6,812 research outputs found
On the nonexistence of certain curves of genus two
We prove that if q is a power of an odd prime then there is no genus-2 curve
over F_q whose Jacobian has characteristic polynomial of Frobenius equal to x^4
+ (2-2q)x^2 + q^2. Our proof uses the Brauer relations in a biquadratic
extension of Q to show that every principally polarized abelian surface over
F_q with the given characteristic polynomial splits over F_{q^2} as a product
of polarized elliptic curves.Comment: LaTeX, 13 page
Isogeny classes of abelian varieties with no principal polarizations
We provide a simple method of constructing isogeny classes of abelian
varieties over certain fields k such that no variety in the isogeny class has a
principal polarization. In particular, given a field k, a Galois extension l of
k of odd prime degree p, and an elliptic curve E over k that has no complex
multiplication over k and that has no k-defined p-isogenies to another elliptic
curve, we construct a simple (p-1)-dimensional abelian variety X over k such
that every polarization of every abelian variety isogenous to X has degree
divisible by p^2. We note that for every odd prime p and every number field k,
there exist l and E as above. We also provide a general framework for
determining which finite group schemes occur as kernels of polarizations of
abelian varieties in a given isogeny class.
Our construction was inspired by a similar construction of Silverberg and
Zarhin; their construction requires that the base field k have positive
characteristic and that there be a Galois extension of k with a certain
non-abelian Galois group.Comment: 13 pages, AMS-TeX, with updated references. To appear in the volume
"Moduli of Abelian Varieties (Texel Island 1999)
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