Isogeny Classes of Abelian Varieties with No Principal Polarizations
نویسندگان
چکیده
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. 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
منابع مشابه
Principally Polarized Ordinary Abelian Varieties over Finite Fields
Deligne has shown that there is an equivalence from the category of ordinary abelian varieties over a finite field A: to a category of Z-modules with additional structure. We translate several geometric notions, including that of a polarization, into Deligne's category of Z-modules. We use Deligne's equivalence to characterize the finite group schemes over k that occur as kernels of polarizatio...
متن کاملQuaternions , polarizations and class numbers
We study abelian varieties A with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms. Moreover, we give an expression for the number π 0 (A) of isomorphism classes of principal polarizations on A in terms of relative class numbers of CM fields by means of Ei...
متن کاملABELIAN VARIETIES OVER FINITE FIELDS WITH A SPECIFIED CHARACTERISTIC POLYNOMIAL MODULO l
We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P (T ) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a rational point of order l.
متن کاملIsogeny Class and Frobenius Root Statistics for Abelian Varieties over Finite Fields
Let I(g, q, N) be the number of isogeny classes of g-dimensional abelian varieties over a finite field Fq having a fixed number N of Fq-rational points. We describe the asymptotic (for q →∞) distribution of I(g, q, N) over possible values of N . We also prove an analogue of the Sato—Tate conjecture for isogeny classes of g-dimensional abelian varieties. 2000 Math. Subj. Class. Primary: 11G25, 1...
متن کاملWeil Numbers Generated by Other Weil Numbers and Torsion Fields of Abelian Varieties
Using properties of the Frobenius eigenvalues, we show that, in a precise sense, “most” isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and a similar result for “most” isogeny classes. Some global cases are also treated.
متن کامل