Weierstrass Points Research Articles

On certain curves of genus three in characteristic two

We study curves of genus 3 over algebraically closed fields of characteristic 2 with the canonical theta characteristic totally supported in one point. We compute the moduli dimension of such curves and focus on some of them which have two Weierstrass points with Weierstrass directions towards the support of the theta characteristic. We answer questions related to order sequence and Weierstrass weight of Weierstrass points and the existence of other Weierstrass points with similar properties.

A remark on Weierstrass points and linear series on multiple covering curves

Here we study multiple coverings of rational and irational curves. We give a theorem about the non-gap sequence on m-gonal curves. We then study general irrational covering f: X → C, and say when h0(X, f*(L)) = h0(C,L) for L line bundle on C.

Siegel Modular Forms of Genus 2 and Level 2: Cohomological Computations and Conjectures

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the \'etale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine' Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures.

Estimation of genus of arithmetic curves and applications

For any element γ∈Γ0(N) and a positive integer N, we find the genus of arithmetic curve 〈Γ1(N),γΦ〉\ℋ*, where \(\Phi=\bigl(\begin{scriptsize}\begin{array}{c@{\ }c}0&-1\\N&0\end{array}\end{scriptsize}\bigr)\) is the Fricke involution. We obtain that the genus of 〈Γ1(N),γΦ〉\ℋ* is zero if and only if 1≤N≤12 or N=14, 15. As its applications, since the genus formula is independent of γ, we determine the Hauptmoduln for the groups 〈Γ1(N),Φ〉 of genus zero which will be used to generate appropriate ray class fields over imaginary quadratic fields, and show that the fixed point of γΦ in ℋ is a Weierstrass point of Γ1(N) for all but finitely many N, which is a direct generalization of Lehner-Newman’s use of Schoeneberg’s Theorem.

Non-hyperelliptic Riemann surfaces of genus five all of whose Weierstrass points have maximal weight

It is proved that any non-hyperelliptic Riemann surface of genus five all of whose Weierstrass points have maximal weight has only Weierstrass points with gap sequence (1,2,3,5,9), and a defining equation of such a surface is given.

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X0(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X0(N), if N=pM with p prime, p∤M and the genus of X0(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.

Revêtements hyperelliptiques d-osculateurs et solitons elliptiques de la hiérarchie KdV

Let d be a positive integer, K an algebraically closed field of characteristic 0 and X an elliptic curve defined over K . We study the hyperelliptic curves equipped with a projection over X, such that the natural image of X in the Jacobian of the curve osculates to order d to the embedding of the curve, at a Weierstrass point. We construct ( d − 1 )-dimensional families of such curves, of arbitrary big genus g, obtaining, in particular, ( g + d − 1 ) -dimensional families of solutions of the KdV hierarchy, doubly periodic with respect to the d-th variable. To cite this article: A. Treibich, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Limit Weierstrass points on nodal reducible curves

In the 1980s D. Eisenbud and J. Harris posed the following question: “What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?” In the present article, we give a partial answer to this question. We consider the case where the limit curve has components intersecting at points in general position and where the degeneration occurs along a general direction. For this case we compute the limits of Weierstrass points of any order. However, for the usual Weierstrass points, of order one, we need to suppose that all of the components of the limit curve intersect each other.

Geometric interplay between function subspaces and their rings of differential operators

We study, in the setting of algebraic varieties, finite-dimensional spaces of functions $V$ that are invariant under a ring $\Dc^V$ of differential operators, and give conditions under which $\Dc^V$ acts irreducibly. We show how this problem, originally formulated in physics \cite{kamran-Milson-Olver:invariant,turbiner:bochner}, is related to the study of principal parts bundles and Weierstrass points \cite{EGA4,laksov-thorup}, including a detailed study of Taylor expansions. Under some conditions it is possible to obtain $V$ and $\Dc^V$ as global sections of a line bundle and its ring of differential operators. We show that several of the published examples of $\Dc^V$ are of this type, and that there are many more - in particular, arising from toric varieties.

Real ramification points and real Weierstrass points of real projective curves

Here we construct real smooth projective curves with prescribed genus, gonality and topological type or with a real Weierstrass point with prescribed rst positive non-gap.

THE WEIERSTRASS SUBGROUP OF A CURVE HAS MAXIMAL RANK

We show that the Weierstrass points of the generic curve of genus g over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian. 2000 Mathematics Subject Classification 11G30, 14H40, 14H10, 14H55, 14Q05.

An optimal systolic inequality for CAT(0) metrics in genus two

We prove an optimal systolic inequality for CAT(0) metrics on a genus 2 surface. We use a Voronoi cell technique, introduced by C. Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y2 = x5 - x. Thus, among all CAT(0) metrics, the one with the best systolic ratio is composed of six flat regular octagons centered at the Weierstrass points of the Bolza surface.

A semigroup at a pair of Weierstrass points on a cyclic 4-gonal curve and a bielliptic curve

We find all candidates for the Weierstrass semigroup of a pair of points on a cyclic 4-gonal curve which are total ramification points of the fixed cyclic 4-gonal map. Then we prove that such semigroups actually occur on some cyclic 4-gonal curve by constructing cyclic 4-gonal curves explicitly. Moreover, we characterize all Weierstrass semigroups of pairs of points with first nonzero nongap 4 on bielliptic curves.

Genus 3 normal coverings of the Riemann sphere branched over 4 points

In this paper we study the 5 families of genus 3 compact Riemann surfaces which are normal coverings of the Riemann sphere branched over 4 points from very different aspects: their moduli spaces, the uniform Belyi functions that factorize through the quotient by the automorphism groups and the Weierstrass points of the non hyperelliptic families.

A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point

The reduced equations for the isomorphism classes of hyperelliptic curves of genus 2 admitting a Weierstrass point over a finite field of arbitrary characteristic, are shown and the number of such classes is included. This work picks up in a unified way a series of previous results published by several authors by using different methodologies. These classifications are of interest in designing and implementing of hyperelliptic curve cryptosystems.

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space $\mathcal {M}_{3}$ of curves of genus 3.

One jump Weierstrass gap sequence

Abstract It has been proved (see [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra 63 (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] and [M. Brundu, G. Sacchiero, On the varieties parametrizing trigonal curves with assigned Weierstrass points. Comm. Algebra 26 (1998), 3291-3312. MR1641619 (99g:14040) Zbl 0937.14016]. ) that a Weierstrass point on a trigonal curve, which is not a ramification point for the morphism induced by a trigonal series, has a gap sequence GP of the form GP = (1,2, …, r, r + 1 + α, …, g + α) for some integers r and α satisfying ≤ r ≤ g − 1 and 1 ≤ α ≤ 2r + 1 − g. Such points will be called one jump Weierstrass points of type (r,α). It was further proved [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra 63 (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] that any one jump numerical sequence of type (r, α), with r and α in the range above, is the Weierstrass gap sequence of a trigonal curve. Here we prove that the property of having an extremal, in some sense, one jump Weierstrass point characterizes trigonal curves. More precisely, we show that if α belongs to the range α4 &lt; α ≤ 2r + 1 − g for a suitable α4, any curve with a Weierstrass point of type (r,α) is a triple covering of a smooth curve of genus p with , and that there exist examples of such coverings. Therefore when 2r + 1 − g − α &lt; 3, such a curve is indeed trigonal. As a consequence, any fourgonal curve of genus g ≥ 10 having a one jump Weierstrass point satisfies α ≤ α4 with few exceptions. Finally, we exhibit examples of fourgonal curves with a Weierstrass point of type (r, α) with ≤ r ≤ and 1 ≤ α ≤ α4.

On Weierstrass points of Hurwitz curves

On Weierstrass points of Hurwitz curves

Explicit Mumford isomorphism for hyperelliptic curves

Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov–Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.

Curves of Genus Three in Characteristic Two

ABSTRACT For an algebraic nonsingular nonhyperelliptic complete curve of genus three defined over a field of characteristic two we show that the number of canonical Weierstrass points is one of the following: 24, 21, 20, 18, 17, 16, 15, 12, 11, 6 or 5. In most cases, we give also an explicit equation for the curve with the given number of Weierstrass points.