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Jaap Top

Publications and source records attributed to Jaap Top.

At least 19 recordsLinked to original sources
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.

math.AG↗
arXiv2025

Explicit Families of Hyperelliptic Curves with CM Jacobians

We construct explicit families of hyperelliptic curves over $\QQ$ whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by \[ v^2 = (u+2)\,\varphi_d(u), \quad d = 2^e \text{ or } d=p \geq 3 \text{ prime}, \] where $\varphi_d(x)$ is the Chebyshev polynomial of degree $d$. We prove that the Jacobians are simple and determine the associated CM-fields explicitly. Our approach exploits the interplay between Chebyshev polynomials and Galois coverings, providing concrete examples of abelian varieties with CM and explicit criteria for their construction.

math.AG↗
arXiv2025

CM theory, maximal hyperelliptic curves, and Chebyshev polynomials

This paper studies hyperelliptic curves $\cH_d$ corresponding to $y^2=\varphi_d(x)$ over finite fields, with $\varphi_d(x)$ a Chebyshev polynomial. Starting from the case where $d=\ell$ is an odd prime number, new cases $(d,q)$ are presented where $\cH_d$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. In addition, new conditions ruling out the possibility that $\cH_d/\FF_{q^2}$ is maximal for given $(d,q)$, are presented. The arguments involve a mix of results on slopes of Frobenius, explicit descriptions of abelian subvarieties of the jacobian of $\cH_d$ with complex multiplication, and a technique from the theory of $2$-descent on jacobians of hyperelliptic curves. In particular, the method used here to prove maximality in characteristics $p\equiv 1\bmod 4$ for $d\equiv 1\bmod 4$ a prime number, deserves attention, as it differs from earlier maximality arguments for other curves. Using the new results as well as extensive calculations with Magma, we pose some questions. A positive answer would completely classify the pairs $(q,d)$ resulting in maximality.

math.NT↗
arXiv2025

Automorphisms of Plane Curves defined from Chebychev polynomials

We investigate the automorphism groups of the algebraic curves \[ \mathcal{C}_d : y^d = \varphi_d(x), \] where $\varphi_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\operatorname{char}(k) \nmid 2d$. We determine the full automorphism group of $\mathcal{C}_d$ in all the cases considered in this paper, namely for $d=4$, and more generally when $2d = p^r+1$ or $4d = p^r+1$. For all other $d>4$, Expectation~\ref{3.19} predicts what the automorphism group should be. As an application, we show that certain maximal curves of the same genus are not isomorphic.

math.AG↗
arXiv2024

Tate-Shafarevich results for quartic twists in characteristic $2$

The aim of this paper is to present elliptic curves defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study elliptic curves arising as quartic twist of a supersingular elliptic curve defined over $\mathbb{F}_2$ using the function field of a maximal curve $C$ that admits an order 4 automorphism. For such elliptic curves we provide a rank formula for its Mordell-Weil group in terms of the genera of $C$ and of another curve covered by $C$.

math.AG↗
arXiv2024

Isomonodromy and Painlev\'e Type Equations, Case Studies

There is an abundance of equations of Painlev\'e type besides the classical Painlev\'e equations. Classifications have been computed by the Japanese school. Here we consider Painlev\'e 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 Painlev\'e 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 Painlev\'e 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.

math.CA↗
arXiv2021

The last chapter of the Disquisitiones of Gauss

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.

math.HO↗
arXiv2021

Moduli Spaces for the Fifth Painlev\'e Equation

Isomonodromy for the fifth Painlev\'e equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlev\'e 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.

math.CA↗
arXiv2021

A remark on prime (non)congruent numbers

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.

math.NT↗
arXiv2021

Two-descent on some genus two curves

For the hyperelliptic curve C_p with equation y^2=x(x-2p)(x-p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we present some results on rational points of C_p.

math.NT↗
arXiv2021

A parametrized set of explicit elements of $\Sha(E/\Q)[3]$

In this paper, we give explicit equations for homogeneous spaces corresponding to a rational isogeny of degree $3$. An explicit set of elliptic curves with elements of order $3$ in their Tate-Shafarevich group is constructed. Combining this gives explicit examples of plane cubics over $\Q$ that have a point everywhere locally, but not globally.

math.NT↗
arXiv2019

Serre's genus 50 example

This note presents explicit equations (up to birational equivalence over $\mathbb{F}_2$) for a complete, smooth, absolutely irreducible curve $X$ over $\mathbb{F}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985 Harvard lecture notes on curves over finite fields, J-P.~Serre already showed the existence of such a curve: he used class field theory to describe the function field $\mathbb{F}_2(X)$ as a certain abelian extension of the function field $\mathbb{F}_2(E)$ of some elliptic curve $E/\mathbb{F}_2$. Although various more recent texts recall Serre's construction, explicit equations as well as a description of intermediate curves $X\to Y\to E$ over $\mathbb{F}_2$ seem to be new. We also describe explicit equations for a curve over $\mathbb{F}_2$ of genus $8$ with $11$ rational points, and for a curve over $\mathbb{F}_2$ of genus $22$ with $21$ rational points.

math.NT↗
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.

math.AG↗
arXiv2019

The level of pairs of polynomials

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator and the notion of `stratification' in the case of hyperelliptic curves. Next we extend the notion of level to that of a pair of polynomials. We prove some basic properties and we compute this level in certain special cases. In particular we present examples of polynomials $g$ and $f$ such that there is no differential operator raising $g/f$ to its $p$th power.

math.AC↗
arXiv2019

On certain maximal hyperelliptic curves related to Chebyshev polynomials

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2 = \varphi_{d}(x)$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. Here $\varphi_{d}(x)$ denotes the Chebyshev polynomial of degree $d$. The same question is studied for the curves corresponding to $y^2=(x\pm 2) \varphi_{d}(x)$, and also for $y^2=(x^2-4)\varphi_d(x)$.

math.AG↗
arXiv2017

Hesse Pencils and 3-Torsion Structures

This paper intends to focus on the universal property of this Hesse pencil and of its twists. The main goal is to do this as explicit and elementary as possible, and moreover to do it in such a way that it works in every characteristic different from three.

math.AG↗
arXiv2017

Twists of Elliptic Curves

In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to the remaining characteristics. For this, we make explicit use of the correspondence between the twists and the Galois cohomology set $H^1\big(\operatorname{G}_{\overline{K}/K}, \operatorname{Aut}_{\overline{K}}(E)\big)$. The results are illustrated by examples.

math.NT↗