On Drinfeld Modular Curves with Many Rational Points over Finite Fields

It is known that Drinfeld modular curves can be used to construct asymptotically optimal towers of curves over finite fields. Using reductions of the Drinfeld modular curves X₀( ), we try to find individual curves over finite fields with many rational points. The main idea is to divide by an Atkin–Lehner involution which has many fixed points in order to obtain a quotient with a better ratio #{rational points}/genus. In a few cases we can improve the known records of rational points.     

Solving the Multi-discrete Logarithm Problems over a Group of Elliptic Curves with Prime Order

In this paper, we discuss the expected number of steps in solving multi-discrete logarithm problems over a group of elliptic curves with prime order by using Pollard＇s rho method and parallel collision search algorithm. We prove that when using these algorithms to compute discrete logarithms, the knowledge gained through computing many logarithms does not make it easier for finding other logarithms. Hence in an elliptic cryptosystem, it is safe for many users to share the same curve, with different private keys.     

Cascadic Multigrid for Finite Volume Methods for Elliptic Problems

In this paper, some effective cascadic multigrid methods are proposed for solving the large scale symmetric or nonsymmetric algebraic systems arising from the finite volume methods for second order elliptic problems. It is shown that these algorithms are optimal in both accuracy and computational complexity. Numerical experiments are reported to support our theory.     

The Selmer Groups of Elliptic Curves

Let E/K be an elliptic curve with K-rational p-torsion points.The p-Selmer group of E is described by the image of a map λk and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.     

关于有限域F_q上矩阵阶的研究

详细地研究了有限域 Fq上的矩阵的阶的问题 ,得到了相当理想的结果 .并给出一类矩阵方幂的极小多项式的求法     

关于椭圆曲线E_D:y~2=x~3-D~2x的BSD猜想

本文证明了形如ｙ~２＝ｘ~３－ｐ-１~２…ｐ-ｎ~２ｘ的一类椭圆曲线的Ⅲ群２-分支在一定条件下同构于 Ｚ／２Ｚ × Ｚ／２Ｚ,其阶为 ４,与Ｌ函数部分的相应结果和 Ｒｕｂｉｎ Ｋ．关于有复乘的椭圆曲线的重要结果一起,我们知道ＢＳＤ猜想对本文定理中的椭圆曲线成立．     

Explicit Classification for Torsion Subgroups of Rational Points of Elliptic Curves

We study the classification of elliptic curves E over the rationals ℚ according to the torsion sugroups E _tors(ℚ). More precisely, we classify those elliptic curves with E _tors(ℚ) being cyclic with even orders. We also give explicit formulas for generators of E _tors(ℚ). These results, together with the recent results of K. Ono for the non-cyclic E _tors(ℚ), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2. Received July 29, 1999, Revised March 9, 2001, Accepted July 20, 2001     

The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.with an Appendix by Jean-Pierre Serre     

On Supersingular Abelian Varieties of Dimension Two over Finite Fields

In this paper we study the characteristic polynomials and the rational point group structure of supersingular varieties of dimension two over finite fields. Meanwhile, we also list allL-functions of supersingular curves of genus two over ₂and determine the group structure of their divisor class groups over all finite algebraic extension of ₂.     

On the size of the Shafarevich--Tate group of elliptic curves over function fields

Let E be a nonconstant elliptic curve, over a global field K of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of E, we show that the order of theShafarevich-Tate group of E, is given by O ^{(N1/2+6 log(2)/ log(q))}, where N is the conductor of E,q isthe cardinality of the finite field of constants of K, and where the constant in the bound depends only on K. The method of proof is to workwith the geometric analog of the Birch-Swinnerton Dyer conjecture for thecorresponding elliptic surface over the finite field, as formulatedby Artin-Tate, and to examine the geometry of this elliptic surface.     

On a Class of Domain Optimization Problems in Elliptic Boundary Value Problems

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On Certain Uniformity Properties of Curves Over Function Fields

A uniform version of the Shafarevich Conjecture for function fields (Theorem of Parshin–Arakelov) is proved, together with a uniform version of the geometric Mordell conjecture (Theorem of Manin). Such results are generalized to base varieties of arbitrary dimension (i.e. function fields of arbitrary transcendence degree).     

Solvability of Elliptic Boundary Value Problems without Standard Landesman-Lazer Condition

Abstract In this paper we prove a very general result concerning solvability of the resonant problem: Δu + λ_κ u + g(x, u) = h (x); u = 0, x ∈∂Ω, which immediately gives three generalized Landesman-Lazer conditions. The most interesting application of the general result is concerned with the problem when λ_κ = λ₁, in which case we prove solvability results for it under conditions which are not the standard Landesman-Lazer condition or only partly enjoy it. Furthermore, we propose a new sign condition and give a comprehensive extension of a main result of Figueiredo and Ni.     

Geometry of Cubic and Quartic Hypersurfaces over Finite Fields

Let YPⁿ be a cubic hypersurface defined over GF(q). Here, we study the Finite Field Nullstellensatz of order [q/3] for the set Y(q) of its GF(q)-points, the existence of linear subspaces of PG(n,q) contained in Y(q) and the possibility to join any two points of Y(q) by the union of two lines of PG(n,q) entirely contained in Y(q). We also study the existence of linear subspaces defined over GF(q) for the intersection of Y with s quadrics and for quartic hypersurfaces.     

On the Number of Points of Elliptic Curves over a Finite Field and a Problem of B. Segre

Several theorems are studied concerning the number of points of an elliptic curve with a Legendre form on a finite field, in order to analyse the distribution of regular and pseudoregular points in relation to a hyperbola in a finite affine plane.     

On Characteristic Polynomials of Formal Groups over Finite Fields

Generalizing the results of Serre, Hill and Koch, we give some classification theorems of higher dimensional simple formal groups over finite fields. A relation between endomorphism rings of formal groups over ?_p and characteristic polynomials of their reductions mod p is studied. A condition of existence of formal groups over ?_p with complex multiplication is given. Some formal groups over ?_p are also constructed.     

On the Diameter of Plane Curves

Recently, differential geometric properties of embedded projective varieties have gained increasing interest. In this note, we consider plane algebraic curves equipped with the Fubini--Study metric from₂ () and give an estimate for the diameter in terms of the degree, initiated in a paper by F. A. Bogomolov.