Hyperelliptic Case Research Articles

Explicit construction of a plane sextic model for genus-five Howe curves, II

A Howe curve is defined as the normalization of the fiber product over a projective line of two hyperelliptic curves. Howe curves are very useful to produce important classes of curves over fields of positive characteristic, e.g., maximal, superspecial, or supersingular ones. Determining their feasible equations explicitly is a basic problem, and it has been solved in the hyperelliptic case and in the non-hyperelliptic case with genus not greater than 4. In this paper, we construct an explicit plane sextic model for non-hyperelliptic Howe curves of genus 5. We also determine the number and type of singularities on our sextic model, and prove that the singularities are generically 4 double points. Our results together with Moriya-Kudo's recent ones imply that for each s∈{2,3,4,5}, there exists a non-hyperelliptic curve H of genus 5 with Aut(H)⊃V4 such that its associated plane sextic has s double points.

The cyclicity of a class of quadratic reversible systems with hyperelliptic curves

This paper is devoted to the cyclicity of a class of quadratic reversible systems with hyperelliptic level curves and two period annuli. By a new Chebyshev criterion which is an extension of the method proposed by Liu and Xiao (2020) [20], we prove that the cyclicity of every period annulus is two and show at the same time the configuration of the limit cycles. This paper is, to the best of our knowledge, the first work on the cyclicity of quadratic reversible systems for the hyperelliptic case.

Efficient Search for Superspecial Hyperelliptic Curves of Genus Four with Automorphism Group Containing C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{C}}_6$$\\end{document}

In arithmetic and algebraic geometry, superspecial (s.sp. for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If g≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g \\ge 4$$\\end{document}, it is not even known whether there exists such a curve of genus g in general characteristic p>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p > 0$$\\end{document}, and in the case of g=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=4$$\\end{document}, several computational approaches to search for those curves have been proposed. In the genus-4 hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp. curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group as a subgroup. In this paper, we propose an algorithm with complexity O~(p4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{O}}(p^4)$$\\end{document} in theory but O~(p3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{O}}(p^3)$$\\end{document} in practice to enumerate s.sp. hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order 6. By executing the algorithm over Magma, we enumerate those curves for p up to 1000. We also succeeded in finding a s.sp. hyperelliptic curve of genus 4 in every p with p≡2(mod3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\equiv 2 \\pmod {3}$$\\end{document}.

Hyperelliptic Functions and Motion in General Relativity

Analysis of black hole spacetimes requires study of the motion of particles and light in these spacetimes. Here exact solutions of the geodesic equations are the means of choice. Numerous interesting black hole spacetimes have been analyzed in terms of elliptic functions. However, the presence of a cosmological constant, higher dimensions or alternative gravity theories often necessitate an analysis in terms of hyperelliptic functions. Here we review the method and current status for solving the geodesic equations for the general hyperelliptic case, illustrating it with a set of examples of genus g=2: higher dimensional Schwarzschild black holes, rotating dyonic U(1)2 black holes, and black rings.

New generalisation of Jacobi’s derivative formula

A stream of new theta relations is obtained. They follow from the general Thomae formula, which is a new result giving expressions for theta derivatives (the zero values of the lowest non-vanishing derivatives of theta functions with singular half-period characteristics) in terms of branch points and the period matrix of a hyperelliptic Riemann surface. The new theta relations contain (i) linear relations on the vector space of first-order theta derivatives which are arranged in gradients, (ii) relations between second-order theta derivatives and symmetric bilinear forms on the vector space of the gradients, (iii) relations between third-order theta derivatives and symmetric trilinear forms on the vector space of the gradients and (iv) a conjecture regarding higher-order theta derivatives. It is shown how the Schottky identity (in the hyperelliptic case) is derived from the obtained relations.

Uniform bounds for the number of rational points on symmetric squares of curves with low Mordell–Weil rank

A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points on a hyperelliptic curve $X$ provided that the rank of the Jacobian of $X$ is at most $g - 3$. Katz, Rabinoff and Zureick-Brown generalized his result to arbitrary curves satisfying the same rank condition. In this paper, we prove conditional uniform bounds on the number of rational points on the symmetric square of $X$ outside its algebraic special set, provided that the rank of the Jacobian is at most $g-4$. We also find rank-favorable uniform bounds (that is, bounds depending on the rank of the Jacobian) in the hyperelliptic case.

Moduli space of meromorphic differentials with marked horizontal separatrices

We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat geometry surfaces "near" the Deligne-Mumford boundary.We compute the number of connected components of the corresponding strata, and give a simple topological invariant that distinguishes them. In particular we see that for $g>0$, there are at most two such components, except in the hyperelliptic case.

General derivative Thomae formula for singular half-periods

The paper develops the result of second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular characteristics of arbitrary multiplicity in terms of branch points and period matrix. We call these values derivative theta constants. First and second Thomae formulas follow as particular cases. Some further results are derived. Matrices of second derivative theta constants (Hessian matrices of zero-values of theta functions with characteristics of multiplicity two) have rank three in any genus. Similar result about the structure of order $3$ tensor of third derivative theta constants is obtained, and a conjecture regarding higher multiplicities is made. As a byproduct a generalization of Bolza formulas are deduced.

Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of solution of the Jacobi inversion problem which provides the way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details comparing with the known in cryptography, and is extended to divisors of arbitrary degrees comparing with the known in the theory of hyperelliptic functions.

DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

AbstractWe provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$-rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$. We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$. On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

On the sections of universal hyperelliptic curves

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves CHg,n/k→Hg,n/k, when g≥3 and the image of the ℓ-adic cyclotomic character Gk→Z× is infinite. Furthermore, we will study the nonabelian phenomena associated to the universal hyperelliptic curves. For example, we will show that the section conjecture holds for CHg/k→Hg/k and the unipotent analogue of the conjecture holds for the pointed cases. This work is an extension of Hain's original work on the rational points of universal curves to the hyperelliptic case.

Graphs of gonality three

In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge- and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.

ON TYPICALITY OF TRANSLATION FLOWS WHICH ARE DISJOINT WITH THEIR INVERSE

In this paper we prove that the set of translation structures for which the corresponding vertical translation flows are disjoint with its inverse contains a $G_{\unicode[STIX]{x1D6FF}}$-dense subset in every non-hyperelliptic connected component of the moduli space ${\mathcal{M}}$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is reversible, i.e., it is isomorphic to its inverse by an involution. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the $G_{\unicode[STIX]{x1D6FF}}$-condition. Moreover, as a by-product we get that in every non-hyperelliptic connected component of the moduli space there is a dense subset of translation structures whose vertical flow is reversible.

Computing -series of geometrically hyperelliptic curves of genus three

Let$C/\mathbf{Q}$be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of$\mathbf{Q}$, but may not have a hyperelliptic model of the usual form over$\mathbf{Q}$. We describe an algorithm that computes the local zeta functions of$C$at all odd primes of good reduction up to a prescribed bound$N$. The algorithm relies on an adaptation of the ‘accumulating remainder tree’ to matrices with entries in a quadratic field. We report on an implementation and compare its performance to previous algorithms for the ordinary hyperelliptic case.

Fields of moduli of some special curves

In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic p-gonal curves by A. Kontogeorgis.

The $$\mathrm {al}$$ al function of a cyclic trigonal curve of genus three

A cyclic trigonal curve of genus three is a \(\mathbb {Z}_3\) Galois cover of \(\mathbb {P}^1\), therefore can be written as a smooth plane curve with equation \(y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)\). Following Weierstrass for the hyperelliptic case, we define an “\(\mathrm {al}\)” function for this curve and \(\mathrm {al}^{(c)}_r\), \(c=0,1,2\), for each one of three particular covers of the Jacobian of the curve, and \(r=1,2,3,4\) for a finite branchpoint \((b_r,0)\). This generalization of the Jacobi \(\mathrm {sn}\), \(\mathrm {cn}\), \(\mathrm {dn}\) functions satisfies the relation: $$\begin{aligned} \sum _{r=1}^4 \frac{\prod _{c=0}^2\mathrm {al}_r^{(c)}(u)}{f'(b_r)} = 1 \end{aligned}$$ which generalizes \(\mathrm {sn}^2u + \mathrm {cn}^2u = 1\). We also show that this can be viewed as a special case of the Frobenius theta identity.

Integrable equations with Ermakov–Pinney nonlinearities and Chiellini damping

We introduce a special type of dissipative Ermakov–Pinney equations of the form vζζ+g(v)vζ+h(v)=0, where h(v)=h0(v)+cv-3 and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h0(v) is a linear function, h0(v)=λ2v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipative function is actually a dissipation-gain function because it can be negative on some intervals. We also examine the nonlinear case h0(v)=Ω02(v-v2) and show that it leads to an integrable hyperelliptic case.

Curves in characteristic $2$ with non-trivial $2$-torsion

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian. We extend their observation to curves given by Laurent polynomials with a fixed Newton polygon, provided that the polygon satisfies a certain combinatorial property. We also show that in each of these cases, the sufficiently general condition is implied by being ordinary. Our treatment includes many classical families, such as hyperelliptic curves of odd genus and $C_{a,b}$ curves. In the hyperelliptic case, we provide alternative proofs using an explicit description of the 2-torsion subgroup.

Szpiro's small points conjecture for cyclic covers

Let X be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro conjectured that X has a «small point». In this paper we prove that if X is a cyclic cover of prime degree of the projective line, then X has infinitely many «small points». In particular, we establish the first cases of Szpiro's small points conjecture, including the genus two case and the hyperelliptic case. The proofs use Arakelov theory for arithmetic surfaces and the theory of logarithmic forms.

Constructing Benney Reductions Associated with Cyclic n, s Curves

Benney’s equations admit “reductions” to systems of dispersionless pdes with only finitely many N dependent variables. These reductions are parametrised by conformal mappings from the half-plane to the half-plane minus N slits. A large family of these can be constructed as Schwartz-Christoffel mappings; an important subclass of these reduce to integrals of a second kind differential on an algebraic curve. Particular examples of these have been constructed; in particular there are two different constructions for the case of a hyperelliptic curve. Other particular examples have been worked out, and the overall structure is similar to the hyperelliptic case, though there are differences of detail. It is thus important to find the general structure of these, independently of the particular curve being studied. We use the Fay Prime Form to investigate this, and sketch how this approach could be applied to general cyclic (n, s) curves.