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Marius van der Put

Publications and source records attributed to Marius van der Put.

At least 19 recordsLinked to original sources
arXiv2026

Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights

Mumford curves generalize the Tate uniformization of elliptic curves with split multiplicative reduction and provide p-adic analogues of the uniformization of Riemann surfaces. In this paper, we present several algorithms for hyperelliptic Mumford curves. For a given hyperelliptic Mumford curve $X$ defined over a finite extension of the field of p-adic numbers for some $p\neq 2$, we first describe how to compute a p-adic Schottky group W that uniformizes X; this is based on our extension to Kadziela's approximation theorem. As applications, we explain how to use this uniformization in order to compute p-adic Abelian integrals and $p$-adic Schneider heights on X; the latter uses Werner's formula expressing the p-part of the Schneider height in terms of theta functions. We illustrate our algorithms with numerical examples computed using the computer algebra system SageMath.

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arXiv2026

Whittaker groups and hyperelliptic curves

Let K be a complete, non-archimedean valued field with a residue field of characteristic different from 2. A Whittaker group G is a discontinuous subgroup of PGL(2,K), freely generated by elements s_0,...,s_g of order two, each defined by a pair of fixed points {a_0,b_0},...,{a_g,b_g}. These fixed points are called ``in good position''. A subgroup W in G of index 2 is a Schottky group and produces a hyperelliptic Mumford curve Omega/W --> Omega/G = P^1, called `Whittaker curve', of genus g and with branch locus B in P^1(K). An explicit parametrization of Whittaker curves in terms of theta functions for W and G and the data of the fixed points, is developed. In particular, this allows one to express the branched points (and other data such as p-adic periods and p-adic heights) in terms of values of theta functions. A central theme of this paper is the relation between the fixed points and the branch locus. For a given configuration (P,m) of $g+1$ pairs of points in P^1, one defines a rigid space Fix_{P,m} of fixed points in good position with that configuration and a rigid space of branched points $ Branch_{P,m} in that configuration. A main result is that the natural morphism FB: Fix_{P,m} --> Branch_{P,m} is a rigid etale covering with Galois group {\pm 1}^{d-1} for some d>0. For all cases of genus g=2,3 (and for some more), an approximation of FB is computed which confirms the main result. Classification of Whittaker groups and analytic reductions of Whittaker curves is another important issue of this paper. The background material in this paper complements the work of L.~Gerritzen, G.~Van Steen, F.~Herrlich and others. It involves re-examination of some proofs, the derivation of properties of semi-stable analytic reductions and studying good position of fixed points.

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arXiv2025

Isomonodromy and Painlevé Type Equations, Case Studies

There is an abundance of equations of Painlevé type besides the classical Painlevé equations. Classifications have been computed by the Japanese school. Here we consider Painlevé type equations induced by isomonodromic families of linear ODE's having at most ${z=0}$ and $z=\infty$ as singularities. Requiring that the formal data at the singularities produce isomonodromic families parametrized by a single variable $t$ leads to a small list of hierarchies of cases. The study of these cases involves Stokes matricesand moduli for linear ODE's on the projective line. Case studies reveal interesting families of linear ODE's and Painlevé type equations. However, rather often the complexity (especially of the Lax pair) is too high for either the computations or for the output. Apart from classical Painlevé equations one rediscovers work of Harnad, Noumi and Yamada. A hierarchy, probably new, related to the classical $P_3(D_8)$, is discovered. Finally, an amusing ''companion'' of $P_1$ is presented.

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arXiv2023

Moduli Spaces for the Fifth Painlevé Equation

Isomonodromy for the fifth Painlevé equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${\rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${\rm P}_5$, equivalent to the one of Okamoto.

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arXiv2019

Variations for Some Painlevé Equations

This paper first discusses irreducibility of a Painlevé equation $P$. We explain how the Painlevé property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian $\mathbb{H}$ to a Painlevé equation $P$. Complete integrability of $\mathbb{H}$ is shown to imply that all solutions to $P$ are classical (which includes algebraic), so in particular $P$ is solvable by ''quadratures''. Next, we show that the variational equation of $P$ at a given algebraic solution coincides with the normal variational equation of $\mathbb{H}$ at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases $P_{2}$ to $P_{5}$ where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the differential Galois group for the first two variational equations. As expected there are no cases where this group is commutative.

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arXiv2019

Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic

Stratifications and iterative differential equations are analogues in positive characteristic of complex linear differential equations. There are few explicit examples of stratifications. The main goal of this paper is to construct stratifications on projective or affine curves in positive characteristic and to determine the possibilities for their differential Galois groups. For the related "differential Abhyankar conjecture" we present partial answers, supplementing the literature. The tools for the construction of regular singular stratifications and the study of their differential Galois groups are $p$-adic methods and rigid analytic methods using Mumford curves and Mumford groups. These constructions produce many stratifications and differential Galois groups. In particular, some information on the tame fundamental groups of affine curves is obtained.

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arXiv2019

Autonomous first order differential equations

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on $D^n$-finiteness of solutions of first order differential equations.

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arXiv2018

Mumford curves and Mumford groups in positive characteristic

A Mumford group is a discontinuous subgroup $Γ$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $Γ$ is a curve of genus 0. As abstract group $Γ$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to our paper "Discontinuous subgroups of PGL(2,K)" published in Journ. of Alg. (2004). Part of it clarifies, corrects and extends work of G.~Cornelissen, F.~Kato and K.~Kontogeorgis.

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arXiv2017

Isomonodromy for the Degenerate Fifth Painlevé Equation

This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto-Painlevé space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlevé equation, for the Bäcklund transformations.

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arXiv2015

The Stokes Phenomenon and Some Applications

Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlevé equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations.

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arXiv2014

Stokes matrices for the quantum differential equations of some Fano varieties

The classical Stokes matrices for the quantum differential equation of projective n-space are computed, using multisummation and the so-called monodromy identity. Thus, we recover the results of D. Guzzetti that confirm Dubrovin's conjecture for projective spaces. The same method yields explicit formulas for the Stokes matrices of the quantum differential equations of smooth Fano hypersurfaces in projective n-space and for weighted projective spaces.

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arXiv2013

Real and p-adic Picard-Vessiot fields

We consider differential modules over real and p-adic differential fields such that their field of constants is real closed (respectively p-adically closed). Using Deligne's work on Tannakian categories and a result of Serre on Galois cohomology, a purely algebraic proof of the existence and unicity of real (respectively p-adic) Picard-Vessiot fields is obtained.

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arXiv2012

A Riemann--Hilbert approach to Painlevé IV

This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlevé equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto--Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on $\mathbb{P}^1$ are introduced, inspired by the symmetric form for ${\rm PIV}$ as was studied by M. Noumi and Y. Yamada.

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arXiv2012

Order one equations with the Painlevé property

Differential equations with the Painlevé property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order one equations appeared before, the connection with the classical theory did not. Using techniques from algebraic geometry we provide the link between the classical and the modern treatment, and with the help of differential Galois theory a new classification is derived, both for characteristic 0 and p.

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arXiv2010

Periodic behaviors

This paper studies behaviors that are defined on a torus, or equivalently, behaviors defined in spaces of periodic functions, and establishes their basic properties analogous to classical results of Malgrange, Palamodov, Oberst et al. for behaviors on R^n. These properties - in particular the Nullstellensatz describing the Willems closure - are closely related to integral and rational points on affine algebraic varieties.

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arXiv2009

Effective descent for differential operators

A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=M\otimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=M\otimes N$, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields.

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