Taniyama's problems
The problems primarily focused on algebraic geometry, number theory, and the connections between modular forms and elliptic curves.
[1][2][3] 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.
[3] During the 1955 international symposium on algebraic number theory at Tokyo and Nikkō—the first symposium of its kind to be held in Japan that was attended by international mathematicians including Jean-Pierre Serre, Emil Artin, Andre Weil, Richard Brauer, K. G. Ramanathan, and Daniel Zelinsky[4]—Taniyama compiled his 36 problems in a document titled "Problems of Number Theory" and distributed mimeographs of his collection to the symposium's participants.
[2][5] Serre later brought attention to these problems in the early 1970s.
This conjecture became central to modern number theory and played a crucial role in Andrew Wiles' proof of Fermat's Last Theorem in 1995.
[2][5] Taniyama's problems influenced the development of modern number theory and algebraic geometry, including the Langlands program, the theory of modular forms, and the study of elliptic curves.
[2] 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.
in a suitable manner, we can obtain a system of Erich Hecke's L-series with Größencharakter
of Hecke to Hilbert modular functions (cf.
[3] 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.
be an elliptic curve defined over an algebraic number field
by the inverse Mellin transformation must be an automorphic form of dimension -2 of a special type (see Hecke[a]).
If so, it is very plausible that this form is an ellipic differential of the field of associated automorphic functions.
Now, going through these observations backward, is it possible to prove the Hasse-Weil conjecture by finding a suitable automorphic form from which
[6][3] 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.
[1][2] 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.
[1] For curves over number fields, the situation is substantially more complex and remains unclear even at a conjectural level today.
[2] To characterize the field of elliptic modular functions of level
of this function field into simple factors up to isogeny.
