Frobenius Action Research Articles

Frobenius actions on Del Pezzo surfaces of degree 2

Frobenius actions on Del Pezzo surfaces of degree 2

Comparison of prismatic cohomology and derived de Rham cohomology

We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when X is a proper smooth formal scheme over \mathcal{O}_K with K being a p -adic field, we improve Breuil–Caruso’s theory on comparison between torsion crystalline cohomology and torsion étale cohomology.

Automorphisms of the supersingular isogeny graph

We provide a condition for which the supersingular l-isogeny graph in characteristic p has only one nontrivial automorphism, given by the action of Frobenius. For a fixed p, our condition is known to hold for a density 1 set of primes l.

Rees algebras and generalized depth‐like conditions in prime characteristic

AbstractIn this paper, we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen–Macaulay singularities. This is achieved by introducing a depth‐like invariant which captures as special cases Lyubeznik's F‐depth and the generalized F‐depth from Maddox–Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F‐nilpotence deforms.

Geometric construction of Heisenberg–Weil representations for finite unitary groups and Howe correspondences

We give a geometric construction of the Heisenberg–Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for (textrm{Sp}_{2n},textrm{O}_2^-) over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.

Keel's base point free theorem and quotients in mixed characteristic

We develop techniques of mimicking the Frobenius action in the study of universal homeomorphisms in mixed characteristic. As a consequence, we show a mixed characteristic Keel's base point free theorem obtaining applications towards the mixed characteristic Minimal Model Program, we generalise Kollár's theorem on the existence of quotients by finite equivalence relations to mixed characteristic, and we provide a new proof of the existence of quotients by affine group schemes.

Holomorphic Frobenius Actions for DQ-Modules

Given a complex manifold endowed with a \mathbb{C}^\times -action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the \mathbb{C}^\times -action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension-three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold.

Integral models for spaces via the higher Frobenius

We give a fully faithful integral model for simply connected finite complexes in terms ofE∞\mathbb {E}_{\infty }-ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory ofpp-completeE∞\mathbb {E}_{\infty }-rings for each primepp. Using this, we show that the data of a simply connected finite complexXXis the data of its Spanier-Whitehead dual, as anE∞\mathbb {E}_{\infty }-ring, together with a trivialization of the Frobenius action after completion at each prime.In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’sQQ-construction acts on the∞\infty-category ofE∞\mathbb {E}_{\infty }-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraicKK-theory which we callpartialKK-theory. We develop the notion of partialKK-theory and give a computation of the partialKK-theory ofFp\mathbb {F}_pup topp-completion.

A toy model for the Drinfeld–Lafforgue shtuka construction

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas à la Drinfeld and of excursion operators à la V. Lafforgue. We take as the point of departure the Hecke action of Rep(Gˇ) on the category Shv(BunG) of sheaves on BunG, and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

LOGARITHMIC DE RHAM–WITT COMPLEXES VIA THE DÉCALAGE OPERATOR

AbstractWe provide a new formalism of de Rham–Witt complexes in the logarithmic setting. This construction generalises a result of Bhatt–Lurie–Mathew and agrees with those of Hyodo–Kato and Matsuue for log-smooth schemes of log-Cartier type. We then use our construction to study the monodromy action and slopes of Frobenius on log crystalline cohomology.

Hasse–Witt matrices for polynomials, and applications

In a classical paper, Manin gives a congruence [15, Theorem 1] for the characteristic polynomial of the action of Frobenius on the Jacobian of a curve C , defined over the finite field \mathbf{F}_{q} , q=p^m , in terms of its Hasse–Witt matrix. The aim of this article is to prove a congruence similar to Manin’s one, valid for any L -function L(f,T) associated to the exponential sums over affine space attached to an additive character of \mathbf{F}_q , and a polynomial f . In order to do this, we define a Hasse–Witt matrix \mathrm{HW}(f) , which depends on the characteristic p , the set D of exponents of f , and its coefficients. We also give some applications to the study of the Newton polygons of Artin–Schreier (hyperelliptic when p=2 ) curves, and zeta functions of varieties.

F-Stable Secondary Representations and Deformation of F-Injectivity

We prove that deformation of F-injectivity holds for local rings \((R,\mathfrak {m})\) that admit secondary representations of \({H}_{\mathfrak {m}}^{i}(R)\) which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when \((R,\mathfrak {m})\) is sequentially Cohen–Macaulay (or more generally when all the local cohomology modules \({H}_{\mathfrak {m}}^{i}(R)\) have no embedded attached primes). We obtain some additional cases if \(R/\mathfrak {m}\) is perfect or if R is \(\mathbb {N}\)-graded.

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.

ON THE DISTRIBUTION OF ORDERS OF FROBENIUS ACTION ON ^-TORSION OF ABELIAN SURFACES

The computation of the order of Frobenius action on the $\ell$-torsion is a part of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve $E$ over a finite field $\mathbb{F}_q$. The idea of Schoof's algorithm is to compute the trace of Frobenius $t$ modulo primes $\ell$ and restore it by the Chinese remainder theorem. Atkin's improvement consists of computing the order $r$ of the Frobenius action on $E[\ell]$ and of restricting the number $t \pmod{\ell}$ to enumerate by using the formula $t^2 \equiv q (\zeta + \zeta^{-1})^2 \pmod{\ell}$. Here $\zeta$ is a primitive $r$-th root of unity. In this paper, we generalize Atkin's formula to the general case of abelian variety of dimension $g$. Classically, finding of the order $r$ involves expensive computation of modular polynomials. We study the distribution of the Frobenius orders in case of abelian surfaces and $q \equiv 1 \pmod{\ell}$ in order to replace these expensive computations by probabilistic algorithms.

Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

The study of Frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure. This paper establishes new connections by developing the notion of relative Frobenius actions on local cohomology. As an application, we show that a ring has F-nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its Frobenius closure.

F-sets and finite automata

It is observed that Derksen’s Skolem–Mahler–Lech theorem is a special case of the isotrivial positive characteristic Mordell-Lang theorem due to the second author and Scanlon. This motivates an extension of the classical notion of a k-automatic subset of the natural numbers to that of an F-automatic subset of a finitely generated abelian group Γ equipped with an endomorphism F. Applied to the Mordell–Lang context, where F is the Frobenius action on a commutative algebraic group G over a finite field, and Γ is a finitely generated F-invariant subgroup of G, it is shown that the “F-subsets” of Γ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X⊆G is a closed subvariety then X∩Γ is F-automatic. Derksen’s notion of a k-normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that F-subsets are F-normal. In particular, the X∩Γ appearing in the Mordell-Lang problem are F-normal. This generalises Derksen’s Skolem–Mahler–Lech theorem to the Mordell–Lang context.

On the uniform bound of Frobenius test exponents

In this paper we prove the existence of a uniform bound for Frobenius test exponents for parameter ideals of a local ring (R,m) of prime characteristic in the following cases:(1)R is generalized Cohen–Macaulay. Our proof is much simpler than the original proof of Huneke, Katzman, Sharp and Yao;(2)The Frobenius actions on all lower local cohomologies Hmi(R), i<dim⁡R, are nilpotent.

Integral models of Shimura varieties with parahoric level structure

For a prime $p > 2$ , we construct integral models over $p$ for Shimura varieties with parahoric level structure, attached to Shimura data $(G,X)$ of abelian type, such that $G$ splits over a tamely ramified extension of ${\mathbf {Q}}_{\,p}$ . The local structure of these integral models is related to certain “local models”, which are defined group theoretically. Under some additional assumptions, we show that these integral models satisfy a conjecture of Kottwitz which gives an explicit description for the trace of Frobenius action on their sheaf of nearby cycles.

Length of Local Cohomology in Positive Characteristic and Ordinarity

Abstract Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ≥ 3 variables with coefficients in a perfect field of characteristic p. We compute the D-module length of the 1st local cohomology module ${H^{1}_{f}}(R)$ with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero D-module whose length is expected to equal that above, for ordinary primes.

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison

We present a unified approach to the study of Hilbert-Kunz multiplicity, F-signature, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F-signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring.