Frobenius Map Research Articles

Frobenius Morphisms and Stability Conditions

We generalize Deng–Du’s folding argument, for the bounded derived category \operatorname{\mathcal{D}}(Q) of an acyclic quiver Q , to the finite-dimensional derived category \operatorname{\mathcal{D}}(\Gamma Q) of the Ginzburg algebra \Gamma Q associated to Q . We show that the F -stable category of \operatorname{\mathcal{D}}(\Gamma Q) is equivalent to the finite-dimensional derived category \operatorname{\mathcal{D}}(\Gamma\mathbb{S}) of the Ginzburg algebra \Gamma\mathbb{S} associated to the species \mathbf{\mathbb{S}} , which is folded from Q . If (Q,\mathbf{\mathbb{S}}) is of Dynkin type, we prove that the space \operatorname{Stab}\operatorname{\mathcal{D}}(\mathbf{\mathbb{S}}) of the stability conditions on \operatorname{\mathcal{D}}(\mathbf{\mathbb{S}}) is canonically isomorphic to the space \operatorname{FStab}\operatorname{\mathcal{D}}(Q) of F -stable stability conditions on \operatorname{\mathcal{D}}(Q) . For the case of Ginzburg algebras, we also prove a similar isomorphism between principal components \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma\mathbb{S}) and \operatorname{FStab}^{\circ}\operatorname{\mathcal{D}}(\Gamma Q) . There are two applications. One is for the space \operatorname{NStab}\operatorname{\mathcal{D}}(\Gamma Q) of numerical stability conditions in \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma Q) . We show that \operatorname{NStab}\operatorname{\mathcal{D}}(\Gamma Q) consists of \operatorname{Br}Q/\operatorname{Br}\mathbf{\mathbb{S}} many connected components, each of which is isomorphic to \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma\mathbb{S}) , for (Q,\mathbf{\mathbb{S}}) is of type (A_{3}, B_{2}) or (D_{4}, G_{2}) . The other is that we relate the F -stable stability conditions to Gepner-type stability conditions.

On tame ramification and centers of F$F$‐purity

AbstractWe introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of center of ‐purity (aka compatibly ‐split subvariety). As an application, we describe the behavior of centers of ‐purity under finite covers — it all comes down to a transitivity property for tame ramification in towers.

Rationality of extended unipotent characters

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof, we use realisations of characters in ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-adic cohomology groups of Deligne–Lusztig varieties as well as block theoretic considerations.

A novel S-box generator using Frobenius automorphism and its applications in image encryption

A novel S-box generator using Frobenius automorphism and its applications in image encryption

Extensions representing Nori-Srinivas obstruction

Let (X,F) be a pair of a smooth variety X over an algebraically closed field k of characteristic p>0 and its Frobenius morphism F. Given a Frobenius Wn(k)-lifting (X¯,F¯) of the pair (X,F) for n≥1, Nori and Srinivas [9] determined the obstruction obsX¯,F¯∈Ext(ΩX/k1,BF⁎ΩX/k1) to Frobenius Wn+1(k)-lifting of (X¯,F¯) in terms of Čech cohomology. The extension representing obsX¯,F¯ has been only known for n=1, which uses the Cartier operator. In this paper, we interpret obsX¯,F¯ in terms of Kato's version of de Rham-Witt Cartier operator [8] and determine the extension representing obsX¯,F¯ for n≥2.

Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g\geq 3$.

Un principe d’Ax–Kochen–Ershov imaginaire

We study interpretable sets in henselian and \sigma -henselian valued fields with value group elementarily equivalent to \mathbb{Q} or \mathbb{Z} . Our first result is an Ax–Kochen–Ershov type principle for weak elimination of imaginaries in finitely ramified characteristic zero henselian fields – relative to value group imaginaries and residual linear imaginaries. We extend this result to the valued difference context and show, in particular, that existentially closed equicharacteristic zero multiplicative difference valued fields eliminate imaginaries in the geometric sorts; the \omega -increasing case corresponds to the theory of the non-standard Frobenius automorphism acting on an algebraically closed valued field. On the way, we establish some auxiliary results on separated pairs of characteristic zero henselian fields and on imaginaries in linear structures, which are also of independent interest.

On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves

We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the l-adic cohomology of the moduli stack in terms of Chern classes.

Characters of [formula omitted] and vertex operators

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GLn(Fq). Green's theory of GLn(Fq) is recovered and enhanced under the realization of the Grothendieck ring of representations RG=⨁n≥0R(GLn(Fq)) as two isomorphic Fock spaces associated to two infinite-dimensional F-equivariant Heisenberg Lie algebras hˆF‾ˆq and hˆF‾q, where F is the Frobenius automorphism of the algebraically closed field F‾q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space RG as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.

Drinfeld’s Lemma for F-isocrystals, I

Abstract We prove that in either the convergent or overconvergent setting, an absolutely irreducible $F$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $p$, further equipped with actions of the partial Frobenius maps, is an external product of $F$-isocrystals over the multiplicands. The corresponding statement for lisse $\overline{{\mathbb{Q}}}_{\ell }$-sheaves, for $\ell \neq p$ a prime, is a consequence of Drinfeld’s lemma on the fundamental groups of absolute products of schemes in characteristic $p$. The latter plays a key role in V. Lafforgue’s approach to the Langlands correspondence for reductive groups with $\ell $-adic coefficients; the $p$-adic analogue will be considered in subsequent work with Daxin Xu.

Shintani descent for standard supercharacters of algebra groups

Let [Formula: see text] be a finite-dimensional nilpotent algebra over a finite field [Formula: see text] with q elements, and let [Formula: see text]. On the other hand, let [Formula: see text] denote the algebraic closure of [Formula: see text], and let [Formula: see text]. Then [Formula: see text] is an algebraic group over [Formula: see text] equipped with an [Formula: see text]-rational structure given by the usual Frobenius map [Formula: see text], and [Formula: see text] can be regarded as the fixed point subgroup [Formula: see text]. For every [Formula: see text], the nth power [Formula: see text] is also a Frobenius map, and [Formula: see text] identifies with [Formula: see text]. The Frobenius map restricts to a group automorphism [Formula: see text], and hence it acts on the set of irreducible characters of [Formula: see text]. Shintani descent provides a method to compare F-invariant irreducible characters of [Formula: see text] and irreducible characters of [Formula: see text]. In this paper, we show that it also provides a uniform way of studying supercharacters of [Formula: see text] for [Formula: see text]. These groups form an inductive system with respect to the inclusion maps [Formula: see text] whenever [Formula: see text], and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group G. Indeed, we show that Shintani descent permits the definition of a certain “superdual algebra” which encodes information about the supercharacters of [Formula: see text] for [Formula: see text].

The quantum Frobenius for character varieties and multiplicative quiver varieties

We prove that quantized multiplicative quiver varieties, quantum character varieties, and Kauffman bracket skein algebras each define sheaves of Azumaya algebras over the smooth loci of their corresponding classical moduli spaces. In the case of skein algebras this establishes a strong form of the Unicity Conjecture of Bonahon and Wong. Our proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as well as a natural nondegeneracy condition satisfied by each of the classical Hamiltonian spaces. We therefore introduce the concepts of Frobenius quantum moment maps, Frobenius Poisson orders and their Hamiltonian reductions, and apply them to the study of Azumaya loci.

Frobenius on the Cohomology of Thickenings

Abstract We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds—that is, independent of the characteristic—on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves.

An image encryption based on confusion-diffusion using two chaotic maps and Frobenius endomorphism

An image encryption based on confusion-diffusion using two chaotic maps and Frobenius endomorphism

O Endomorfismo de Frobenius para Curvas Elípticas e algumas propriedades

In this work we present the Frobenius endomorphism for Elliptical Curves defined over finite fields and some of their properties relative to their kernel and the condition of separability.

Plectic structures in p-adic de Rham cohomology

Given a Hilbert modular form for a totally real field F, and a prime p split completely in F, the f-eigenspace in p-adic de Rham cohomology has a family of partial filtrations and partial Frobenius maps, indexed by the primes of F above p. The general plectic conjectures of Nekovář and Scholl suggest a “plectic comparison isomorphism” comparing these structures to étale cohomology. We prove this conjecture in the case [F:Q]=2 under some mild assumptions; and for general F we prove a weaker statement which is strong evidence for the conjecture, showing that the plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii.

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

Using properties of certain character sums over finite fields, we here obtain some transformations for finite field Appell series F2⁎ and F4⁎ in terms of F34-Gaussian hypergeometric series. These motivate us to find a finite field analog for certain transformations satisfied by the classical Appell series F2 and F4. We also deduce an expression for the finite field Appell series F4⁎ in terms of F23-Gaussian hypergeometric series. Consequently, we investigate connections of trace of Frobenius endomorphism for certain families of elliptic curves and the finite field Appell series F4⁎.

Differential operators on classical invariant rings do not lift modulo p

Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with very few exceptions, these hypersurfaces—and, more generally, classical invariant rings—do not admit a mod p2 lift of the Frobenius endomorphism.

Projective varieties with nef tangent bundle in positive characteristic

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

We study embeddings of groups of Lie type H H in characteristic p p into exceptional algebraic groups G \mathbf {G} of the same characteristic. We exclude the case where H H is of type P S L 2 \mathrm {PSL}_2 . A subgroup of G \mathbf {G} is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of G \mathbf {G} . With a few possible exceptions, we prove that there are no Lie primitive subgroups H H in G \mathbf {G} , with the conditions on H H and G \mathbf {G} given above. The exceptions are for H H one of P S L 3 ( 3 ) \mathrm {PSL}_3(3) , P S U 3 ( 3 ) \mathrm {PSU}_3(3) , P S L 3 ( 4 ) \mathrm {PSL}_3(4) , P S U 3 ( 4 ) \mathrm {PSU}_3(4) , P S U 3 ( 8 ) \mathrm {PSU}_3(8) , P S U 4 ( 2 ) \mathrm {PSU}_4(2) , P S p 4 ( 2 ) ′ \mathrm {PSp}_4(2)’ and 2 B 2 ( 8 ) {}^2\!B_2(8) , and G \mathbf {G} of type E 8 E_8 . No examples are known of such Lie primitive embeddings. We prove a slightly stronger result, including stability under automorphisms of G \mathbf {G} . This has the consequence that, with the same exceptions, any almost simple group with socle H H , that is maximal inside an almost simple exceptional group of Lie type F 4 F_4 , E 6 E_6 , 2 E 6 {}^2\!E_6 , E 7 E_7 and E 8 E_8 , is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group. The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.