Modular elliptic curve

About: Modular elliptic curve is a research topic. Over the lifetime, 1959 publications have been published within this topic receiving 57111 citations.

Advanced Topics in the Arithmetic of Elliptic Curves

Abstract: In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. This book continues the study of elliptic curves by presenting six important, but somewhat more specialized topics: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-N ron classification of special fibres, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.

Modular elliptic curves and Fermat’s Last Theorem

Abstract: When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

Speeding the Pollard and elliptic curve methods of factorization

Abstract: Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1 and Monte Carlo methods. More recently, Williams published a p + 1 method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of p ? 1 and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with n/2 + o(n) multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3 1og2 n multiplications for arbitrary P.

Modular Functions and Dirichlet Series in Number Theory

Abstract: This is the second volume of a 2-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years The second volume presupposes a background in number theory com¬ parable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis Most of the present volume is devoted to elliptic functions and modular functions with some of their number-theoretic applications Among the major topics treated are Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular functionj( r), and Hecke's theory of entire forms with multiplicative Fourier coefficients The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series Both volumes of this work emphasize classical aspects of a subject wh ich in recent years has undergone a great deal of modern development It is hoped that these volumes will help the nonspecialist become acquainted with an important and fascinating part of mathematics and, at the same time, will provide some of the background that belongs to the repertory of every specialist in the field This volume, like the first, is dedicated to the students who have taken this course and have gone on to make notable contributions to number theory and other parts of mathematics T M A January, 1976

Modular curves and the Eisenstein ideal

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