (such that the only ``infinite point'' of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). We say that an integer $m>1$ is $(n,d)$-reachable over $K_0$ if there exists a polynomial $f(x)$ as above such that $\mathcal{C}_{f,d}(K_0)$ contains a torsion point of order $m$. Let us put $\ell_0:=[(n+d)/d], \ m_0:=\ell_0 d$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m = n$ or $m \ge m_0$ (in addition, both $d$ and $n$ are $(n,d)$-reachable over every $K_0$). We also proved that if $m_0$ is $(n,d)$-reachable over some $K_0$ then $n-m_0+\ell_0\ge 0$. In the present paper we discuss the $(n,d)$-reachability of $m_0$ when $n-m_0+\ell_0=0$ or $1$.
Let $d\geq 2$ be an integer, $K_0$ a perfect field such that $char(K_0)$ does not divide $d$, $n > d$ an integer prime to $d$, $f(x)\in K_0[x]$ a degree $n$ monic polynomial without repeated roots, and $C_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(C_{f,d})$ be the Jacobian of the $K_0$-curve $C_{f,d} $. We identify $C_{f,d}$ with its canonical image in $J(C_{f,d})$ (such that the infinite point of $C_{f,d}$ goes to the zero of the group law on $J(C_{f,d})$). We say that an integer $m>1$ is $(n,d)$-reachable over $K_0$ if there exists a polynomial $f(x)$ as above such that $C_{f,d}(K_0)$ contains a torsion point of order $m$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m \geq n$ (in addition, both $d$ and $n$ are $(n,d)$-reachable). In the present paper we prove the following. If $n n$, then $d\cdot [(n+d)/d]$ is $(n,d)$-reachable if and only if $n-(d-1)\cdot [(n+d)/d]\ge 0$. If $char(K_0)=0$, then $n+d$ is $(n,d)$-reachable if and only if $d^2-2d<n$. If $d=2$ (the hyperelliptic case) and $char(K_0)=0$, then $m$ is $(n,d)$-reachable if $n+1 \le m \le 2n+1$. (The case when $n \le m \le 3(n-1)/2$ was done earlier by E.V. Flynn.)
In this paper we study principally polarized abelian varieties that admit an automorphism of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves.
We call an order $O$ in a quadratic field $K$ odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve $E$ over the field $C$ of complex numbers with CM odd (resp. even) if its endomorphism ring $End(E)$ is an odd (resp. even) order in the corresponding imaginary quadratic field. Suppose that $j(E)$ is a real number and let us consider the set $J(R,E)$ of all $j(E')$ where $E'$ is any elliptic curve that enjoys the following properties. 1) $E'$ is isogenous to $E$; 2) $j(E')$ is a real number; 3) $E'$ has the same parity as $E$. We prove that the closure of $J(R,E)$ in the set $R$ of real numbers is the closed semi-infinite interval $(-\infty,1728]$ (resp. the whole $R$) if $E$ is odd (resp. even). This paper was inspired by a question of Jean-Louis Colliot-Thélène and Alena Pirutka about the distribution of $j$-invariants of certain elliptic curves of CM type.
Let $d\geq 2$ be a positive integer, $K$ an algebraically closed field of characteristic not dividing $d$, $n\geq d+1$ a positive integer that is prime to $d$, $f(x)\in K[x]$ a degree $n$ monic polynomial without multiple roots, $C_{f,d}: y^d=f(x)$ the corresponding smooth plane affine curve over $K$, $\mathcal{C}_{f,d}$ a smooth projective model of $C_{f,d}$ and $J(\mathcal{C}_{f,d})$ the Jacobian of $\mathcal{C}_{f,d} $. We identify $\mathcal{C}_{f,d}$ with the image of its canonical embedding into $J(\mathcal{C}_{f,d})$ (such that the infinite point of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). Earlier the second named author proved that if $d=2$ and $n=2g+1 \ge 5$ then the genus $g$ hyperelliptic curve $\mathcal{C}_{f,2}$ contains no points of orders lying between $3$ and $n-1=2g$. In the present paper we generalize this result to the case of arbitrary $d$. Namely, we prove that if $P$ is a point of order $m>1$ on $\mathcal{C}_{f,d}$, then either $m=d$ or $m\geq n$. We also describe all curves $\mathcal{C}_{f,d}$ having a point of order $n$.
We study the endomorphism algebra, the automorphism group and the Hodge group of complex tori of complex tori, whose second rational cohomology group enjoys a certain Hodge property introduced by F. Campana.
Let $\ell$ be an odd prime and $K$ a field of characteristic different from $\ell$. Let $\bar{K}$ be an algebraic closure of $K$. Assume that $K$ contains a primitive $\ell$th root of unity. Let $n \ne \ell$ be another odd prime. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider superelliptic curves $C_{f,\ell}: y^{\ell}=f(x)$ and $C_{h,\ell}: y^{\ell}=h(x)$ of genus $(n-1)(\ell-1)/2$, and their jacobians $J^{(f,\ell)}$ and $J^{(h,\ell)}$, which are $(n-1)(\ell-1)/2$-dimensional abelian varieties over $\bar{K}$. Suppose that one of the polynomials is irreducible and the other reducible over $K$. We prove that if $J^{(f,\ell)}$ and $J^{(h,\ell)}$ are isogenous over $\bar{K}$ then both endomorphism algebras $\mathrm{End}^{0}(J^{(f,\ell)})$ and $\mathrm{End}^{0}(J^{(h,\ell)})$ contain an invertible element of multiplicative order $n$.
We deal with $g$-dimensional abelian varieties $X$ over finite fields. We prove that there is an universal constant (positive integer) $N=N(g)$ that depends only on $g$ that enjoys the following properties. If a certain self-product of $X$ carries an exotic Tate class then the self-product $X^{2N}$of $X$ also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.
This is an exposition of results of R.C. Vaughan and the author (Mathematika 70 (2024), no. 4). We discuss how often the squarefree values of an integral polynomial do occur. We discuss interrelations between our results and results of B. Poonen (arXiv:math/0203292 [math.NT]).
We construct absolutely simple jacobians of non-hyperelliptic genus 4 curves, using Del Pezzo surfaces of degree 1. This paper is a natural continuation of author's paper math.AG/0405156.
Let $O$ be an order of odd discriminant $D$ in an imaginary quadratic field $K$. Let $Cl(O)$ be the group of proper $O$-ideals and $Cl(O)[2]$ the kernel of multiplication by $2$ in $Cl(O)$. We describe explicitly the group $Cl(O)[2]$. In particular, we prove that its order is $2^{s_D-1}$ where $s_D$ is the number of prime divisors of $D$.
Let f(x) be a polynomial of degree at least 5 with complex coefficients and without repeated roots. Let p be an odd prime. Suppose that all the coefficients of f(x) lie in a subfield K such that: 1) K contains a primitive p-th root of unity; 2) f(x) is irreducible over K; 3) the Galois group \Gal(f) of f(x) acts doubly transitively on the set of roots of f(x); 4) the index of every maximal subgroup of Gal(f) does not divide deg(f)-1. Then the endomorphism ring of the Jacobian of the superelliptic curve y^p=f(x) is isomorphic to the pth cyclotomic ring for all primes p>deg(f).
Using Galois theory, we construct explicitly (in all complex dimensions >1) an infinite family of simple complex tori of algebraic dimension 0 with Picard number 0.
In this survey we discuss holomorphic $\mathbb{P}^1$-bundles $p: X \to Y$ over a non-uniruled complex compact Kähler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We discuss so called Jordan properties of the groups $Aut(X)$ and $Bim(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively.
We construct jacobians of plane quartics without complex multiplication, using Del Pezzo surfaces of degree 2.
Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is an (odd) prime $\ell$ dividing $n$ such that the endomorphism algebras of both $J(C_f)$ and $J(C_h)$ contain a subfield that is isomorphic to the field of $\ell$th roots of $1$.
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.