Irreducible Jacobian derivations in positive characteristic

We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an (n-1)-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.     

Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection D of k-derivations on A, that we interpret as algebraic vector fields on , we study the group spanned by the hypersurfaces V(f) of X invariant under D modulo the rational first integrals of D. We prove that this group is always a finite dimensional F_p-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between A^p and A, we show that the kernel of the pull-back morphism is a finite F_p-vector space. In particular, if A is a UFD, then the Picard group of B is finite.     

Maximal wild monodromy in unequal characteristic

Let R be a complete discrete valuation ring of mixed characteristic $(0,p)$ with fraction field K. We study stable models of p-cyclic covers of ${\mathbb{P}}_K^1$. First, we determine the monodromy extension, the monodromy group, its filtration and the Swan conductor for special covers of arbitrarily high genus with potential good reduction. In the case $p=2$ we consider hyperelliptic curves of genus 2.     

Krichever formal groups

On the basis of the general Weierstrass model of the cubic curve with parameters μ = (μ₁, μ₂, μ₃, μ₄, μ₆), the explicit form of the formal group that corresponds to the Tate uniformization of this curve is described. This formal group is called the general elliptic formal group. The differential equation for its exponential is introduced and studied. As a consequence, results on the elliptic Hirzebruch genus with values in ℤ[μ] are obtained.     

Commutative formal groups

The formulations of several results on the structure of commutative formal groups of any dimensional over rings without -torsion are given, and also a geometric construction of a class of formal groups arising from algebraic manifolds is given. Proofs are only outlined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 117–124, 1976.     

Displays and formal p-divisible groups

Over p-adic Nagata rings, formal p-divisible groups are classified by nilpotent displays according to T. Zink. We extend this result to arbitrary p-adic rings. The proof uses the Grothendieck–Illusie deformation theory of truncated p-divisible groups.     

Convex polytopes and formal groups

Some constructions of commutative formal groups proceeding from convex polytopes and Laurent polynomials are studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 87–90, 1978.     

Scalar restriction for formal groups

The aim of this work is to generalize the results on the scalar restriction functor for formal groups stated in the paper “Explicit classification of formal groups over local fields” by the first two authors. These results are extended to formal groups over an arbitrary commutative ring, which we assume to be an integral domain or a ring of characteristic zero if necessary. Bibliography: 6 titles. Deceased. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 59–70.     

Integrable orbit equivalence rigidity for free groups

It is shown that every accessible group which is integrable orbit equivalent to a free group is virtually free. Moreover, we also show that any integrable orbit-equivalence between finitely generated groups extends to their end compactifications.     

Integrable magnetic geodesic flows on lie groups

On Lie group manifolds, we consider right-invariant magnetic geodesic flows associated with 2-cocycles of the corresponding Lie algebras. We investigate the algebra of the integrals of motion of magnetic geodesic flows and also formulate a necessary and sufficient condition for their integrability in quadratures, giving the canonical forms of 2-cocycles for all four-dimensional Lie algebras and selecting integrable cases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 189–206, August, 2008.     

Unbounded derivations tangential to compact groups of automorphisms

We consider unbounded derivations in C¹-algebras commuting with compact groups of ¹-automorphisms. A closed ¹-derivation δ in a C¹-algebra $\text{U}$ is said to be a generator if there exists a strongly continuous one-parameter subgroup t∈$\text{R}$→τ(t)? Aut($\text{U}$) such that $\text{δ =}\text{d}\text{dt}\text{τ(t)¦}{}_{\mathrm{t\; =\; 0}}$. If δ is known to commute with a compact abelian action α:G→Aut($\text{U}$), and if δ(a) = 0 for all a in the fixed point algebra $\text{U}$^α of the action G, then we show that δ is necessarily a generator. Moreover, in any faithful G-covariant representation, there is a commutative operator field γ ∈ ? → v(γ) such that $\text{v(γ)}{}^1\text{= ?v(γ), v(γ)}$ is possibly unbounded but affiliated with the center of {$\text{U}$^α}″, and e^tδ(x) = xe^tv(γ) for all x in the Arveson spectral subspace $\text{U}$^α(γ). In particular, if $\text{U}$ is the CAR algebra over an infinite-dimensional Hilbert space and α is the gauge group, then any such derivation δ is a scalar multiple of the generator of the gauge group.     

Integer valued polynomials and Lubin-Tate formal groups

If R is an integral domain and K is its field of fractions, we let Int(R) stand for the subring of K[x] which maps R into itself. We show that if R is the ring of integers of a p-adic field, then Int(R) is generated, as an R-algebra, by the coefficients of the endomorphisms of any Lubin-Tate group attached to R.     

Curves and formal (Co) groups

We prove and extend the results of [7], thereby obtaining a generalization of [4], [5] to the case of noncommutative formal (co)groups.     

Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Let be a differentiably simple Noetherian commutative ring of characteristic (then is local with ). A short proof is given of the Theorem of Harper (1961) on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there exists a nilpotent simple derivation of the ring such that if , then for some . The derivation is given explicitly, and it is unique up to the action of the group of ring automorphisms of . Let be the set of all such derivations. Then . The proof is based on existence and uniqueness of an iterative -descent (for each ), i.e., a sequence in such that , and for all . For each , and .