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Explicit Classification for Torsion Subgroups of rational points of Elliptic Curves

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Acta Mathematica Sinica,English Series 2002年第3期18卷 539-548页

作者： De Rong QIU Center for Advanced Study, Tsinghua University, Beijing 100084. P. R. China Xian Ke ZHANG Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China Center for Advanced Study Tsinghua University Beijing P. R. China Department of Mathematical Sciences Tsinghua University Beijing P. R. China

We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups 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.

关键词： Elliptic curve rational point Torsion group Mordell group Parameterization

Remarks on the Number of rational points on a Class of Hypersurfaces over Finite Fields

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Algebra Colloquium 2018年第3期25卷 533-540页

作者： Hua Huang Wei Gao Wei Cao Department of Mathematics Ningbo University Ningbo Zhejiang 315211 China

Let Fq be the finite field of q elements and f be a nonzero polynomial over Fq. For each b ∈ Fq, let Nq（f = b） denote the number of Fq-rational points on the affine hypersurface f = b. We obtain the formula of Nq（f= b） for a class of hypersurfaces over Fq by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.

关键词： finite field polynomial rational point degree matrix

The Number of rational points of a Family of Algebraic Varieties over Finite Fields

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Algebra Colloquium 2017年第4期24卷 705-720页

作者： Shuangnian Hu Junyong Zhao School of Mathematics and Statistics Nanyang Institute of Tehnology Nanyang Henan 473004 China School of Mathematics and Physics Zhengzhou University Zhengzhou 450001 China School of Mathematics and Statistics Nanyang Institute of Technology Nanyang Henan 473004 China

Let Fq stand for the finite field of odd characteristic p with q elements （q= p^n, n∈ N） and Fq denote the set of all the nonzero elements of Fq. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations……positive integer. Further, under some natural conditions, we arrive at an explicit Iormula for the number of Fq-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 （2015） 135-153].

关键词： exponent matrix finite field rational point Smith normal form variety

rational points on Genus 3 Real Hyperelliptic Curves

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Open Journal of Discrete Mathematics 2021年第4期11卷 103-113页

作者： Brice M. Miayoka Regis F. Babindamana Basile G. R. Bossoto Faculté Des Sciences et Techniques Université Marien Ngouabi Brazzaville Congo

We compute rational points on real hyperelliptic curves of genus 3 defined on whose Jacobian have Mordell-Weil rank r=0. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C (). We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.

关键词： Hyperelliptic Curve Jacobian Coleman Integration rational point

Counting Multiplicities in a Hypersurface over Number Fields

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Algebra Colloquium 2018年第3期25卷 437-458页

作者： Hao Wen Chunhui Liu Department of iVlathematicM Sciences Tsinghua University Beijing 100084 China Department of Mathematics Faculty of Science Kyoto University Kyoto 606-8502 Japan

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.

关键词： algebraic point counting multiplicities height function over projective space intersection tree rational point

The value of polynomials represented by norm forms

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Science China Mathematics 2014年第12期57卷 2525-2528页

作者： WANG Xin State Key Laboratary of Mathematical Engineering and Advanced Computing

Let P(t) be a product of(possibly repeated) linear factors over Q and K/Q an abelian extension. Under a strict condition, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the ... 详细信息

关键词： Brauer group Brauer-Manin obstruction rational point

Note on the Number of Solutions of Cubic Diagonal Equations over Finite Fields

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Wuhan University Journal of Natural Sciences 2023年第5期28卷 369-372页

作者： HU Shuangnian WANG Shihan LI Yanyan NIU Yujun School of Mathematics and Physics Nanyang Institute of TechnologyNanyang 473004HenanChina Faculty of Science and Technology Beijing Normal University-Hong Kong Baptist University United International CollegeZhuhai 519087GuangdongChina School of Information Engineering Nanyang Institute of TechnologyNanyang 473004HenanChina

Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive *** F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.

关键词： finite field rational point diagonal equations Jacobi sums

Research on a Class of Equations overFinite Fields

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Wuhan University Journal of Natural Sciences 2018年第6期23卷 493-497页

作者： HU Shuangnian DIAO Tianbo NIU Yujun WU Honge School of Mathematics and Statistics Nanyang Institute of Technology

Let Fq stand for the finite field of odd characteristic p with q elements（q=pn,n∈N）and Fq＊ denote the set of all the nonzero elements of *** this paper,by using the augmented degree matrix and the result given by Cao,we obtain a formula for the number of rational points of the following equation over Fq：f（x 1,x 2,...,x n）=（a1 x1 x2d＋a2 x2 x3d...＋a（n-1）x（n-1）xnd＋an xn x1d）λ-bx1（d1）x2d2...xn（dn）,with ai,b∈Fq＊,n≥2,λ〉0 being positive integers,and d,di being nonnegative integers for 1≤i *** technique can be applied to the polynomials of the form h1λ=h2 with λ being positive integer and h1,h2∈Fq[x 1,x 2,...,x n].It extends the results of the Markoff-Hurwitz-type equations.

关键词： finite field rational point degree matrix

The Number of Solutions of Certain Equations over Finite Fields

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Wuhan University Journal of Natural Sciences 2022年第1期27卷 49-52页

作者： HU Shuangnian LIU Jianghan QIN Zhentao School of Mathematics and Statistics Nanyang Institute of TechnologyNanyang 473004HenanChina College of Computer and Information Science Southwest UniversityChongqing 400715China School of Mathmatics and Computer Panzhihua UniversityPanzhihua 617000SichuanChina

Let s be a positive integer,p be an odd prime,q=p^(s),and let F_(q)be a finite field of q *** N_(q)be the number of solutions of the following equations:(x_(1)^(m_(1))+x_(2)^(m_(2))+…+x_(n)^(m_(n)))^(k)=x_(1)x_(2)…x_(n)x^(k_(n+1))_(n+1)…x^(k_(t))_(t)over the finite field F_(q),with n≥2,t>n,k,and k_(j)(n+1≤j≤t),m_(i)(1≤i≤n)are positive *** this paper,we find formulas for N_(q)when there is a positive integer l such that dD|(p^(l)+1),where D=1 cm[d_(1),…,d_(n)],d=gcd(n∑i=1M/m_(i)-kM,(q-1)/D),M=1 cm[m_(1),…,m_(n)],d_(j)=gcd(m_(j),q-1),1≤j≤*** we determine N_(q)explicitly under certain *** extends Markoff-Hurwitz-type equations over finite field.

关键词： finite field rational point diagonal equation Markoff-Hurwitz-type equations