What Is the Mordell Conjecture (Faltings’s Theorem)?

Diophantine geometry is the field of mathematics that concerns integer solutions and rational solutions of polynomial equations. It is named after Diophantus of Alexandria from around the third century who wrote a series of books called Arithmetica. Diophantine geometry is one of the oldest fields of mathematics, and it continues to be a major field in number theory and arithmetic geometry. If integer solutions and rational solutions are put aside, then polynomial equations determine an algebraic variety. Since around the start of the twentieth century, algebro-geometric methods have played an important role in the study of Diophantine geometry. In 1922, Mordell (Figure 1.1) made a surprising conjecture in a paper where he proved the so-called Mordell–Weil theorem for elliptic curves (see Theorem 3.42). This conjecture, called the Mordell conjecture before Faltings’s proof appeared, states that the number of rational points is finite on any geometrically irreducible algebraic curve of genus at least 2 defined over a number field. It is not certain on what grounds Mordell made this conjecture, but it was audacious at the time, and attracted the attention of many mathematicians. While some partial results were obtained, the Mordell conjecture stood as an unclimbed mountain before the proof by Faltings. Thus, when Faltings (Figure 1.2) proved the Mordell conjecture in a paper published in 1983, the news was circulated around the globe with much enthusiasm. Faltings’s proof was momentous, using sophisticated and profound theories of arithmetic geometry. He proved the Shafarevich conjecture, the Tate conjecture, and the Mordell conjecture concurrently, and he was awarded the Fields Medal in 1986. Nevertheless, first-year students at universities can understand the statement of the Mordell conjecture, except for the notion of genus. Let f (X,Y ) be a two-variable polynomial with coefficients in a number field K (e.g., the field Q of rational numbers). We assume the following: