Taniyama's problems

Taniyama's problems are a set of 36 mathematical problems posed by Japanese mathematician Yutaka Taniyama in 1955. The problems primarily focused on algebraic geometry, number theory, and the connections between modular forms and elliptic curves. Taniyama's twelfth and thirteenth problems were the precursor to the Taniyama–Shimura conjecture, also known as the modularity theorem, which would be used in Andrew Wiles' proof of Fermat's Last Theorem in 1995.

History

In the 1950s post-World War II period of mathematics, there was renewed interest in the theory of modular curves due to the work of Taniyama and Goro Shimura. During the 1955 international symposium on algebraic number theory at Tokyo and Nikkō, Taniyama compiled his 36 problems in a document titled "Problems of Number Theory" and distributed mimeographs of his collection to the symposium's participants. Serre later brought attention to these problems in the early 1970s.
The most famous of Taniyama's problems are his twelfth and thirteenth problems. These problems led to the formulation of the Taniyama–Shimura conjecture, which states that every elliptic curve over the rational numbers is Modular [elliptic curve|modular]. This conjecture played a major role in Andrew Wiles' proof of Fermat's Last Theorem in 1995.
Taniyama's problems influenced the development of the Langlands program, the theory of modular forms, and the study of elliptic curves.

Taniyama's tenth problem addressed Dedekind zeta functions and Hecke L-series, and while distributed in English at the 1955 Tokyo-Nikkō conference attended by both Serre and André Weil, it was only formally published in Japanese in Taniyama's collected works.
According to Serge Lang, Taniyama's eleventh problem deals with elliptic curves with complex multiplication, but is unrelated to Taniyama's twelfth and thirteenth problems.
Taniyama's twelfth problem's significance lies in its suggestion of a deep connection between elliptic curves and modular forms. While Taniyama's original formulation was somewhat imprecise, it captured a profound insight that would later be refined into the modularity theorem. The problem specifically proposed that the L-functions of elliptic curves could be identified with those of certain modular forms, a connection that seemed surprising at the time.
Fellow Japanese mathematician Goro Shimura noted that Taniyama's formulation in his twelfth problem was unclear: the proposed Mellin transform method would only work for elliptic curves over rational numbers. For curves over number fields, the situation is substantially more complex and remains unclear even at a conjectural level today.