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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.

History

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 – these problems would become well-known in mathematical folklore.

The most influential Taniyama's problems led to the formulation of the Taniyama–Shimura conjecture (now known as the modularity theorem), which states that every elliptic curve over the rational numbers is modular. This conjecture became central to modern number theory and played a crucial role in Andrew Wiles' proof of Fermat's Last Theorem in 1995.

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.

Problems

The most famous of Taniyama's problems are his twelfth and thirteenth problems.

Let ${\displaystyle C}$ be an elliptic curve defined over an algebraic number field ${\displaystyle k}$, and ${\displaystyle L_{C}(s)}$ the L-function of ${\displaystyle C}$ over ${\displaystyle k}$ in the sense that ${\displaystyle \zeta _{C}(s)=\zeta _{k}(s)\zeta _{k}(s-1)/L_{C}(s)}$ is the zeta function of ${\displaystyle C}$ over ${\displaystyle k}$. If the Hasse–Weil conjecture is true for ${\displaystyle \zeta _{C}(s)}$, then the Fourier series obtained from ${\displaystyle L_{C}(s)}$ by the inverse Mellin transformation must be an automorphic form of dimension -2 of a special type (see Hecke). 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 ${\displaystyle L_{C}(s)}$ can be obtained?

In particular, 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.

To characterize the field of elliptic modular functions of Stufe ${\displaystyle N}$, and especially to decompose the Jacobian variety ${\displaystyle J}$ of this function field into simple factors up to isogeny. Also it is well known that if ${\displaystyle N=q}$, a prime, and ${\displaystyle q\equiv 3{\pmod {4}}}$, then ${\displaystyle J}$ contains elliptic curves with complex multiplication. What can one say for general ${\displaystyle N}$?

Notes

1. The reference to Hecke in Problem 12 was to his paper, "fiber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung", which involves not only congruence subgroups of ${\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}$ but also some Fuchsian groups not commensurable with it.

See also

• Hilbert's problems
• List of unsolved problems in mathematics
• Wiles's proof of Fermat's Last Theorem

References

• Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", The Bulletin of the London Mathematical Society, 21 (2): 186–196, doi:10.1112/blms/21.2.186, ISSN 0024-6093, MR 0976064
• Taniyama, Yutaka (1956), "Problem 12", Sugaku (in Japanese), 7: 269
• Mazur, B. (1991), "Number Theory as Gadfly", The American Mathematical Monthly, 98 (7): 593–610

• Number theory
• Algebraic geometry
• Mathematical problems