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class number and computer —— A note on an S. Chowla's conjecture

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Progress in Natural Science:Materials International 1995年第4期

作者：计光恒陆洪文 China China. Department of Mathematics Computer Centre Department of Mathematics Hefei 230026 Shanghai 200092 Tongji University University of Science and Technology of China

We use LEO 386/25 to achieve some class number formulae if the real quadratic number field Q(P1/2) has class number 1 for a prime p = 4N2+1 (N is a positive integer).

关键词： quadratic number fields class number continued fractions.

DETERMINATION OF ALGEBRAIC FUNCTION FIELDS OF TYPE (2,2,…,2) WITH class number ONE

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Science in China,Ser.A 1988年第8期 908-915页

作者：张贤科 Department of Mathematics University of Science and Technology of China Hefei

Let k=Fq（t） be the rational function field of variable t over the q-element field Fq of characteristic different from 2. Suppose that L=k(D11/2, …, Dn1/2 is an extension of k of degree 2π with constant field Fq, n>1, Di∈Fq[t]. The following are proved: （1） Fields L with divisor-class-number one are just k(P11/2, P21/2) and k(P1P21/2, P1P31/2 ), where Pi∈Fq[t] are linear and distinct. （2） Imaginary fields L=k(D11/2, D21/2) with ideal-class-nurnber one （i. e. ring of the integers of L is a unique factorization domain） are just those with D1=t; D2=t3-t-1（q=3）, t2-t-1（q=3）, t2+2（q=5）, or t+c（c∈Fq） up to transformation t（?）at+b（a, b∈Fq）.

关键词： algebraic function field genus class number unique factorization.

Fundamental Unit System and class number for Real number Fields of Type (2,2,2)

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Tsinghua Science and Technology 2000年第2期5卷 150-153页

作者：王鲲鹏张贤科 Department of Mathematical Sciences Tsinghua University Beijing 100084 China

Let k=Q((D 2+md)(D 2+nd)(D 2+rd)), this paper proves firstly that the fundamental unit of k is ε=((D 2+md)(D 2+nd)+D 2(D 2+rd)) 2/(|mn|d 2), where D,d,m,n, and r are rational integers satisfying certain conditions. Consequently, we describe the fundamental unit system of K=Q(D 2+md,D 2+nd,D 2+rd) explicitly by the fundamental unit of all the quadratic subfields and the class number h K explicitly by the class numbers of all the quadratic subfields. We also provide the fundamental unit system of some fields of (2,2) type.

关键词： number field octic field fundamental unit system class number

The Problem on class numbers of Quadratic number Fields

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Chinese Quarterly Journal of Mathematics 1996年第3期11卷 1-7页

作者：陆洪文

It is a survey of the problem on class numbers of quadratic number fields.

关键词： quadratic number fields class number elliptic curves

Quadratic number Fields with class numbers Divisible by a Prime q

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Tsinghua Science and Technology 2004年第4期9卷 475-481页

作者：杨东张贤科 Department of Mathematical Sciences Tsinghua University Beijing 100084 China

Let q 5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1)+). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unramified cyclic extension of degree q (which implies the class number of k is divisible by q), and W is the splitting field of some irreducible trinomial f(X) = XqaXb with integer coefficients, k Df=(())with D(f) the discriminant of f(X). Then f(X) must be of the form f(X) = Xquq2wXuq1 in a cer-tain sense where u,w are certain integers. Therefore, k d=() with d =-----qqqqqquwuq(1)122(1)((1)+). Moreover, the above two results are both generalized for certain kinds of general polynomials.

关键词： quadratic field class number unramified Newton抯 polygon

Upper bound and formula for class numbers of abelian function fields

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Progress in Natural Science:Materials International 2006年第3期16卷 321-323页

作者： MA Lianrong and ZHANG Xianke (Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China) Department of Mathematical Sciences Tsinghua University Beijing 100084 China

For any abelian function field K (i.e. any subfield of a cyclotomic function field L=k (∧P) over the rational function field k ) with conductor being an irreducible polynomial over a finite field of odd characteristic, by studying the Carlitz-module structure and the character group of K, an explicit upper bound and a calculating formula of the relative divisor class number h- (K) of K are given. Our calculated results of K develop Rosen’s recent results of L .

关键词： Carlitz module cyclotomic function field class number character abelian function field.

Congruence formulae modulo powers of 2 for class numbers of cyclic quartic fields

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Science China Mathematics 2009年第3期52卷 417-426页

作者： MA LianRong LI Wei ZHANG XianKe Department of Mathematics Tsinghua UniversityBeijing 100084China

Let K = $ k(\sqrt \theta ) $ be a real cyclic quartic field, k be its quadratic subfield and $ \tilde K = k(\sqrt { - \theta } ) $ be the corresponding imaginary quartic field. Denote the class numbers of K, k and $ \tilde K $ by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h ? = h K /h K and $ \tilde h^ - = h_{\tilde K} /h_k $ are obtained via studying the p-adic L-functions of the fields.

关键词： class number regulator unit group character p-adic L-function conductor 11M9 11R29

Rational Points of Elliptic Curve y2 = x3＋ k3

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Algebra Colloquium 2018年第1期25卷 133-138页

作者： Xia Wu Yan Qin School of Mathematics Southeast University Nanjing 210096 China Nanjing University Business School Nanjing 210093 China

Let E be an elliptic curve defined over the field of rational numbers ~. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed（Q） is zero. Let E（k） denote the elliptic curve y2 = x3 ＋ k. Then the quadratic twist E（1）d of E（1） by d is the elliptic curve E（d3）： y2 = x3＋ k3. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d = r （rood 24） such that rankE（-d3）（Q） = 0, using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E（k3）（Q） has rank zero.

关键词： elliptic curve rational point class number Pell equation

ON THE EXCEPTIONAL FIELDS FOR A class OF REAL QUADRATIC FIELDS

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Acta Mathematica Scientia 2011年第3期31卷 1179-1188页

作者：刘丽陆洪文 School of Mathematics Hefei University of Technology Department of Mathematics Tongji University

In this paper, we give a lower bound exp（2.2 × 10~8 ） for those discriminants of real quadratic fields Q（√ d） with d= N^2-4 and h（d）=1.

关键词： quadratic field class number discriminant Zeta-function lower bound