Frobenius Map Research Articles

Invariants of polynomials mod Frobenius powers

Lewis, Reiner, and Stanton conjectured a Hilbert series for a space of invariants under an action of finite general linear groups using (q,t)-binomial coefficients. This work gives an analog in positive characteristic of theorems relating various Catalan numbers to the representation theory of rational Cherednik algebras. They consider a finite general linear group as a reflection group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. We prove a variant of their conjecture in the local case, when the group acting fixes a reflecting hyperplane.

Hazewinkel Functional Lemma and Classification of Formal Groups

The main fields of application of formal groups are algebraic geometry and class field theory. The later uses both the classical Hilbert symbol (the norm-residue symbol) and its generalization. One of the most important problems is finding explicit formulas for various modifications of this symbol related to formal groups. There are two approaches to constructing formal groups (i.e., power series satisfying certain conditions). The functional lemma proved by Hazewinkel allows one to make formal groups with coefficients from a ring of characteristic zero by means of functional equations using a certain ideal of this ring, an overfield of the ring, and a ring homomorphism with specified properties (e.g., identical; for a local field, the Frobenius homomorphism can be chosen). There is a convenient criterion for the isomorphism of formal groups, constructed by Hazewinkel’s formula, as well as a formula for logarithms (in particular, the Artin–Hasse logarithm). At the same time, Lubin and Tate construct formal groups over local fields, using isogeny, and Honda, when constructing formal groups over the ring of integers of a discrete valued field of characteristic zero, introduces a certain noncommutative ring induced by the original ring and a fixed homomorphism. The paper relates the classical classification of formal groups (standard, generalized, relative Lubin–Tate formal groups, and formal Honda groups) to their classification using the Hazewinkel functional lemma. For each type, the corresponding functional equations are composed and logarithms, as well as series used to construct an explicit formula for the Hilbert symbol, are studied.

Arithmetically rigid schemes via deformation theory of equivariant vector bundles

We analyze the deformation theory of equivariant vector bundles. In particular, we provide an effective criterion for verifying whether all infinitesimal deformations preserve the equivariant structure. As an application, using rigidity of the Frobenius homomorphism of general linear groups, we prove that projectivizations of Frobenius pullbacks of tautological vector bundles on Grassmanians are arithmetically rigid, that is, do not lift over rings where $$p \ne 0$$ . This gives the same conclusion for Totaro’s examples of Fano varieties violating Kodaira vanishing. We also provide an alternative purely geometric proof of non-liftability mod $$p^2$$ and to characteristic zero of the Frobenius homomorphism of a reductive group of non-exceptional type. In the appendix, written jointly with Piotr Achinger, we provide examples of non-liftable Calabi–Yau varieties in every characteristic $$p \geqslant 5$$ .

Computing endomorphism rings and Frobenius matrices of Drinfeld modules

Let Fq[T] be the polynomial ring over a finite field Fq. We study the endomorphism rings of Drinfeld Fq[T]-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.

Pushforwards of p-adic differential equations

Given a differential equation on a smooth $p$-adic analytic curve, one may construct a new one by pushing forward by an \'etale morphism. The main result of the paper provides an explicit formula that relates the radii of convergence of the solutions of the two differential equations using invariants coming from the topological behavior of the morphism. We recover as particular cases the known formulas for Frobenius morphisms and tame morphisms.

Immaculate line bundles on toric varieties

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.

Noise Free Fully Homomorphic Encryption Scheme Over Non-Associative Algebra

Among several approaches to privacy-preserving cryptographic schemes, we have concentrated on noise-free homomorphic encryption. It is a symmetric key encryption that supports homomorphic operations on encrypted data. We present a fully homomorphic encryption (FHE) scheme based on sedenion algebra over finite $Z_{n}$ rings. The innovation of the scheme is the compression of a 16-dimensional vector for the application of Frobenius automorphism. For sedenion, we have $p^{16}$ different possibilities that create a significant bijective mapping over the chosen 16-dimensional vector that adds permutation to our scheme. The security of this scheme is based on the assumption of the hardness of solving a multivariate quadratic equation system over finite $Z_{n}$ rings. The scheme results in $256n$ multivariate polynomial equations with $256+16n$ unknown variables for $n$ messages. For this reason, the proposed scheme serves as a security basis for potentially post-quantum cryptosystems. Moreover, after sedenion, no newly constructed algebra loses its properties. This scheme would therefore apply as a whole to the following algebras, such as 32-dimensional trigintadunion.

On Kauffman bracket skein modules of marked 3–manifolds and the Chebyshev–Frobenius homomorphism

In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary components. The addition of unmarked components allows us to develop a surgery theory which enables us to extend the Chebyshev homomorphism of Bonahon and Wong between skein algebras of unmarked surfaces to a "Chebyshev-Frobenius homomorphism" between skein modules of marked 3-manifolds. We show that the image of the Chebyshev-Frobenius homomorphism is either transparent or skew-transparent. In addition, we make use of the Muller algebra method to calculate the center of the skein algebra of a marked surface when the quantum parameter is not a root of unity.

On Congruence Relations and Equations of Shimura Curves

On a Shimura curve, the reduction modulo a prime $p$ of the Hecke correspondence $T(p)$ yields the congruence relation $\Pi\cup\Pi'$ with $\Pi$ being the graph of the Frobenius mapping from the Shimura curve modulo $p$ to itself, and $\Pi'$ its transpose. Starting with a curve $C$ of genus $g \geq 2$ over $\mathbb{F}_p$ together with a subset $\mathfrak{S}\subset C(\mathbb{F}_{p^2})$, Ihara studied the liftability to characteristic $0$ of $\Pi\cup\Pi'$ so that $\Pi$ and $\Pi'$ are separated outside $\mathfrak{S}$ in the lifting. In some case, Ihara obtained the uniqueness of the liftability to characteristic $0$ and gave some necessary and sufficient condition, described by some differential form on $C$, for $(C,\mathfrak{S})$ to be liftable to modulo $p^2$. In this paper, in case when $C$ is defined over $\mathbb{F}_{p^2}$, we compute complete tables of such $(C,{\mathfrak S})$ liftable to modulo $p^2$ for $g=2$ and $3\leq p \leq 13$ using computer, and as an application of this uniqueness, we identify some particular Shimura curve by its equation.

Pseudofinite difference fields

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 establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

Suzuki–Ree groups and Tits mixed groups over rings

It is shown that Suzuki–Ree groups can be easily defined by means of comparing two fundamental representations of the ambient Chevalley group in characteristic 2 or 3. This eliminates the distinction between the Suzuki–Ree groups over perfect and imperfect fields and gives a natural definition for the analogs of such groups over commutative rings. As an application of the same idea, we explicitly construct a pair of polynomial maps between the groups of types and in characteristic 2 that compose to the Frobenius endomorphism. This, in turn, provides a simple definition for the Tits mixed groups over rings.

On higher direct images of convergent isocrystals

Let $k$ be a perfect field of characteristic $p>0$ and let $\operatorname{W}$ be the ring of Witt vectors of $k$. In this article, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over $k$ relative to $\operatorname{W}$. This proof allows us to deduce an analogue of the de Rham complexes comparison theorem of Berthelot [$\mathscr{D}$-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000)] without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot’s conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of $k$-varieties.

Frobenius and homological dimensions of complexes

It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if $${\text {Tor}}_i^R({}^{e}\!R, M)=0$$ for $${\text {dim}}\,R$$ consecutive positive values of i and infinitely many e. Here $${}^{e}\!R$$ denotes the ring R viewed as an R-module via the eth iteration of the Frobenius endomorphism. In the case R is Cohen–Macualay, it suffices that the Tor vanishing above holds for a single $$e\geqslant \log _p e(R)$$, where e(R) is the multiplicity of the ring. This improves a result of Dailey et al. (J Commut Algebra), as well as generalizing a theorem due to Miller (Contemp Math 331:207–234, 2003) from finitely generated modules to arbitrary modules. We also show that if R is a complete intersection ring then the vanishing of $${\text {Tor}}_i^R({}^{e}\!R, M)$$ for single positive values of i and e is sufficient to imply M has finite flat dimension. This extends a result of Avramov and Miller (Math Res Lett 8(1–2):225–232, 2001).

Generating functions associated to Frobenius algebras

We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this generating function explicitly as a rational function.

Irreducible generalized numerical semigroups and uniqueness of the Frobenius element

Let $\mathbb{N}^{d}$ be the $d$-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid $ S\subseteq \mathbb{N}^d$ such that $H(S)=\mathbb{N}^d \setminus S$ is a finite set. We introduce irreducible generalized numerical semigroups and characterize them in terms of the cardinality of a special subset of $H(S)$. In particular, we describe relaxed monomial orders on $\mathbb N^d$, define the Frobenius element of $S$ with respect to a given relaxed monomial order, and show that the Frobenius element of $S$ is independent of the order if the generalized numerical semigroup is irreducible.

Splittings and Symbolic Powers of Square-free Monomial Ideals

Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

A unified viewpoint for upper bounds for the number of points of curves over finite fields via euclidean geometry and semi-definite symmetric Toeplitz matrices

We provide an infinite sequence of upper bounds for the number of rational points of absolutely irreducible smooth projective curves X X over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many n n -th order Weil bounds, and ending asymptotically at the Drinfeld-Vlăduţ bound. We relate this set of bounds to those of Oesterlé, proving that these are inverse functions in some sense. We explain how the Riemann hypothesis for the curve X X can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface X × X X\times X together with the general theory of symmetric Toeplitz matrices. We also give some interpretation for the defect of asymptotically exact towers. This is achieved by pushing further the classical Weil proof in terms of euclidean relationships between classes in the euclidean part F X \mathcal {F}_X of the numerical group Num ⁡ ( X × X ) \operatorname {Num}(X\times X) generated by classes of graphs of iterations of the Frobenius morphism. The noteworthy Toeplitz shape of their intersection matrix takes a central place by implying a very strong cyclic structure on F X \mathcal {F}_X .

Irreducibility of infinite dimensional Steinberg modules of reductive groups with Frobenius maps

Let G be a connected reductive group over an algebraic closure F¯q of a finite field Fq. In this paper it is proved that the infinite dimensional Steinberg module of kG defined by N. Xi in 2014 is irreducible when k is a field of positive characteristic and chark≠charFq. For certain special linear groups, we show that the Steinberg modules of the groups are not quasi-finite with respect to some natural quasi-finite sequences of the groups.

Galois group action and Jordan decomposition of characters of finite reductive groups with connected center

Let G be a connected reductive group with connected center defined over Fq, with Frobenius morphism F. Given an irreducible complex character χ of GF with its Jordan decomposition, and a Galois automorphism σ∈Gal(Q‾/Q), we give the Jordan decomposition of the image χσ of χ under the action of σ on its character values.

Periodic points of algebraic functions and Deuring’s class number formula

The exact set of periodic points in $\overline{\mathbb{Q}}$ of the algebraic function $\widehat{F}(z)=(-1\pm \sqrt{1-z^4})/z^2$ is shown to consist of the coordinates of certain solutions $(x,y)=(\pi, \xi)$ of the Fermat equation $x^4+y^4=1$ in ring class fields $\Omega_f$ over imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of odd conductor $f$, where $-d \equiv 1$ (mod $8$). This is shown to result from the fact that the $2$-adic function $F(z)=(-1+ \sqrt{1-z^4})/z^2$ is a lift of the Frobenius automorphism on the coordinates $\pi$ for which $|\pi|_2<1$, for any $d \equiv 7$ (mod $8$), when considered as elements of the maximal unramified extension $\textsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. This gives an interpretation of the case $p=2$ of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations $H_{-d}(x)$ is given that is applicable for small periods. The pre-periodic points of $\widehat{F}(z)$ in $\overline{\mathbb{Q}}$ are also determined.