Taniyama's problems are a set of 36

mathematical problems A mathematical problem is a problem that can be Representation (mathematics), represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the Orbit#Planetary orbits, orbits of the ...

posed by Japanese

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Yutaka Taniyama was a Japanese mathematician known for the Taniyama–Shimura conjecture. Life Taniyama was born on 22 November 1927 in Kisai, a town in Saitama. He was the sixth of eight children born to a doctor's family. He studied at Urawa High School (pre ...

in 1955. The problems primarily focused on

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, and the connections between

modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

s and

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

History

In the 1950s

post-World War II The aftermath of World War II saw the rise of two global superpowers, the United States (U.S.) and the Soviet Union (U.S.S.R.). The aftermath of World War II was also defined by the rising threat of nuclear warfare, the creation and implementati ...

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 Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

Tokyo Tokyo, officially the Tokyo Metropolis, is the capital of Japan, capital and List of cities in Japan, most populous city in Japan. With a population of over 14 million in the city proper in 2023, it is List of largest cities, one of the most ...

and

Nikkō is a Cities of Japan, city in Tochigi Prefecture, Japan. , the city's population was 80,239, in 36,531 households. The population density was 55 persons per km2. The total area of the city is . Nikkō is a popular destination for Japanese and ...

—the first symposium of its kind to be held in

Japan Japan is an island country in East Asia. Located in the Pacific Ocean off the northeast coast of the Asia, Asian mainland, it is bordered on the west by the Sea of Japan and extends from the Sea of Okhotsk in the north to the East China Sea ...

that was attended by international mathematicians including

Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inau ...

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...

, Andre Weil,

Richard Brauer Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation t ...

, K. G. Ramanathan, and Daniel Zelinsky—Taniyama compiled his 36 problems in a document titled ''"Problems of Number Theory"'' and distributed

mimeograph A mimeograph machine (often abbreviated to mimeo, sometimes called a stencil duplicator or stencil machine) is a low-cost duplicating machine that works by forcing ink through a stencil onto paper. The process is called mimeography, and a co ...

s of his collection to the symposium's participants. These problems would become well known in

mathematical folklore In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, folk mathematics, or ...

. 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 (now known as the

modularity theorem In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic c ...

), 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 Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

in 1995. Taniyama's problems influenced the development of modern

and

, including the

Langlands program In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number t ...

, the theory of

s, and the study of

The problems

Taniyama's tenth problem addressed Dedekind zeta functions and Hecke L-series, and while distributed in English at the 1955

conference attended by both Serre and

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...

, it was only formally published in Japanese in Taniyama's collected works. According to

Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...

, Taniyama's eleventh problem deals with

elliptic curves In mathematics, an elliptic curve is a Smoothness, smooth, Projective variety, projective, algebraic curve of Genus of an algebraic curve, genus one, on which there is a specified point . An elliptic curve is defined over a field (mathematics), ...

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

and

modular forms In mathematics, a modular form is a holomorphic function on the Upper half-plane#Complex plane, complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the Group action (mathematics), group action of the ...

. While Taniyama's original formulation was somewhat imprecise, it captured a profound insight that would later be refined into the

. 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 In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used ...

method would only work for elliptic curves over

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...

. For curves over

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

s, the situation is substantially more complex and remains unclear even at a conjectural level today.

Notes

References

{{reflist Number theory Algebraic geometry Mathematical problems

History

The problems

See also

Notes

References