Weierstrass Points Research Articles

On hyperelliptic curves of odd degree and genus g with 6 torsion points of order 2g + 1

Let a hyperelliptic curve C of genus g defined over an algebraically closed field K of characteristic 0, given by the equation y2=f(x), where the polynomial f(x)∈K[x]is square-free and has odd degree 2g+1. The curve Ccontains a single “infinite” point O, which is the Weierstrass point. There is a classical embedding of C(K)into the group of K-points J(K)of the Jacobian variety Jof the curve C, identifying the point Owith the unit element of the group J(K). For 2≤g≤5, the article explicitly found representatives of birational equivalence classes such hyperelliptic curves Cwith a marked unique point at infinity Othat the set C(K)∩J(K)contains at least 6 torsion points of order 2g+1. It was previously known that for g=2there are exactly 5 such equivalence classes, and for g≥3an upper bound was known that depended only on the genus of g. We improve the previously known upper bound by almost 36 times.

Local complete intersections and Weierstrass points

This work presents a simple proof that the moduli space of complete integral Gorenstein curves with a prescribed symmetric Weierstrass semigroup becomes a weighted projective space, even for fields of positive characteristic, when the associated monomial curve is a local complete intersection.

Finite simple permutation groups acting with fixity 4

Abstract Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we classify all finite simple groups that have a faithful transitive action with fixity 4. We also give details about all possible such actions.

Infinitely many Riemann surfaces with a transitive action on the Weierstrass points

AbstractIn this short note, we prove the existence of infinitely many pairwise nonisomorphic, non‐hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also find all compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are simple.

The distribution of Weierstrass points on a tropical curve

The distribution of Weierstrass points on a tropical curve

On nonnegatively graded Weierstrass points

On nonnegatively graded Weierstrass points

THE CERESA CLASS: TROPICAL, TOPOLOGICAL AND ALGEBRAIC

Abstract The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for nonhyperelliptic curves is in general not easy. We present a ‘combinatorialization’ of this problem, explaining how to define a Ceresa class for a tropical algebraic curve and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over $\mathbb {C}(\!(t)\!)$ and show that the Ceresa class in each of these settings is torsion.

GENUS CURVES WITH BAD REDUCTION AT ONE ODD PRIME

Abstract The problem of classifying elliptic curves over $\mathbb Q$ with a given discriminant has received much attention. The analogous problem for genus $2$ curves has only been tackled when the absolute discriminant is a power of $2$ . In this article, we classify genus $2$ curves C defined over ${\mathbb Q}$ with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many $1$ -parameter families of genus $2$ curves. In particular, we provide genus $2$ analogues to Neumann–Setzer families of elliptic curves over the rationals.

Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus

In this article we explicitly determine the Weierstrass semigroup at any point and the full automorphism group of a known Fq2-maximal curve X3 having the third largest genus. This curve arises as a Galois subcover of the Hermitian curve, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough X3 has many different types of Weierstrass semigroups and the set of its Weierstrass points is much richer than its set of Fq2-rational points. This makes the curve X3 the first explicitly known maximal curve having non-rational Weierstrass points. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, Aut(X3) is exactly the automorphism group inherited from the Hermitian curve, apart from small values of q.

ON THE NUMBER OF POINTS OF GIVEN ORDER ON ODD-DEGREE HYPERELLIPTIC CURVES

For integers N≥2 and g≥1, we study bounds on the cardinality of the set of points of order dividing N lying on a hyperelliptic curve of genus g embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.

Équation de Pell–Abel et applications

In this paper, we show that there are solutions of degree r of the equation of Pell–Abel on some real hyperelliptic curve of genus g if and only if r>g. This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive k-differential on an hyperelliptic curve of genus g with a unique zero of order k(2g-2) for every (k,g)≠(2,2). Moreover, we show that there exists a non Weierstrass point of order n modulo a Weierstrass point on a hyperelliptic curve of genus g if and only if n>2g.

Explicit two-cover descent for genus 2 curves

Given a genus 2 curve C with a rational Weierstrass point defined over a number field, we construct a family of genus 5 curves that realize descent by maximal unramified abelian two-covers of C, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus 2 quintic curves over $$\mathbb {Q}$$ of Mordell–Weil rank 2 or 3 whose sets of rational points have not previously been provably computed. We analyze how often this method succeeds in computing the set of rational points and what obstacles lead it to fail in some cases.

An elementary abelian p-cover of the Hermitian curve with many automorphisms

The full automorphism group of a certain elementary abelian p-cover of the Hermitian curve in characteristic $$p>0$$ is determined. It is remarkable that the order of Sylow p-groups of the automorphism group is close to Nakajima’s bound in terms of the p-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated.

Hyperelliptic odd coverings

We investigate a class of odd (ramification) coverings C → ℙ1 where C is hyperelliptic, its Weierstrass points map to one fixed point of ℙ1 and the covering map makes the hyperelliptic involution of C commute with an involution of ℙ1. We show that the total number of hyperelliptic odd coverings of minimal degree 4g is \(\left({\matrix{{3g} \cr {g - 1} \cr}} \right){2^{2g}}\) when C is general. Our study is approached from three main perspectives: if a fixed effective theta characteristic is fixed they are described as a solution of a certain class of differential equations; then they are studied from the monodromy viewpoint and a deformation argument that leads to the final computation.

Torsion points of small order on hyperelliptic curves

Let $$\mathscr {C}$$ be a hyperelliptic curve of genus $$g>1$$ over an algebraically closed field K of characteristic zero and $$\mathscr {O}$$ one of the $$(2g{+}2)$$ Weierstrass points in $$\mathscr {C}(K)$$ . Let J be the Jacobian of $$\mathscr {C}$$ , which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of $$\mathscr {C}$$ into J that sends $$\mathscr {O}$$ to the zero of the group law on J. This embedding allows us to identify $$\mathscr {C}(K)$$ with a certain subset of the commutative group J(K). A special case of the famous theorem of Raynaud (Manin–Mumford conjecture) asserts that the set of torsion points in $$\mathscr {C}(K)$$ is finite. It is well known that the points of order 2 in $$\mathscr {C}(K)$$ are exactly the “remaining” $$(2g{+}1)$$ Weierstrass points. One of the authors (Zarhin in Izv Math 83:501–520, 2019) proved that there are no torsion points of order n in $$\mathscr {C}(K)$$ if $$3\leqslant n\leqslant 2g$$ . So, it is natural to study torsion points of order $$2g+1$$ (notice that the number of such points in $$\mathscr {C}(K)$$ is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually non-isomorphic pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least four points of order $$2g+1$$ . In the present paper we prove that (for a given g) there are at most finitely many (up to an isomorphism) pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least six points of order $$2g+1$$ .

Weierstrass points of modular curves at cusps

We prove that all non-exact cusps on $X_1(N)$ of genus $\geq 2$ are Weierstrass points except for $X_1(18)$. Also, for any positive integer $N$ of the form $p^2M$ with a prime $p$ and a positive integer $M$, we obtain some results on when the cusps o

Weierstrass points on modular curves X0(N) fixed by the Atkin–Lehner involutions

PurposeThe authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ on X0(N) for N ≤ 50 are Weierstrass points or not.Design/methodology/approachThe design is by using Lawittes's and Schoeneberg's theorems.FindingsFinding all Weierstrass points on X0(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.Originality/valueThe Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X0(N).

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

For any smooth Hurwitz curve Hn:XYn+YZn+XnZ=0 over the finite field Fp, an explicit description of its Weierstrass points for the morphism of lines is presented. As a consequence, bounds on the number of rational points of Hn are obtained via Stöhr-Voloch Theory. Further, the full automorphism group Aut(Hn), as well as the genera of all Galois subcovers of Hn, with n≠3,pr, are computed. Finally, a question by F. Torres on plane nonsingular maximal curves is answered.

Weierstrass points on bielliptic intermediate modular curves

We find all Weierstrass points on some of the bielliptic intermediate modular curves XΔ(N) whose bielliptic involutions are of Atkin–Lehner type.

From Picard groups of hyperelliptic curves to class groups of quadratic fields

Let C C be a hyperelliptic curve defined over Q \mathbb {Q} , whose Weierstrass points are defined over extensions of Q \mathbb {Q} of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case of elliptic curves), we prove that any line bundle of degree 0 0 on C C which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large. This gives a positive answer, for such curves, to a question by Agboola and Pappas.