This book arises from the conference "Elliptic Curves and Modular Forms in Arithmetic Geometry, Celebrating Massimo Bertolini's 60th birthday" held in Milano in September 2022. Massimo Bertolini is one of the most influential number theorists of the last 30 years, whose results and ideas have been a source of inspiration for many mathematicians working in the fascinating area of the Birch and Swinnerton-Dyer conjecture, a Millennium Problem of the Clay Institute. The beauty of the subject, combined with the deep mathematics involved, attracts some of the most brilliant mathematicians in all the world. The book of Massimo Bertolini opened the way to study these problems, using several different techniques, especially of p-adic nature, and is recognized as a leading mathematician in this area. Because of the special position of Massimo in this area of number theory, many influential mathematicians attended the conference in Milano and the Summer School in Essen in his honor on the occasion of his 60th birthday.