Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

On Weierstrass elliptic function solutions for a (N+1) dimensional potential KdV equation

This paper is concerned with several aspects of travelling wave solutions for a (N+1) dimensional potential KdV equation. The Weierstrass elliptic function solutions, the Jaccobi elliptic function solutions, solitary wave solutions, periodic wave solutions to the equation are acquired under certain circumstances. It is shown that the coefficients of derivative terms in the equation cause the qualitative changes of physical structures of the solutions.     

Construction of high-rank elliptic curves with a nontrivial torsion point

We construct a family of infinitely many elliptic curves over with a nontrivial rational 2-torsion point and with rank 6, which is parametrized by the rational points of an elliptic curve of rank 1.

     

Extension of elliptic curves on Krasner hyperfields

The aim of the article is to initiate a new study in the framework of algebraic hyperfields, with an applicative impact in cryptography. First we define the notions of generalized Weierstrass equation and elliptic hypercurve on Krasner hyperfields, as a generalization of the notion of elliptic curve on fields. Then we investigate properties of the hypergroups derived from elliptic hypercurves and of the associated H_v-groups. Finally, we present a class of canonical hypergroups, which can be used as an alphabet in a special cryptographic system.     

Torsion on elliptic curves in isogeny classes

Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .

     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Selmer groups of elliptic curves that can be arbitrarily large

In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals, where g is the genus of X₀(p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p=5,7 and 13 (the cases p?7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N=9,10,12,16 and 25.     

Eventually minimal curves

A curve defined over a finite field is maximal or minimal according to whether the number of rational points attains the upper or the lower bound in Hasse-Weils theorem, respectively. In the study of maximal curves a fundamental role is played by an invariant linear system introduced by Rück and Stichtenoth in [6]. In this paper we define an analogous invariant system for minimal curves, and we compute its orders and its Weierstrass points. In the last section we treat the case of curves having genus three in characteristic two.     

Pure Weierstrass gaps from a quotient of the Hermitian curve

In this paper, by employing some results on Kummer extensions, we give an arithmetic characterization of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve, including the well-studied Hermitian curve as a special case. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results.     

Existence of the non-primitive Weierstrass gap sequences on curves of genus 8

We show that for any possible Weierstrass gap sequence L on a non-singular curve of genus 8 with twice the smallest positive non-gap is less than the largest gap there exists a pointed non-singular curve (C, P) over an algebraically closed field of characteristic 0 such that the Weierstrass gap sequence at P is L. Combining this with the result in [6] we see that every possible Weierstrass gap sequence of genus 8 is attained by some pointed non-singular curve. *Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. **Partially supported by Grant-in-Aid for Scientific Research (17540030), Japan Society for the Promotion of Science.     

The eigenvalue problem for elliptic partial differential equation with multi-point nonlocal conditions

We study the spectral problem for the system of difference equations of a two-dimensional elliptic partial differential equation with nonlocal conditions. A new form of two-point nonlocal conditions that involve interior points is proposed. The matrix of the difference system is nonsymmetric thus different types of eigenvalues occur. The conditions for the existence of the eigenvalues and their corresponding eigenvectors are presented for the one dimensional problem. Then, these relations are generalized to the two-dimensional problem by the separation of variables technique.     

On the Tate-Shafarevich groups of certain elliptic curves

The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.     

On the modularity of elliptic curves over -adic exercises

We complete the proof that every elliptic curve over the rational numbers is modular.

     

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

We consider the quotient of the Hermitian curve defined by the equation y^q + y = x^m over where m > 2 is a divisor of q+1. For 2≤ r ≤ q+1, we determine the Weierstrass semigroup of any r-tuple of -rational points on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, we prove that there are r-point codes, that is codes of the form where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which is obtained by taking m=q +1 in the above equation Communicated by: J.W.P. Hirschfeld     

Weierstrass Semigroups of a Pair of Points Whose First Nongaps are Three

We found all candidates for a Weierstrass semigroup at a pair of Weierstrass points whose first nongaps are three. We prove that such semigroups are actually Weierstrass semigroups by constructing examples.     

Existence of solution for a singular critical elliptic equation

In this paper, a singular semilinear elliptic problem involving the critical Sobolev exponent is studied by variational method, the existence of a solution is proved under certain conditions. The Hardy inequality is used and plays an important role in the discussion.     

Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds

In this paper, we study the local gradient estimate for the positive solution to the following equation:

     

Hausdorff Dimension of a Class of Weierstrass Functions

It was proved by Shen that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log \lambda/\log b$, for every integer $b\geq 2$ and every $\lambda\in (1/b,1)$ [Hausdorff dimension of the graph of the classical Weierstrass functions, Math. Z., 289 (2018), 223–266]. In this paper, we prove that the dimension formula holds for every integer $b\geq 3$ and every $\lambda\in (1/b,1)$ if we replace the function $\cos$ by $\sin$ in the definition of Weierstrass function. A class of more general functions are also discussed.