P-adic Completion Research Articles

Lie invariant Frobenius lifts

We begin with the observation that the p-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts ϕ that are “Lie invariant mod p” in the sense that the “normalized” action of ϕ on 1-forms preserves mod p the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be “infinitesimally deformed” in the sense that the above mod p result has a mod p2 analogue. We end by showing that, in contrast with the case of elliptic curves, the following holds: if G is a linear algebraic group over a number field and if G is not a torus then for all but finitely many primes p the p-adic completion of G does not possess a Frobenius lift that is Lie invariant mod p.

The Picard group of vertex affinoids in the first Drinfeld covering

AbstractLetFbe a finite extension of${\mathbb Q}_p$. Let$\Omega$be the Drinfeld upper half plane, and$\Sigma^1$the first Drinfeld covering of$\Omega$. We study the affinoid open subset$\Sigma^1_v$of$\Sigma^1$above a vertex of the Bruhat–Tits tree for$\text{GL}_2(F)$. Our main result is that$\text{Pic}\!\left(\Sigma^1_v\right)[p] = 0$, which we establish by showing that$\text{Pic}({\mathbf Y})[p] = 0$for${\mathbf Y}$the Deligne–Lusztig variety of$\text{SL}_2\!\left({\mathbb F}_q\right)$. One formal consequence is a description of the representation$H^1_{{\acute{\text{e}}\text{t}}}\!\left(\Sigma^1_v, {\mathbb Z}_p(1)\right)$of$\text{GL}_2(\mathcal{O}_F)$as thep-adic completion of$\mathcal{O}\!\left(\Sigma^1_v\right)^\times$.

P-Adic metric preserving functions and their analogues

Abstract The p-adic completion ℚ p of the rational numbers induces a different absolute value |⋅| p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ+ → ℝ+, for which the composition with the p-adic metric dp generated by |⋅| p is still a metric on ℚ p , with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ d p is still an ultrametric on ℚ p if and only if there is a function g such that f ∘ d p = g ∘ d p and g ∘ d is still an ultrametric for every ultrametric d. Some general variants of the last statement are also proved.

One application on hypergeometic series and values of g-adic functions algebraic independence investigation methods

The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔 ∼=Ω𝑝1⊕...⊕Ω𝑝𝑛, a subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Also, hypergeometric series $$𝑓(𝑧) =∞Σ𝑗=0((𝛾1)𝑗 . . . (𝛾𝑟)𝑗)/((𝛽1)𝑗 . . . (𝛽𝑠)𝑗)(𝑧𝑡)^𝑡𝑗 $$, and their formal derivatives are under consideration. Sufficient conditions are obtained under which the values of the series 𝑓(𝛼) and formal derivatives satisfy global relation of algebraic independence, if 𝛼 =∞Σ𝑗=0 𝑎_𝑗𝑔^(𝑟_𝑗), where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded.

Semistable models for modular curves and power operations for Morava E-theories of height 2

We construct an integral model for Lubin–Tate curves. These curves arise as moduli of finite subgroups of deformations of formal groups. In particular, they are p-adic completions of the modular curves X0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer–Lashof algebras for Morava E-theories of height 2. These algebras are local moduli of power operations in elliptic cohomology.

Arithmetic differential operators on a semistable model of $${\mathbb {P}}^1$$ P 1

In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one.

Arithmetic differential equations on $$GL_n$$ G L n , II: arithmetic Lie–Cartan theory

Motivated by the search of a concept of linearity in the theory of arithmetic differential equations (Buium in Arithmetic differential equations. Math. surveys and monographs, vol 118. American Mathematical Society, Providence, 2005), we introduce here an arithmetic analogue of Lie algebras, of Chern connections, and of Maurer–Cartan connections. Our arithmetic analogues of Chern connections are certain remarkable lifts of Frobenius on the p-adic completion of \(GL_n\) which are uniquely determined by certain compatibilities with the “outer” involutions defined by symmetric (respectively, antisymmetric) matrices. The Christoffel symbols of our arithmetic Chern connections will involve a matrix analogue of the Legendre symbol. The analogues of Maurer–Cartan connections can then be viewed as families of “linear” flows attached to each of our Chern connections. We will also investigate the compatibility of lifts of Frobenius with the inner automorphisms of \(GL_n\); in particular, we will prove the existence and uniqueness of certain arithmetic analogues of “isospectral flows” on the space of matrices.

Some subgroups of the general linear group of order two over the ring of p-adic integers

Let p be a prime number, ℤ p the ring of p-adic integers and ℚ p its field of fractions. In this paper we are mainly concerned with the profinite structure of the general linear group GL(2, ℤ p ) and some of its subgroups. In particular, we study the subgroup SL(2, ℤ p ), the special linear group with p-adic integer entries. We determine the p-adic completion of the general linear group GL(2, ℤ)with integer entries. In contrastwith the padic completion of SL(2, ℤ) which is equal to SL(2, ℤ p ) that of GL(2, ℤ) is far to be equal to GL(2, ℤ p ). It is the semi-direct product {−1, 1} ⋉ SL(2, ℤ p ) and is a normal subgroup of GL(2, ℤ p ). It is well known that the special linear group SL(2, ℤ) is generated by two transvections. Hence SL(2, ℤ p ) is topological generated by these transvections. Moreover since ℤ p is a local ring, one can apply O. Litoff’s theorem to get short decomposition of any element of SL(2, ℤ p ). We revisit here the algebraic and profinite structures of these groups. The explicit description of these structures will be applied elsewhere to their continuous linear representations in ultrametric Banach spaces. In particular from the profinite structure of the general linear group as well as the special linear group, one can state when there exists or not a Haar measure on these groups with values in a given complete valued field. As already said, the special linear group SL(2, ℤ p ) has two topological generators, we shall prove that the general linear group GL(2, ℤ p ) admits four topological generators. As a first consequence we deduce here that any subgroup of these groups that is of finite index is open. On the other hand, the fact that these groups are finitely generated profinite groups will be applied elsewhere to their ultrametric continuous linear representations.

A FEW COMMENTS ON MATLIS DUALITY

For a Noetherian local ring (R, m) with p ∈ Spec(R), we denote the R-injective hull of R/p by ER(R/p). We show that it has an Rˆp -module structure, and there is an isomorphism ER(R/p) ∼= ERˆp (Rˆp/pRˆp ), where Rˆp stands for the p-adic completion of R. Moreover, for a complete Cohen-Macaulay ring R, the module D(ER(R/p)) is isomorphic to Rˆp provided that dim(R/p) = 1, where D(·) denotes the Matlis dual functor HomR(·, ER(R/m)). Here, Rˆp denotes the completion of Rp with respect to the maximal ideal pRp. These results extend those of Matlis (see [11]) shown in the case of the maximal ideal m.

On the p-adic completion of the units of a real abelian number field

TextLet E be the units of a real abelian number field k. We investigate whether E⊗Zp is a cyclic Zp[G]-module. If p∤#G, then E⊗Zp is cyclic. If p|#G and G is cyclic, then the cyclicality of E⊗Zp depends on the capitulation of ideals through the p-part of the extension k/Q among other things. There are cases when p|#G and we can unequivocally say that E⊗Zp is cyclic, for example, if #G=p. This has application to the annihilation of ideal classes, specifically, we get results analogous to Thaine [6, Theorem 3] but applicable in some non-semi-simple cases. VideoFor a video summary of this paper, please click here or visit http://youtu.be/-JwNNOxzwDI.

On the syntomic regulator for products of elliptic curves

We consider the syntomic regulator on the integral motivic cohomology of a smooth proper surface over a p-adic field and apply a recent formula of Besser that uses p-adic integration theory, in particular his theory of triple indices on Coleman integrals, to the case of a self-product of an elliptic curve. The method is suitable to separate decomposable from indecomposable elements in the (integral) motivic cohomology. As an interesting example, we construct an element that, though not given in decomposable form, becomes decomposable after taking p-adic completion.

Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms

Let Goo be a semisimple real Lie group with unitary dual Goo- We produce new upper bounds for the multiplicities with which representations ^ e of cohomological type appear in certain spaces of cusp forms on Goo* The main new idea is to apply noncommutative Iwasawa theory to certain p-adic completions of the cohomology of locally symmetric spaces.

Arithmetic Algebraic Geometry

The workshop covered developments in the field in the last four years. Roughly speaking arithmetic geometry consider algebraic schemes over rings of integers of numberfields. However an important tool is to first extend the base to a p-adic completion. Although both global and local problems matter this time there was a heavy emphasis on p-adic topics. One of them is the deformation-theory of Galois-representations, leading to a proof of Serre's conjecture. here one starts with a global Galois-representation modulo p, then lifts modulo p^2 , etc. For the lifts one requires certain local conditions (like being unramified outside a given set of places), and the most important and difficult such conditions arise at primes dividing p . Here the most important tool is J.M.Fontaine's theory which relates Galois-representations to filtered Frobenius-crystals. Another spectacular progess is the proof (by Ngo) of the fundamental lemma in the theory of automorphic representations. It postulates identities of p-adicorbital integrals and is reduced to a geometric statement about perverse sheaves on Hitchin-fibrations in positive characteristic. Concerning p-adic cohomology theories we are getting closer to a p-adic theory of D-modules, and of overconvergent crystals, over singular schemes. Also the long awaited etale coverings of p-adic period domains have finally been constructed, after it has been understood that they have "holes" which are visible in the Berkovich-space but not in the conventional rigid space. That they exist is suggested by Fontaine's theory. These period domains classify p-divisible groups. Some of them (Drinfeld, Lubin-Tate) can be covered by explicit affinoid domains, thus giving some type of reduction theory for p-divisible groups. There are attempts to extend this to finite flat group-schemes. Concerning K-theory the classical Borel-regulator from K-theory to Deligne-cohomology has been extended to syntomic cohomology, as well as the computation of its values on Eisenstein-symbols. For the l-adic etale theory general finiteness theorems can now be shown for quasi-excellent schemes. A further topic was the theory of p-adic Banach-representations of p-adic Lie-groups. On a more global level we had talks about (p-adic!) constructions of rational points on elliptic curves, the association of K-classes to abelian varieties, and the theory of tame fundamental groups. Finally the theory of small points has been extended to function fields (over numberfields it leads to equidistribution) using tropical geometry.

The p-adic quantum plane algebras and quantum Weyl algebra

Let q be a principal unit of the ring of valuation of a complete valued field K, extension of the field of p-adic numbers. Generalizing Mahler basis, K. Conrad has constructed orthonormal basis, depending on q, of the space of continuous functions on the ring of p-adic integers with values in K. Attached to q there are two models of the quantum plane and a model of the quantum Weyl algebra, as algebras of bounded linear operators on the space of p-adic continuous functions. For q not a root of unit, interesting orthonormal (orthogonal) families of these algebras are exhibited and providing p-adic completion of quantum plane and quantum Weyl algebras.

Polynomial Maps over p -Adics and Residual Properties of Mapping Tori of Group Endomorphisms

We continue our study of residual properties of mapping tori of free group endomorphisms. In this paper, we prove that each of these groups are virtually residually (finite p)-groups for all but finitely many primes p. The method involves further studies of polynomial maps over finite fields and p-adic completions of number fields.

A Segal conjecture for p-completed classifying spaces

We formulate and prove a new variant of the Segal conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic completion of the Grothendieck group A p ( G , K ) of finite principal ( G , K ) -bundles whose isotropy groups are p-groups. Collecting the result for different primes p, we get a new and simple description of the group of homotopy classes of stable maps between (uncompleted) classifying spaces of groups. This description allows us to determine the kernel of the map from the Grothendieck group A ( G , K ) of finite principal ( G , K ) -bundles to the group of homotopy classes of stable maps from BG to BK.

Almost fully decomposable infinite rank lattices over orders

Let Λ be an order over a Dedekind domain R with quotient field K. An object of Λ-Lat, the category of R-projective Λ-modules, is said to be fully decomposable if it admits a decomposition into (finitely generated) Λ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105–128], we give a necessary and sufficient criterion for R-orders Λ in a separable K algebra A with the property that every L∈Λ-Lat is fully decomposable. In the present paper, we assume that A/RadA is separable, but that the p-adic completion Ap is not semisimple for at least one p∈SpecR. We show that there exists an L∈Λ-Lat, such that KL admits a decomposition KL=M0⊕M1 with M0∈A-mod finitely generated, where L∩M1 is fully decomposable, but L itself is not fully decomposable.

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.

Finitude de la dimension homologique d'algèbres d'opérateurs différentiels faiblement complètes et à coefficients surconvergents

We prove that the sheaf of arithmetic differential operators with overconvergent coefficients, introduced by P. Berthelot, has finite cohomological dimension. A similar geometrical proof shows that the weak p-adic completion of the Weyl algebra has also finite cohomological dimension. Moreover, this algebra can be naturally endowed with a filtration which is compatible with the Frobenius.

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups