Adic Differential Equations Research Articles

On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents

We consider a differential system x\frac{d}{dx} Y=GY , where G is a m\times m matrix whose coefficients are power series which converge and are bounded on the open unit disc D(0,1^-) . Assume that G(0) is a diagonal matrix with p -adic integer coefficients. Then there exists a solution matrix of the form Y=F \exp(G(0)\log x) at x=0 if all differences of exponents of the system are p -adically non-Liouville numbers. We give an example where F is analytic on the p -adic open unit disc and has log-growth greater than m . Under some conditions, we prove that if a solution matrix at a generic point has log-growth \delta , then F has log-growth \delta .

Logarithmic growth filtrations for -modules over the bounded Robba ring

In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$-adic differential equations $Dx=0$ on the $p$-adic open unit disc $|t|<1$, which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$. Then, Dwork calculated the log-growth filtration for $p$-adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$-modules over $K[\![t]\!]_0$, which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$-modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.

On Log-growth of Solutions of $p$-adic Differential Equations at a Logarithmic Singular Point

We consider a differential system $x\frac{d}{dx} Y=GY$ where $G(0)$ is a nilpotent matrix. Then there exists a solution matrix of the form $Y=F \exp(G(0)\log x)$. If a solution matrix at a generic point is of log-growth $\delta$, then we prove that $F$ is of log-growth $\delta$.

ON THE DENOMINATORS OF THE TAYLOR COEFFICIENTS OF <i>G</i>-FUNCTIONS

Let $\sum_{n=0}^\infty a_n z^n\in \overline{\mathbb Q}[[z]]$ be a $G$-function, and, for any $n\ge0$, let $\delta_n\ge 1$ denote the least integer such that $\delta_n a_0, \delta_n a_1, ..., \delta_n a_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ such that $\delta_n\le c^{n+1}$ for all $n\ge 0$. In practice, it is observed that $\delta_n$ always divides $D_{bn}^{s} C^{n+1}$ where $D_n=lcm\{1,2, ..., n\}$, $b, C$ are positive integers and $s\ge 0$ is an integer. We prove that this observation holds for any $G$-function provided the following conjecture is assumed: {\em Let $\mathbb{K}$ be a number field, and $L\in \mathbb{K}[z,\frac{d }{d z}]$ be a $G$-operator; then the generic radius of solvability $R_v(L)$ is equal to 1, for all finite places $v$ of $\mathbb{K}$ except a finite number.} The proof makes use of very precise estimates in the theory of $p$-adic differential equations, in particular the Christol-Dwork Theorem. Our result becomes unconditional when $L$ is a geometric differential operator, a special type of $G$-operators for which the conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts that any $G$-operator is of geometric type, hence it implies the above conjecture.

Des $\pi$-exponentielles I: vecteurs de Witt annulés par Frobenius et algorithme de (leur) rayon de convergence

Our object is the theory of “ \pi -exponentials” Pulita developed in his thesis, generalising Dwork's and Robba's exponentials and extending Matsuda's work: We start with an abstract algebra statement about the structure of the kernel of iterations of the Frobenius endomorphism on the ring of Witt vectors with coordinates in the ring of integers of an ultrametric extension of \mathbb Q_p . Provided sufficiently (ramified) roots of unity are available, it is, unexpectedly simply, a principal ideal with respect to an explicit generator essentially given by Pulita's \pi -exponential. This result is a consequence and a reformulation of core facts of Pulita's theory. It happened to be simpler to prove directly than reformulating Pulita's results. Its translation in terms of series is very elementary, and gives a criterion for solvabilty and integrality for p -adic exponential series of polynomials. We explain how to deduce an explicit formula of their radius of convergence, and even the function radius of convergence. We recover this way, in elementary terms, with a new proof, and important simplifications, an algorithm of Christol based similarly on Pulita's work. One concrete advantage is: one can easily prove rigorous complexity bounds about the implied algorithm from our explicit formula. We also add there and there refinements and observation, notably hinting some of the finer informations that can also given by the algorithm. One of the appendix produce a computation which gives finer estimates on the coefficients of these series. It should provide useful in proving complexity bounds for various computational use involving these series. It is not apparent yet in the present work, but should be in latter projected developments, the series under consideration are the base object for some exponential sums on finite fields via p -adic approach, namely via rigid cohomology with rank one coefficients. Convergence radius and coefficients estimates are involved studying the efficiency of computational implementations of these objects. The understanding of convergence radius of p -adic differential equations is a subject undergoing active developments, and we here provide a fine theoretical and computational study of the simplest of cases. This initiate a projected series of articles. We start here, with the case of the affine line as a base space, a Witt vectors paradigm. This provides an alternative purely algebraic approach of Pulita's theory; the richness of Witt vectors theory allow suppleness and efficiency in working with \pi -exponentials, which will prove efficient later in the series.

On $$p$$ -adic differential equations on semistable varieties

In this paper we prove a comparison theorem between the category of certain modules with integrable connection on the complement of a normal crossing divisor of the generic fiber of a proper semistable variety over a DVR and the category of certain log overconvergent isocrystals on the special fiber of the same open.

Riccati Differential Equation for Hypergeometric Differential Equation

In this paper, we study the solutions of Riccati differential equation corresponding to p -adic differential equations which are solvable on the generic disc. As an application, we consider the Grothendieck conjecture for Riccati differential equations. We see that the Riccati differential equations for some globally nilpotent differential equation with coefficients in \mathbb{Q}(t) have, for almost all prime, a solution in rational function field over the finite field \mathbb{F}_p , but do not have any algebraic solutions.

Book Review: $p$-adic differential equations

Book Review: $p$-adic differential equations

On ramification filtrations and p-adic differential equations, II: mixed characteristic case

AbstractLet K be a complete discrete valuation field of mixed characteristic (0,p), with possibly imperfect residue field. We prove a Hasse–Arf theorem for the arithmetic ramification filtrations on GK, except possibly in the absolutely unramified and non-logarithmic case, or the p=2 and logarithmic case. As an application, we obtain a Hasse–Arf theorem for filtrations on finite flat group schemes over 𝒪K.

A note on non-Robba $p$-adic differential equations

Let $\mathcal{M}$ be a differential module, whose coefficients are analytic elements on an open annulus $I$ ($\subset \mathbf{R}_{>0}$) in a valued field, complete and algebraically closed of inequal characteristic, and let $R(\mathcal{M}, r)$ be the radius of convergence of its solutions in the neighborhood of the generic point $t_{r}$ of absolute value $r$, with $r\in I$. Assume that $R(\mathcal{M}, r)<r$ on $I$ and, in the logarithmic coordinates, the function $r\longrightarrow R(\mathcal{M}, r)$ has only one slope on $I$. In this paper, we prove that for any $r\in I$, all the solutions of $\mathcal{M}$ in the neighborhood of $t_{r}$ are analytic and bounded in the disk $D(t_{r},R(\mathcal{M},r)^{-})$.

P-adic differential equations and overconvergent isocrystals

<I>p</I>-adic differential equations and overconvergent isocrystals

Logarithmic growth and Frobenius filtrations for solutions ofp-adic differential equations

AbstractFor a ∇-moduleMover the ringK[[x]]0of bounded functions over ap-adic local fieldKwe define the notion of special and generic log-growth filtrations on the base of the power series development of the solutions and horizontal sections. Moreover, ifMalso admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case ofMa ϕ–∇-module of rank 2, the Frobenius polygon forMand the log-growth polygon for its dual,Mv, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork.

Dwork’s conjecture on the logarithmic growth of solutions of p-adic differential equations

AbstractThis is a study of the asymptotic behaviour of solutions of p-adic linear differential equations near the boundary of their convergence disks. We prove Dwork’s conjecture on the logarithmic growth of solutions in generic versus special disks.

Fourier transforms and $p$-adic ‘Weil II’

We give a purity theorem in the manner of Deligne's ‘Weil II’ theorem for rigid cohomology with coefficients in an overconvergent $F$ -isocrystal; the proof mostly follows Laumon's Fourier-theoretic approach, transposed into the setting of arithmetic $\mathcal{D}$ -modules. This yields in particular a complete, purely $p$ -adic proof of the Weil conjectures when combined with recent results on $p$ -adic differential equations by Andre, Christol, Crew, Kedlaya, Matsuda, Mebkhout and Tsuzuki.

La théorie des équations différentielles $p$-adiques et le Théorème de la monodromie $p$-adique

In this lecture we introduce the reader to the proof of the p -adic monodromy theorem linking the p -adic differential equations theory and the local Galois p -adic representations theory.

On $p$-adic differential equations. The Frobenius structure of differential equations

On $p$-adic differential equations. The Frobenius structure of differential equations

A note on logarithmic growth of solutions of $p$-adic differential equations without solvability

For a $p$-adic differential equation solvable in an open disc (in a $p$-adic sense), around 1970, Dwork proves that the solutions satisfy a certain growth condition on the boundary. Dwork also conjectures that a similar phenomenon should be observed without assuming the solvability. In this paper, we verify Dwork's conjecture in the rank two case, which is the first non-trivial result on the conjecture. The proof is an application of Kedlaya's decomposition theorem of $p$-adic differential equations defined over annulus.