Related papers: Endomorphisms of superelliptic jacobians

We prove that in odd characteristic the jacobian of a hyperelliptic curve $y^2=f(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field if the Galois group of the polynomial $f$ of even degree is ``very big". The…

Let $K$ be a number field, $n>4$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Suppose $C:y^2=f(x)$ is the corresponding…

We classify the endomorphism algebras of factors of the Jacobian of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field…

Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$.…

We prove that the jacobian of a hyperelliptic curve y^2=f(x) has no nontrivial endomorphisms over an algebraic closure of the ground field of characteristic zero if the Galois group of the polynomial f is ``very big''.

Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular graph whose nodes are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of…

This is (mostly) a survey article. We use an information about Galois properties of points of small order on an abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in…

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-power torsion…

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…

Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B \in O_K is…

It is proved that the Jacobian of a k-endomorphism of k[x_1,...,x_n] over a field k of characteristic zero taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an…

We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t \in K$ and the Galois group of the polynomial $h(x)$ over $K$…

Let $n=2g+2$ be a positive even integer, $f(x)$ a degree $n$ complex polynomial without multiple roots and $C_f: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over the field $\C$ of complex numbers. Let a $(g-1)$-dimensional…

In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be…

Let $K$ be a global field of characteristic different from 2 and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group over $K$ is either the full symmetric group $S_{2g}$ or the alternating group…

We mainly study a polynomial $f_{1,n}(x)=x^{n-1} + 2x^{n-2} + 3x^{n-3} + \cdots + kx^{n-k} + \cdots + (n-1)x + n$ over $\mathbb{Z}$ and the Galois group of the minimal splitting field. First, we show that an arbitrary root $\alpha_{n}$ of…

Let p be an odd prime. Let K = \Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a \Q-isomorphism of the extension K/\Q generating the Galois group G of K/\Q. For n in Z, the notation…

Let $K$ be an imaginary quadratic field and $p$ be an odd prime number. Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. We study the Iwasawa theory of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$…

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[\zeta_m+\zeta_m^{-1}])$ and $K_{2k}(\mathbb{Z}[\zeta_m])$,…

Let $k$ be a field of characteristic zero, and let $f: k[x,y] \to k[x,y]$, $f: (x,y) \mapsto (p,q)$, be a $k$-algebra endomorphism having an invertible Jacobian. Write $p=a_ny^n+\cdots+a_1y+a_0$, where $n=deg_y(p) \in \mathbb{N}$, $a_i \in…