arXiv subjects

Yuri G. Zarhin

Publications and source records attributed to Yuri G. Zarhin.

94 records · Page 6Linked to original sources

arXiv2000

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.

math.AG↗

arXiv1999

Hyperelliptic jacobians without complex multiplication

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''.

math.AG↗

arXiv1999

Polarizations on abelian varieties and self-dual ell-adic representations of inertia groups

It is well-known that every finite subgroup of GL_d(Q_{\ell}) is conjugate to a subgroup of GL_d(Z_{\ell}). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type if G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if \ell>d+1, and G is a subgroup of Sp_{2d}(Q_{\ell}) of inertia type, then G is conjugate in GL_{2d}(Q_{\ell}) to a subgroup of \Sp_{2d}(Z_{\ell}). Despite the fact that G can fail to be conjugate in \GL_{2d}(Q_\ell) to a subgroup of \Sp_{2d}(Z_\ell), we prove that it can nevertheless be embedded in \Sp_{2d}(F_\ell) in such a way that the characteristic polynomials are preserved (mod \ell), as long as \ell>3. The latter result holds for arbitrary finite groups, not necessarily of inertia type, and holds also for symmetric forms, not just alternating forms. We give examples which show that the bounds are sharp. We apply these results to construct, for every odd prime \ell, isogeny classes of abelian varieties all of whose polarizations have degree divisible by \ell. This paper is a revised version of ANT-0151, titled `Self-dual ell-adic representations of finite groups'.

math.NT↗

arXiv1997

Torsion of abelian varieties, Weil classes and cyclotomic extensions

Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that under certain conditions on $X$ and $K$ the existence of infinitely many L-rational points of finite order on $X$ implies that the intersection of $L$ and $K(c)$ has infinite degree over $K$.

alg-geom↗