Weil Polynomial Research Articles

Weil polynomials of abelian varieties over finite fields with many rational points

We consider the finite set of isogeny classes of [Formula: see text]-dimensional abelian varieties defined over the finite field [Formula: see text] with endomorphism algebra being a field. We prove that the class within this set whose varieties have the maximal number of rational points is unique, for any prime even power [Formula: see text] big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on [Formula: see text]. In dimension [Formula: see text], we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, for any prime even power [Formula: see text].

Polarizations of Abelian Varieties Over Finite Fields via Canonical Liftings

AbstractWe describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical lifting to characteristic zero, that is, a lifting for which the reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between abelian varieties over finite fields and fractional ideals in étale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations.

Groups of points on abelian threefolds over finite fields

In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given Fq-isogeny class. This work completes partial classification given in [9].

Elliptic curves, random matrices and orbital integrals

An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny class. In this paper, we give a new, transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. This answers a question posed by N. Katz.

On the parity of supersingular Weil polynomials

Let q be an odd power of a prime p and let A/Fq be a supersingular abelian variety of dimension g. We show that if p>2g+1, then the characteristic polynomial of the q-Frobenius is an even polynomial. This generalizes the well-known result on the vanishing of the trace of the p-Frobenius when p>3 for supersingular elliptic curves over Fp.

Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields

AbstractConsider a quartic q-Weil polynomial ƒ. Motivated by equidistribution considerations, we define, for each prime ℓ, a local factor that measures the relative frequency with which ƒ mod ℓ occurs as the characteristic polynomial of a symplectic similitude over 𝔽ℓ. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over 𝔽q with Weil polynomial ƒ.

Abelian surfaces over finite fields with prescribed groups

Let A be an abelian surface over F q , the field of q elements. The rational points on A / F q form an abelian group A ( F q ) ≃ Z / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z . We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups Z / n 1 Z × Z / n 1 n 2 Z × Z / n 1 n 2 n 3 Z × Z / n 1 n 2 n 3 n 4 Z do not occur if n 1 is very large with respect to n 2 , n 3 , n 4 (Theorem 1.1), and occur with density zero in a wider range of the variables (Theorem 1.2).

Finite group subschemes of abelian varieties over finite fields

Let A be an abelian variety over a finite field k. The k-isogeny class of A is uniquely determined by the Weil polynomial fA. For a given prime number ℓ≠chark we give a classification of group schemes B[ℓ], where B runs through the isogeny class, in terms of certain Newton polygons associated to fA. As an application we classify zeta functions of Kummer surfaces over k.

The groups of points on abelian varieties over finite fields

Abstract Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).

Principally polarizable isogeny classes of abelian surfaces over finite fields

Let $\calA$ be an isogeny class of abelian surfaces over $\fq$ with Weil polynomial $x^4 + ax^3 + bx^2 + aqx + q^2$. We show that $\calA$ does not contain a surface that has a principal polarization if and only if $a^2 - b = q$ and $b < 0$ and all prime divisors of $b$ are congruent to $1$ modulo~$3$.

Gauge Interpretation of Characteristic Classes

Let π:P→M be a principal G-bundle. Then one can consider the following diagram of fibre bundles: \CD J^{1}(P) @>\pi_{10}>> P\\ @VqVV @VV\pi V\\ C(P) @>p>> M\endCD where p is the bundle of connections of π. As is well known, q is also a principal G-bundle, and the canonical contact form θ on J1(P) can be considered as a connection form on q, with curvature form Θ. One defines aut P as the Lie algebra of G-invariant vector fields on P and gau P as the ideal of π-vertical G-invariant vector fields on P. If X∈autP⊂X(P), then one defines the infinitesimal contact transformation associated to X, X1∈X(J1(P)), and its q-projection XC∈X(C(P)). A differential form Ω on C(P) is said to be aut P-invariant [resp. gauge invariant] if LXCΩ=0 for every X∈autP [resp. X∈gauP]. On the other hand, let us denote by g the Lie algebra of G. An element of the symmetric algebra of g∗ will be called a Weil polynomial. The main result of the paper is the following theorem: If G is connected, for every gauge invariant form Ω on C(P) there exist differential forms ω1,…,ωk on M and Weil polynomials f1,…,fk such that Ω=p∗(ω1)∧f1(Θ)+⋯+p∗(ωk)∧fk(Θ). As a consequence, the authors prove that a differential form Ω on C(P) is aut P-invariant iff Ω=f(Θ), where f is a Weil polynomial, and then Ω is closed. Explicit examples are shown and the link between the above theorem and the geometric formulation of Utiyama's theorem is explained.

Unified prescription for the generation of electroweak and gravitational gauge field Lagrangian on a principal fiber bundle

It is essentially known that useful gauge field Lagrangians arise as Weil polynomials of the curvature of the gauge connection. The deeper implications and details of this fact are worked out in two widely differing cases. The Glashow–Weinberg–Salam gauge field Lagrangian for electroweak theory and the Townsend–Zardecki action for gravitation are obtained from the same type of ‘‘Yang–Mills’’ Weil form on a principal fiber bundle over space-time, with symmetry group U(2) and SO(2,3), respectively. The unified geometrical approach given here shows that fiber bundle reduction and symmetry breaking are essential not only in electroweak theory but also in the SO(2,3) gauge theory for gravitation. In fact, the process of symmetry breaking in electroweak theory and the soldering of the anti-de Sitter bundle, essential in the interpretation of SO(2,3) gauge theory as a theory for gravitation, are corresponding geometrical concepts.

Fibre bundle generated theory of higher dimensional gravity

With the help of Weil polynomials on the Lie algebra of the Poincare group in D dimensions, we construct a Lagrangian form for higher dimensional gravity on a principal fibre bundle whose base space is an even-dimensional Riemann-Cartan spacetime and whose structure group is the Poincare group.

Construction of N=2 Supergravity on Principal Fibre Bundle. II

Making use of the Weil polynomial and the generalized commutator, we reformulate on shell N=2 supergravity based on OSp(2,4)⊃SO(1,3). The resulting action becomes a theory with OSp(2,4) ⊃SO(1,3) ×SO(2). Detrivialization terms used in the group manifold approach are derived from the Weil polynomial and the generalized commutator. Also in N=2 Poincaré supergravity we have the same results. Symmetries of underlying groups (for example, rotations and scale transformations) are useful tools for analysing theories.

Differential characters and Borel cohomology

61. THE MAIN RESULTS THE PURPOSE of this note is to show that the differential characters associated to a compact Lie group G (as defined by Chern, Simons and Cheeger) comprise its Bore1 cohomology H&,,(G,R/Z). Along the way we obtain some interesting formulae for these characters as cochains in the bar resolution for G, and obtain some results on the cohomology of certain bundles with discrete structure group. Let % be the Lie algebra of G and let I(%, R) be the ring of Weil polynomials, i.e. symmetric multilinear functions on Yl which are invariant under the Adjoint action of G. In 121 Chern and Simons showed how to use a connection 8 on a principal G-bundle E-B to determine a characteristic form TP(8) in A*(E, R) for every P E I(%, R). These forms satisfy the equation