Hyperelliptic jacobians and projective linear Galois groups

In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure $K_a$ of the ground field $K$ if the Galois group $Gal(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $S_n$ or the alternating group $A_n$. Here $n>4$ is the degree of $f$. In math.AG/0003002 we extended this result to the case of certain ``smaller'' Galois groups. In particular, we treated the infinite series $n=2^r+1, Gal(f)=L_2(2^r)$ and $n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$. In the present paper we prove that $J(C)$ has only trivial endomorphisms over $K_a$ if the set of roots of $f$ could be identified with the $(m-1)$-dimensional projective space $P^{m-1}(F_q)$ over a finite field $F_q$ of odd characteristic in such a way that $Gal(f)$, viewed as its permutation group, becomes either the projective linear group $PGL(m,F_q)$ or the projective special linear group $L_m(q):=PSL(m,F_q)$. Here we assume that $m>2$.Comment: LaTeX2e, 8 pages. We include a discussion of the characteristic $p$ cas

Polynomials in one variable and ranks of certain tangent maps

We study a map that sends a monic degree n complex polynomial f(x) without multiple roots to the collection of n values of its derivative at the roots of f(x). We give an answer to a question posed by Ju.S. Ilyashenko.Comment: Remark 3.5 has been corrected/update

Homomorphisms of abelian varieties

We discuss Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications to hyperelliptic jacobians are given. In particular, we improve some of our earlier results.Comment: 23 pages, to appear in the Proceedings of ``Arithmetic, Geometry and Coding Theory - 9" conference (Luminy, May 2003

Absolutely simple Prymians of trigonal curves

Using Galois Theory, we construct explicitly absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. Our approach is based on the previous papers math/0610138 [math.AG] and math/0605028 [math.AG] .Comment: 12 page

Abelian varieties over fields of finite characteristic

The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.Comment: 18 pages. Some typos have been correcte

Division by 2 on hyperelliptic curves and jacobians

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over $K$ and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$). For each point $P=(a,b)\in C(K)$ there are $2^{2g}$ points $\frac{1}{2}P \in J(K)$. We describe explicitly the Mumford represesentations of all $\frac{1}{2}P$. The rationality questions for $\frac{1}{2}P$ are also discussed.Comment: 24 pages. We added results concerning the absence of torsion points of certain order on certain subvarieties of hyperelliptic jacobian

Hyperelliptic jacobians and \U_3(2^m)

In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure $K_a$ of the ground field $K$ if the Galois group $Gal(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $S_n$ or the alternating group $A_n$. Here $n>4$ is the degree of $f$. In math.AG/0003002 we extended this result to the case of certain ``smaller'' Galois groups. In particular, we treated the infinite series $n=2^r+1, Gal(f)=L_2(2^r)$ and $n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$. In this paper we do the case of Gal(f)=\U_3(2^m) and $n=2^{3m}+1$

Jordan groups and elliptic ruled surfaces

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir L. Popov.Comment: 14 page

Division by 2 on odd degree hyperelliptic curves and their jacobians

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over K, and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the identity element of $J$). It is well known that for each $\mathfrak{b} \in J(K)$ there are exactly $2^{2g}$ elements $\mathfrak{a} \in J(K)$ such that $2\mathfrak{a}=\mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $\mathfrak{a}$, in terms of the Mumford representation of $\mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $\mathfrak{a}$, when $\mathfrak{b}\in J(K)$ is given by $P=(a,b) \in C(K)\subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.Comment: 18 pages. The paper overlaps with arXiv:1606.05252 and arXiv:1807.0700

Hyperelliptic jacobians and modular representations

In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure of the ground field $K$ if the Galois group $Gal(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $S_n$ or the alternating group $A_n$. Here $n>4$ is the degree of $f$. In the present paper we extend this result to the case of certain ``smaller'' Galois groups. In particular, we treat the case when $n=11$ or 12 and $Gal(f)$ is the Mathieu group $M_{11}$ or $M_{12}$ respectively. The infinite series $n=2^r+1, Gal(f)=L_2(2^r)$ and $n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$ are also treated.Comment: The paper will appear in Texel volume "Moduli of abelian varieties" (Texel Island 1999), Birkh\"ause