Analogy between arithmetic of elliptic curves and conics
نویسنده
چکیده
In this brief note we bring out the analogy between the arithmetic of elliptic curves and the Riemann zeta-function. · · · The aim of this note is to point out the possibility of developing the theory of elliptic curves so that the arithmetical and analytic aspects are developed in strict analogy with the classical theory of the Riemann zeta-function. This has been triggered by Lemmermeyer’s perceptive analysis of conics [3-6]. The Riemann zeta-function has two fundamental representations. ∞ ∑
منابع مشابه
Conics - a Poor Man’s Elliptic Curves
Introduction 2 1. The Group Law on Pell Conics and Elliptic Curves 2 1.1. Group Law on Conics 2 1.2. Group Law on Elliptic curves 3 2. The Group Structure 3 2.1. Finite Fields 3 2.2. p-adic Numbers 3 2.3. Integral and Rational Points 4 3. Applications 4 3.1. Primality Tests 4 3.2. Factorization Methods 5 4. 2-Descent 5 4.1. Selmer and Tate-Shafarevich Group 5 4.2. Heights 6 5. Analytic Methods ...
متن کاملELLIPTIC FUNCTIONS AND ELLIPTIC CURVES (A Classical Introduction)
(0.0) Elliptic curves are perhaps the simplest 'non-elementary' mathematical objects. In this course we are going to investigate them from several perspectives: analytic (= function-theoretic), geometric and arithmetic. Let us begin by drawing some parallels to the 'elementary' theory, well-known from the undergraduate curriculum. Elementary theory This course arcsin, arccos elliptic integrals ...
متن کاملHigher Descent on Pell Conics. Iii. the First 2-descent
In [Lem2003b] we have sketched the historical development of problems related to Legendre’s equations ar−bs = 1 and the associated Pell equation x−dy = 1 with d = ab. In [Lem2003c] we discussed certain “non-standard” ideas to solve the Pell equation. Now we move from the historical to the modern part: below we will describe the theory of the first 2-descent on Pell conics and explain its connec...
متن کاملHypergeometric Series and Periods of Elliptic Curves
In [7], Greene introduced the notion of general hypergeometric series over finite fields or Gaussian hypergeometric series, which are analogous to classical hypergeometric series. The motivation for his work was to develop the area of character sums and their evaluations through parallels with the theory of hypergeometric functions. The basis for this parallel was the analogy between Gauss sums...
متن کاملOn the Tate-shafarevich Group of a Number Field
For an elliptic curve E defined over a fieldK, the Tate-Shafarevich group X(E/K) encodes important arithmetic and geometric information. An important conjecture of Tate and Shafarevich states X(E/K) is always finite. Supporting this conjecture is a cohomological analogy between Mordell-Weil groups of elliptic curves and unit groups of number fields. In this note, we follow the analogy to constr...
متن کامل