Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

https://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

The Arithmetic of Elliptic Curves

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

A Course in Computational Algebraic Number Theory

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.

/pdf/modular-elliptic-curves-and-fermat-s-last-theorem-2n9m9l6b36.pdf

Modular elliptic curves and Fermat’s Last Theorem

Propriétés galoisiennes des points d'ordre fini des courbes elliptiques

Introduction to the Arithmetic Theory of Automorphic Functions

The recent development of the theory of modular forms and associated zeta functions, together with all its arithmetic significance, is quite pleasing, and our knowledge in this field is evergrowing, but the forms of half integral weight have attracted only casual attention, in spite of their importance and ancientness. Indeed, the connection of such forms with zeta functions was never clarified. When Hecke developed his theory of Euler product forthe forms of integral weight, he pointed out the impossibility of a similar theory for the forms of half integral weight, and that only partial information could be obtained for the Fourier coefficients of such forms (Werke, p. 639). He explained this in more detail in his last paper [3], which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases. A treatment of a more general type of modular form was given by Wohlfahrt [12]. In fact, he defined Hecke operators whose degree is the square of a prime, and showed a certain multiplicative relation, as predicted by Hecke, for the Fourier coefficients, but discussed neither Euler product, nor connection with zeta functions. In the present paper, we try to reveal a more affirmative aspect of the subject. To be specific, put, for each positive integer N,

On Modular Forms of Half Integral Weight

The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function 

$$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty {e^{2\pi in^2 z} } $$

and the Dedekind eta function 

$$ \eta \left( z \right) = e^{\pi iz/12} \prod _{n = 1}^\infty (1 - e^{2\pi inz} ) $$

are classical examples of such forms. (For some practical reasons, we take $ e^{2\pi in^2 z} $ instead of the usual $ e^{\pi in^2 z} $ in the definition of θ.) In fact, the function θ satisfies 

$$ \theta (\gamma (z)) = j(\gamma ,z)\theta (z)for all\gamma \in \Gamma _0 (4) $$

(1.1)

with 

$$ j([\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} ],z) = (\frac{c} {d})\varepsilon _d^{ - 1} (cz + d)^{1/2}. $$

(1.2)

.

Goro Shimura

Papers

Introduction to the Arithmetic Theory of Automorphic Functions

On Modular Forms of Half Integral Weight

Modular Forms of Half Integral Weight

On the Holomorphy of Certain Dirichlet Series

The special values of the zeta functions associated with cusp forms