Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

On the Equation xp + y q = z r : A Survey

Let p, q and r be three integers 2. During the last ten years many new ideas have emerged for the study of the Diophantine equation x + y = z. This study is divided into three parts according whether (p, q, r) = is negative, zero or positive. For instance, if we have (p, q, r) < 0, this study is closely connected to the theory of modular representations of dimension 2 in finite characteristic. The approach of associating to this equation Galois representations given by the division points of elliptic curves is very efficient. The aim of this paper is to present this circle of ideas and the known results concerning this equation.     

The smallest solutions to the diophantine equation

In this paper we discuss a method used to find the smallest nontrivial positive integer solutions to . The method, which is an improvement over a simple brute force approach, can be applied to search the solution to similar equations involving sixth, eighth and tenth powers.

     

Modular Arithmetic on Elements of Small Norm in Quadratic Fields

We describe an algorithm which rapidly computes the coefficients of elements of small norm in quadraticfields modulo a positive integer. Our method requires that an approximation of the natural logarithm of thatquadratic field element is known to sufficient accuracy. To demonstrate the efficiency and utility of our method,we apply it to eliminate a number of exceptional cases of a theorem of Dujella and Peth [9]involving Diophantine triples. In particular, we are able to show that Theorem 1.2 of [9] isunconditionally true for all k 100 with the possible exception of k = 37, for whichthe theorem holds under the assumption of the Extended Riemann Hypothesis.     

Nonexistence conditions of a solution for the congruence

We obtain nonexistence conditions of a solution for of the congruence , where , and are integers, and is a prime power. We give nonexistence conditions of the form for , , , , , and of the form for , , , . Furthermore, we complete some tables concerned with Waring's problem in -adic fields that were computed by Hardy and Littlewood.

     

The Diophantine equation

If and are given positive integers with , then we show that the equation of the title possesses at most one solution in positive integers . Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.

     

Evaluation of discrete logarithms in a group of

We show that to solve the discrete log problem in a subgroup of order of an elliptic curve over the finite field of characteristic one needs operations in this field.

     

The Problem on Class Numbers of Quadratic Number Fields

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

也谈实二次域类数的可除性

设ｄ无平方因子，ｈ（ｄ）是二次域Ｑ（ｄ）的类数，本文证明了：若１＋４ｋ２ｎ＝ｄａ２，ａ，ｋ＞１，ｎ＞２为正整数，且ａ＜０．９ｋ３５ｎ或ｎ的奇素因子ｐ和ｋ的素因子ｑ均适合（ｐ，ｑ－１）＝１，则除（ａ，ｄ，ｋ，ｎ）＝（５，４１，２，４）以外，ｈ（ｄ）≡０（ｍｏｄｎ）．同时，我们猜测：上述结果中的条件（ｐ，ｑ－１）＝１是不必要的．     

On the Diophantine equation

Let be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if and , then the equation , and the equation , have no integral solutions respectively. Here is th Bernoulli number.

     

Solving constrained Pell equations

Consider the system of Diophantine equations , , where is a given integer polynomial. Historically, such systems have been analyzed by using Baker's method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases and , which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

     

On Eisenstein series and

In this paper, we derive some new identities satisfied by the series using Ramanujan's identities for , and . Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.

     

Mod representations of elliptic curves

Explicit equations are given for the elliptic curves (in characteristic ) with mod representation isomorphic to that of a given one.

     

Iwasawa invariants and class numbers of quadratic fields for the prime

Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.

     

On the largest element in D(n)-quadruples

Let $n$ be a nonzero integer. A set of nonzero integers $\{a_1,\dots ,a_m\}$ such that $a_i{a}_j+n$ is a perfect square for all $1\le i<j\le m$ is called a $D\left(n\right)$-$m$-tuple. In this paper, we consider the question, for a given integer $n$ which is not a perfect square, how large and how small can be the largest element in a $D\left(n\right)$-quadruple. We construct families of $D\left(n\right)$-quadruples in which the largest element is of order of magnitude ${\left|n\right|}^3$, resp. ${\left|n\right|}^{2∕5}$.     

Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers

Using Poonen's version of the ``weak vertical method' we produce new examples of ``large' and ``small' rings of algebraic numbers (including rings of integers) where and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.

     

On searching for solutions of the Diophantine equation

We propose an efficient search algorithm to solve the equation for a fixed value of 0$">. By parametrizing , this algorithm obtains and (if they exist) by solving a quadratic equation derived from divisors of . Thanks to the use of several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of below 1000 for which no solution had previously been found. We found three new integer solutions for and 931 in the range of .

     

关于二次域理论求解一类Diophantus方程的整数解

本文研究了两个典型Diophantus方程在实二次域中整数解的问题.利用二次域中的理论和二次代数整数环中算术基本定理,得到了该类方程的一般解法和在实二次域中的所有整数解的相关结论,推广了文献[1]和[2]的结果.     

Polynomial factorization over

We describe algorithms for polynomial factorization over the binary field , and their implementation. They allow polynomials of degree up to to be factored in about one day of CPU time, distributing the work on two processors.