Polarizations of Abelian Varieties Over Finite Fields via Canonical Liftings
نویسندگان
چکیده
Abstract We describe all polarizations for abelian varieties over a finite field in fixed isogeny class corresponding to squarefree Weil polynomial, when one variety the admits canonical lifting characteristic zero, that is, which reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between fields and fractional ideals étale algebras enable us explicitly compute classes polarized satisfying some mild conditions. also implement algorithms perform these computations.
منابع مشابه
Abelian 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 کاملEndomorphisms of Abelian Varieties over Finite Fields
Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k its algebraic closure, and A an abelian variety defined over k, of dimension g. For each integer m > 1, let A m denote the group of elements aeA(k) such that ma=O. Let l be a prime number different fro...
متن کاملHomomorphisms of Abelian Varieties over Finite Fields
The aim of this note is to give a proof of Tate’s theorems on homomorphisms of abelian varieties over finite fields [22, 8], using ideas of [26, 27]. We give a unified treatment for both l 6= p and l = p cases. In fact, we prove a slightly stronger version of those theorems with “finite coefficients”. I am grateful to Frans Oort and Bill Waterhouse for useful discussions. My special thanks go t...
متن کاملSupersingular Abelian Varieties over Finite Fields
Let A be a supersingular abelian variety defined over a finite field k. We give an approximate description of the structure of the group A(k) of k-rational points of A in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. Write f = > gi i for distinct monic irreducible polynomials gi and positive integers ei . We show that there is a group homomorphism .:...
متن کاملGroups of Points on Abelian Varieties over Finite Fields
Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of k-rational points on varieties from this class in terms of Newton polygons of fA(1− t).
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2021
ISSN: ['1687-0247', '1073-7928']
DOI: https://doi.org/10.1093/imrn/rnab333