Weil conjectures for abelian varieties over finite fields
نویسنده
چکیده
This is an expository paper on zeta functions of abelian varieties over finite fields. We would like to go through how zeta function is defined, and discuss the Weil conjectures. The main purpose of this paper is to fill in more details to the proofs provided in Milne. Subject to length constrain, we will not include a detailed proof for Riemann hypothesis in this paper. We will mainly be following Milne’s notes [2].
منابع مشابه
2 . Points over Finite Fields and the Weil Conjectures
In this chapter we will relate the topology of smooth projective varieties over the complex numbers with counting points over finite fields, via the Weil conjectures. If X is a variety defined over a finite field Fq, one can count its points over the various finite extensions of Fq; denote Nm = |X(Fmq )| (for instance, if X ⊂ AnFq is affine, given by equations f1, . . . , fk, then Nm = |{x ∈ Fm...
متن کاملAbelian varieties over finite fields
A. Weil proved that the geometric Frobenius π = Fa of an abelian variety over a finite field with q = pa elements has absolute value √ q for every embedding. T. Honda and J. Tate showed that A 7→ πA gives a bijection between the set of isogeny classes of simple abelian varieties over Fq and the set of conjugacy classes of q-Weil numbers. Higher-dimensional varieties over finite fields, Summer s...
متن کاملHypersurfaces and the Weil conjectures
We give a proof that the Riemann hypothesis for hypersurfaces over finite fields implies the result for all smooth proper varieties, by a deformation argument which does not use the theory of Lefschetz pencils or the `-adic Fourier transform.
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 کاملPrincipally 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 کامل