Reference. The arithmetic of elliptic curves [silverman2009ec]

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel’s theorem on finiteness of integral points.

@book{silvermanArithmeticEllipticCurves2009,
  title = {The Arithmetic of Elliptic Curves},
  author = {Silverman, Joseph H.},
  year = 2009,
  series = {Graduate Texts in Mathematics},
  volume = {106},
  publisher = {Springer New York},
  address = {New York, NY},
  doi = {10.1007/978-0-387-09494-6},
  isbn = {978-0-387-09493-9 978-0-387-09494-6},
}