Frobenius Map Research Articles

Global Frobenius liftability I

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo p^2 – we expect that such varieties, after a finite étale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, which states that in characteristic zero a smooth image of a projective toric variety is a toric variety. To this end we analyse the behaviour of toric varieties in families showing some generalization and specialization results. Furthermore, we prove a positive characteristic analogue of Winkelmann’s theorem on varieties with trivial logarithmic tangent bundle (generalizing a result of Mehta–Srinivas), and thus obtaining an important special case of our conjecture. Finally, using deformations of rational curves we verify our conjecture for homogeneous spaces, solving a problem posed by Buch–Thomsen–Lauritzen–Mehta.

Stated Skein Modules of Marked 3-Manifolds/Surfaces, a Survey

We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces) and describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representation theory of the skein algebra. In particular, we show that the skein algebra of non-closed marked surface at any root of 1 is a maximal order. We give a full description of the Azumaya locus of the skein algebra of the puncture torus and give partial results for closed surfaces.

The structure of the group of rational points of an abelian variety over a finite field

Let $A$ be a simple abelian variety of dimension $g$ defined over a finite field $\mathbb{F}_q$ with Frobenius endomorphism $\pi$. This paper describes the structure of the group of rational points $A(\mathbb{F}_{q^n})$, for all $n \geq 1$, as a module over the ring $R$ of endomorphisms which are defined over $\mathbb{F}_q$, under certain technical conditions. If $[\mathbb{Q}(\pi) : \mathbb{Q}]=2g$ and $R$ is a Gorenstein ring, then ${A(\mathbb{F}_{q^n}) \cong R/R(\pi^n-1)}$. This includes the case when $A$ is ordinary and has maximal real multiplication. Otherwise, if $Z$ is the center of $R$ and $(\pi^n - 1)Z$ is the product of invertible prime ideals in $Z$, then $A(\mathbb{F}_{q^n})^d \cong R/R(\pi^n - 1)$ where $d = 2g/[\mathbb{Q}(\pi):\mathbb{Q}]$. Finally, we deduce the structure of $A(\overline{\mathbb{F}}_q)$ as a module over $R$ under similar conditions. These results generalize results of Lenstra for elliptic curves.

Abstract-Induced Modules for Reductive Algebraic Groups With Frobenius Maps

Abstract Let ${\textbf{G}}$ be a connected reductive algebraic group defined over a finite field $\mathbb{F}_q$ of $q$ elements and $\textbf{B}$ be a Borel subgroup of ${\textbf{G}}$ defined over $\mathbb{F}_q$. Let $\mathbb{k}$ be a field and we assume that $\mathbb{k}=\bar{\mathbb{F}}_q $ when $\textrm{char}\ \mathbb{k}=\textrm{char} \ \mathbb{F}_q$. We show that the abstract-induced module $\mathbb{M}(\theta )=\mathbb{k}{\textbf{G}}\otimes _{\mathbb{k}\textbf{B}}\theta $ (here $\mathbb{k}\textbf{H}$ is the group algebra of $\textbf{H}$ over the field $\mathbb{k}$ and $\theta $ is a character of $\textbf{B}$ over $\mathbb{k}$) has a composition series (of finite length) if $\textrm{char}\ \mathbb{k}\ne \textrm{char} \ \mathbb{F}_q$. In the case $\mathbb{k}=\bar{\mathbb{F}}_q$ and $\theta $ is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of $\mathbb{M}(\theta )$. We determine all the composition factors whenever a composition series exists. Thus we obtain a large class of abstract infinite-dimensional irreducible $\mathbb{k}{\textbf{G}}$-modules.

Quantum symmetric pairs at roots of 1

A quantum symmetric pair is a quantization of the symmetric pair of universal enveloping algebras. Recent development suggests that most of the theory for quantum groups can be generalized to the setting of quantum symmetric pairs. In this paper, we study the ıquantum group at roots of 1. We generalize Lusztig's quantum Frobenius morphism in this new setting. We define the small ıquantum group and compute its dimension.

Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings

We construct complex projective schemes with Lyubeznik numbers of their cones depending on the choices of projective embeddings. This answers a question of G. Lyubeznik in the characteristic 0 case. It contrasts with a theorem of W. Zhang in the positive characteristic case where the Frobenius endomorphism is used. Reducibility of schemes is essential in our argument. B. Wang recently constructed examples of irreducible projective schemes (which are not normal) from our examples of reducible ones. So the question is still open in the normal singular case.

On almost-symmetry in generalized numerical semigroups

In this work, we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that this class yields a new family of Frobenius generalized numerical semigroups and extends the class of irreducible generalized numerical semigroups. This investigation allows us to provide a method of computing all almost symmetric generalized numerical semigroup having a fixed Frobenius element and organizing them in a rooted tree depending on a chosen monomial order.

Sophisticated analysis of a method to eliminate fruitless cycles for Pollard's rho method with skew Frobenius mapping over a Barreto-Naehrig curve

In this paper, the authors focus on and propose an approach to attack a kind of pairing-friendly curves, the Barreto-Naehring (BN) curve, to accelerate the evaluation of the security level concerning the elliptic curve discrete logarithm problem (ECDLP). More precisely, this paper targets the BN curve, which is known to be a pairing-friendly curve, and Pollard's rho method based on the random-walk is adopted to attack the curve. Though Pollard's rho method with skew Frobenius mapping is known to solve the ECDLP efficiently, this approach sometimes induces the unsolvable cycle, called the fruitless cycle, and such trials must restart with a different starting point. However, any effective method to eliminate such fruitless cycles has not been proposed. Therefore, the authors focus and give the sophisticated analysis to propose an effective approach to eliminate such cycles to optimize Pollard's rho method furthermore. In addition, we confirm the effectiveness of the method by applying it to a BN curve with 12, 17, and 33-bit parameters.

Frobenius groups which are the automorphism groups of orientably-regular maps

In this paper, we prove that a Frobenius group (except for those which are dihedral groups) can only be the automorphism group of an orientably-regular chiral map. The necessary and sufficient conditions for a Frobenius group to be the automorphism group of an orientably-regular chiral map are given. Furthermore, these orientably-regular chiral maps with Frobenius automorphisms are proved to be normal Cayley maps. Frobenius groups conforming to these conditions are also constructed.

Local Smooth Conjugations of Frobenius Endomorphisms

A generalization of the Bottcher equation is considered. It turned out that the parametrized Poisson integral, as a function of its parameters, satisfies an equation of the type described. The structure theorem for splitting maps of Frobenius endomorphisms in a ring and in an algebra over it is proved. The real field case is considered. The generalized Bottcher equation is solved for classical two-dimensional algebras and for the Poisson algebra.

An explicit version of the correspondence between Noetherian modules with Cartier maps and Artinian modules with Frobenius maps

In this paper, we introduce an explicit version of the correspondence between Noetherian modules with Cartier maps and Artinian modules with Frobenius maps which recovers the correspondence given by R. Y. Sharp and Y. Yoshino, and the correspondence given by M. Blickle and G. Böckle.**********************************************************************************************************************************************************************************************************************************

Counting points on hyperelliptic curves of type y2 = x2g+1 + axg+1 + bx

In this work, we investigate hyperelliptic curves of type C:y2=x2g+1+axg+1+bx over the finite field Fq,q=pn,p>2. For the case of g=3 we propose an algorithm to compute the number of points on the Jacobian of the curve with complexity O˜(log4⁡p) over Fp. In case of g=4 we present a point counting algorithm with complexity O˜(log8⁡q) over Fq. The Jacobian JC splits over an extension of the field Fq on the Jacobians of the curves defined by the Dickson polynomials Dg(x,1) of degree g. For these curves of genus 2,3,5 with equation y2=Dg(x,1)+a and curves of genus 2,4 with equation y2=(x+2)(Dg(x,1)+a), we give the lists of possible characteristic polynomials of the Frobenius endomorphism modulo p.

On the restricted Jordan plane in odd characteristic

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given.

BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$

Let $GR(p^r,m)$ denote the Galois ring of characteristic $p^r$ and cardinality $p^{rm}$ seen as a free module of rank $m$ over the integer ring $\mathbb{Z}_{p^r}$. A general formula for the sum of the homogeneous weights of the $p^r$-ary images of elements of $GR(p^r,m)$ under any basis is derived in terms of the parameters of $GR(p^r,m)$. By using a Vandermonde matrix over $GR(p^r,m)$ with respect to the generalized Frobenius automorphism, a constructive proof that every basis of $GR(p^r,m)$ has a unique dual basis is given. It is shown that a basis is self-dual if and only if its automorphism matrix is orthogonal, and that a basis is normal if and only if its automorphism matrix is symmetric.

Computing representation matrices for the action of Frobenius on cohomology groups

In algebraic geometry, the Frobenius map (or called Frobenius action, or Frobenius operator) F⁎ on cohomology groups plays an important role in the classification of algebraic varieties over a field of positive characteristic. In particular, representation matrices for F⁎ give rise to many important invariants such as p-rank and a-number. Several methods for computing representation matrices for F⁎ have been proposed for specific curves.In this paper, we present an algorithm to compute representation matrices for F⁎ of general projective schemes over a perfect field of positive characteristic. We also propose an efficient algorithm specific to complete intersections; it requires to compute only certain coefficients in a power of a multivariate polynomial. Our algorithms shall derive fruitful applications such as computing Hasse-Witt matrices, and enumerating superspecial curves. In particular, the second algorithm for complete intersections provides a useful tool to judge the superspeciality of an algebraic curve, which is a key ingredient to prove main results in Kudo and Harashita (2017a,b, 2020) on the enumeration of superspecial genus-4 curves.

A Theorem About Maximal Cohen–Macaulay Modules

Abstract It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

Counting invariant subspaces and decompositions of additive polynomials

The functional (de)composition of polynomials is a topic in pure and computer algebra with many applications. The structure of decompositions of (suitably normalized) polynomials f=g∘h in F[x] over a field F is well understood in many cases, but less well when the degree of f is divisible by the positive characteristic p of F. This work investigates the decompositions of r-additive polynomials, where every exponent and also the field size is a power of r, which itself is a power of p.The decompositions of an r-additive polynomial f are intimately linked to the Frobenius-invariant subspaces of its root space V in the algebraic closure of F. We present an efficient algorithm to compute the rational Jordan form of the Frobenius automorphism on V. A formula of Fripertinger (2011) then counts the number of Frobenius-invariant subspaces of a given dimension and we derive the number of decompositions with prescribed degrees.

Pseudofinite difference fields and counting dimensions

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.

On the interplay between the Frobenius functor and its dual

For a commutative Noetherian ring [Formula: see text] of prime characteristic, denote by [Formula: see text] the ring [Formula: see text] with the left structure given by the Frobenius map. We develop Thomas Marley’s work on the property of the Frobenius functor [Formula: see text] and show some interplays between [Formula: see text] and its dual [Formula: see text], which is introduced by Jürgen Herzog.

Alvis–Curtis duality for representations of reductive groups with Frobenius maps

Abstract We generalize the Alvis–Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis–Curtis duality of infinite type, which we define in this paper, also interchanges the irreducible representations in the principal representation category.