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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.^[1]^[2]^[3]

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.^[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. These problems would become well known in mathematical folklore.^[2]^[5] Serre later brought attention to these problems in the early 1970s.^[3]

The most famous of Taniyama's problems are his twelfth and thirteenth problems.^[3]^[2] These 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.^[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]

The problems

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.^[3]

Taniyama's tenth problem (translated)

Let $k$ be a totally real number field, and $F(\tau )$ be a Hilbert modular form to the field $k$ . Then, choosing $F(\tau )$ in a suitable manner, we can obtain a system of Erich Hecke's L-series with Größencharakter $\lambda$ , which corresponds one-to-one to this $F(\tau )$ by the process of Mellin transformation. This can be proved by a generalization of the theory of operator $T$ of Hecke to Hilbert modular functions (cf. Hermann Weyl).^[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.^[3]

Taniyama's twelfth problem (translated)

Let $C$ be an elliptic curve defined over an algebraic number field $k$ , and $L_{C}(s)$ the L-function of $C$ over $k$ in the sense that $\zeta _{C}(s)=\zeta _{k}(s)\zeta _{k}(s-1)/L_{C}(s)$ is the zeta function of $C$ over $k$ . If the Hasse–Weil conjecture is true for $\zeta _{C}(s)$ , then the Fourier series obtained from $L_{C}(s)$ 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 $L_{C}(s)$ can be obtained?^[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]

Taniyama's thirteenth problem (translated)

To characterize the field of elliptic modular functions of level $N$ , and especially to decompose the Jacobian variety $J$ of this function field into simple factors up to isogeny. Also it is well known that if $N=q$ , a prime, and $q\equiv 3{\pmod {4}}$ , then $J$ contains elliptic curves with complex multiplication. What can one say for general $N$ ?^[1]

Notes

^ 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 ${\text{SL}}_{2}(\mathbb {Z} )$ but also some Fuchsian groups not commensurable with it.

References

^ ^a ^b ^c ^d 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
^ ^a ^b ^c ^d ^e ^f ^g Mazur, B. (1991), "Number Theory as Gadfly", The American Mathematical Monthly, 98 (7): 593–610
^ ^a ^b ^c ^d ^e ^f ^g ^h Lang, Serge (1995), "Some History of the Shimura-Taniyama Conjecture", Notices of the AMS, 42 (11): 1301–1307
^ Proceedings of the International Symposium on Algebraic Number Theory, The Organizing Committee International Symposium on Algebraic Number Theory, 1955
^ ^a ^b "Taniyama-Shimura Conjecture". Wolfram MathWorld. Retrieved December 27, 2024.
^ Taniyama, Yutaka (1956), "Problem 12", Sugaku (in Japanese), 7: 269

[6] 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 ${\text{SL}}_{2}(\mathbb {Z} )$ but also some Fuchsian groups not commensurable with it.

[Shimura-1] 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

[Mazur-2] ^ ^a ^b ^c ^d ^e ^f ^g Mazur, B. (1991), "Number Theory as Gadfly", The American Mathematical Monthly, 98 (7): 593–610

[Lang-3] ^ ^a ^b ^c ^d ^e ^f ^g ^h Lang, Serge (1995), "Some History of the Shimura-Taniyama Conjecture", Notices of the AMS, 42 (11): 1301–1307

[4] Proceedings of the International Symposium on Algebraic Number Theory, The Organizing Committee International Symposium on Algebraic Number Theory, 1955

[Wolfram_Mathworld-5] "Taniyama-Shimura Conjecture". Wolfram MathWorld. Retrieved December 27, 2024.

[7] Taniyama, Yutaka (1956), "Problem 12", Sugaku (in Japanese), 7: 269

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[2]

[3]

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-[[File:Colorized upscaled photograph of Yutaka Taniyama.jpg|thumb|right|Colorized upscaled photograph of [[Yutaka Taniyama]]]]
 {{short description|Set of 36 mathematics problems posed by Yutaka Taniyama}}
-'''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 form]]s and [[elliptic curve]]s.
+'''Taniyama's problems''' are a set of 36 [[list of unsolved problems in mathematics|mathematical problems]] posed by [[Japanese people|Japanese]] [[mathematician]] [[Yutaka Taniyama]] in 1955. The problems primarily focused on [[algebraic geometry]], [[number theory]], and the connections between [[modular form]]s and [[elliptic curve]]s.<ref name="Shimura">{{citation |  last1=Shimura | first1=Goro | title=Yutaka Taniyama and his time. Very personal recollections | doi=10.1112/blms/21.2.186 | year=1989 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=21 | issue=2 | pages=186–196 |  mr=976064| doi-access=free }}</ref><ref name="Mazur">{{citation | last=Mazur|first=B.|title=Number Theory as Gadfly|journal=The American Mathematical Monthly|volume=98|number=7|pages=593–610|year=1991}}</ref><ref name="Lang">{{citation | last=Lang|first=Serge|title=Some History of the Shimura-Taniyama Conjecture|journal=Notices of the AMS|pages=1301-1307|year=1995|volume=42|number=11}}</ref>
 == History ==
+{{See also|Modularity theorem#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 &ndash; these problems would become well-known in [[mathematical folklore]].
+[[File:Jean-Pierre Serre 2003.jpg|right|thumb|French mathematician [[Jean-Pierre Serre]], a participant in the 1955 international symposium, brought attention to Taniyama's problems in the early 1970s.]]
+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]].<ref name="Lang"/> During the 1955 international symposium on [[algebraic number theory]] at [[Tokyo]] and [[Nikkō]]&mdash;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 (mathematician)|Daniel Zelinsky]]<ref>{{citation | title=Proceedings of the International Symposium on Algebraic Number Theory|publisher=The Organizing Committee International Symposium on Algebraic Number Theory|year=1955}}</ref>&mdash;Taniyama compiled his 36 problems in a document titled ''"Problems of Number Theory"'' and distributed [[mimeograph]]s of his collection to the symposium's participants. These problems would become well known in [[mathematical folklore]].<ref name="Mazur"/><ref name="Wolfram Mathworld">{{cite web|url=https://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html|title=Taniyama-Shimura Conjecture|website=Wolfram MathWorld|access-date=December 27, 2024}}</ref> Serre later brought attention to these problems in the early 1970s.<ref name="Lang"/>
-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]]' [[Wiles's proof of Fermat's Last Theorem|proof]] of [[Fermat's Last Theorem]] in 1995.
+The most famous of Taniyama's problems are his twelfth and thirteenth problems.<ref name="Lang"/><ref name="Mazur"/> These 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]]' [[Wiles's proof of Fermat's Last Theorem|proof]] of [[Fermat's Last Theorem]] in 1995.<ref name="Mazur"/><ref name="Wolfram Mathworld"/>
-Taniyama's problems influenced the development of modern [[number theory]] and [[algebraic geometry]], including the [[Langlands program]], the theory of [[modular form]]s, and the study of [[elliptic curve]]s.
+Taniyama's problems influenced the development of modern [[number theory]] and [[algebraic geometry]], including the [[Langlands program]], the theory of [[modular form]]s, and the study of [[elliptic curve]]s.<ref name="Mazur"/>
-== Problems ==
+== The problems ==
+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 [[Jean-Pierre Serre|Serre]] and [[André Weil]], it was only formally published in Japanese in Taniyama's collected works.<ref name="Lang"/>
-The most famous of Taniyama's problems are his twelfth and thirteenth problems.
-{{Math proof|title=Taniyama's twelfth problem (translated)|proof=Let <math>C</math> be an [[elliptic curve]] defined over an [[algebraic number field]] <math>k</math>, and <math>L_C(s)</math> the [[L-function|''L''-function]] of <math>C</math> over <math>k</math> in the sense that <math>\zeta_C(s) = \zeta_k(s) \zeta_k(s-1) /L_C(s)</math> is the zeta function of <math>C</math> over <math>k</math>. If the [[Hasse–Weil conjecture]] is true for <math>\zeta_C(s)</math>, then the [[Fourier series]] obtained from <math>L_C(s)</math> by the inverse [[Mellin transformation]] must be an [[automorphic form]] of dimension -2 of a special type (see [[Erich Hecke|Hecke]]{{efn|The reference to [[Erich Hecke|Hecke]] in Problem 12 was to his paper, "fiber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung", which involves not only congruence subgroups of <math>\text{SL}_2(\mathbb Z)</math> but also some [[Fuchsian group]]s not [[commensurability (mathematics)|commensurable]] with it.}}). 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 <math>L_C(s)</math> can be obtained?}}
+{{Math proof|title=Taniyama's tenth problem (translated)|proof=Let <math>k</math> be a totally real [[number field]], and <math>F(\tau)</math> be a [[Hilbert modular form]] to the field <math>k</math>. Then, choosing <math>F(\tau)</math> in a suitable manner, we can obtain a system of [[Erich Hecke]]'s ''L''-series with [[Größencharakter]] <math>\lambda</math>, which [[one-to-one correspondence|corresponds one-to-one]] to this <math>F(\tau)</math> by the process of [[Mellin transformation]]. This can be proved by a generalization of the theory of operator <math>T</math> of [[Hecke operator|Hecke]] to [[Hilbert modular functions]] (cf. [[Hermann Weyl]]).<ref name="Lang"/>}}
+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.<ref name="Lang"/>
-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 field]]s, the situation is substantially more complex and remains unclear even at a conjectural level today.
+{{Math proof|title=Taniyama's twelfth problem (translated)|proof=Let <math>C</math> be an [[elliptic curve]] defined over an [[algebraic number field]] <math>k</math>, and <math>L_C(s)</math> the [[L-function|''L''-function]] of <math>C</math> over <math>k</math> in the sense that <math>\zeta_C(s) = \zeta_k(s) \zeta_k(s-1) /L_C(s)</math> is the zeta function of <math>C</math> over <math>k</math>. If the [[Hasse–Weil conjecture]] is true for <math>\zeta_C(s)</math>, then the [[Fourier series]] obtained from <math>L_C(s)</math> by the inverse [[Mellin transformation]] must be an [[automorphic form]] of dimension -2 of a special type (see [[Erich Hecke|Hecke]]{{efn|The reference to [[Erich Hecke|Hecke]] in Problem 12 was to his paper, "fiber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung", which involves not only congruence subgroups of <math>\text{SL}_2(\mathbb Z)</math> but also some [[Fuchsian group]]s not [[commensurability (mathematics)|commensurable]] with it.}}). 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 <math>L_C(s)</math> can be obtained?<ref>{{citation|last=Taniyama|first=Yutaka |journal=Sugaku|volume=7|page=269|year=1956|title=Problem 12|language=ja}}</ref><ref name="Lang"/>}}
-{{Math proof|title=Taniyama's thirteenth problem (translated)|proof=To characterize the field of elliptic modular functions of ''Stufe''{{cfn|date=December 2024}} <math>N</math>, and especially to decompose the [[Jacobian variety]] <math>J</math> of this [[function field of an algebraic variety|function field]] into simple factors up to [[isogeny]]. Also it is well known that if <math>N = q</math>, a [[prime number|prime]], and <math>q \equiv 3 \pmod 4</math>, then <math>J</math> contains elliptic curves with complex multiplication. What can one say for general <math>N</math>?}}
+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]].<ref name="Shimura"/><ref name="Mazur"/> 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.
-== Notes ==
-{{notelist}}
+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]].<ref name="Shimura"/> For curves over [[number field]]s, the situation is substantially more complex and remains unclear even at a conjectural level today.<ref name="Mazur"/>
+{{Math proof|title=Taniyama's thirteenth problem (translated)|proof=To characterize the field of elliptic modular functions of [[Modular elliptic curve#Modularity theorem|level]] <math>N</math>, and especially to decompose the [[Jacobian variety]] <math>J</math> of this [[function field of an algebraic variety|function field]] into simple factors up to [[isogeny]]. Also it is well known that if <math>N = q</math>, a [[prime number|prime]], and <math>q \equiv 3 \pmod 4</math>, then <math>J</math> contains elliptic curves with complex multiplication. What can one say for general <math>N</math>?<ref name="Shimura"/>}}
 == See also ==
 * [[Hilbert's problems]]
+* [[Thurston's 24 questions]]
 * [[List of unsolved problems in mathematics]]
 * [[Wiles's proof of Fermat's Last Theorem]]
+== Notes ==
+{{notelist}}
 == References ==
+{{reflist}}
-*{{citation |  last1=Shimura | first1=Goro | title=Yutaka Taniyama and his time. Very personal recollections | doi=10.1112/blms/21.2.186 | year=1989 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=21 | issue=2 | pages=186–196 |  mr=976064| doi-access=free }}
-*{{citation|last=Taniyama|first=Yutaka |journal=Sugaku|volume=7|page=269|year=1956|title=Problem 12|language=ja}}
-*{{citation | last=Mazur|first=B.|title=Number Theory as Gadfly|journal=The American Mathematical Monthly|volume=98|number=7|pages=593-610|year=1991}}
 [[Category:Number theory]]