Misplaced Pages

Taniyama's problems: Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editNext edit →Content deleted Content addedVisualWikitext
Revision as of 16:26, 27 December 2024 editGregariousMadness (talk | contribs)Extended confirmed users1,409 editsNo edit summary← Previous edit Revision as of 16:27, 27 December 2024 edit undoGregariousMadness (talk | contribs)Extended confirmed users1,409 editsNo edit summaryNext edit →
Line 5: Line 5:
}} }}


'''Taniyama's problems''' are a set of 36 mathematical problems posed by ] ] ] in 1955. The problems primarily focused on ], ], and the connections between ]s and ]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> '''Taniyama's problems''' are a set of 36 mathematical problems posed by ] ] ] in 1955. The problems primarily focused on ], ], and the connections between ]s and ]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>


== History == == History ==
During the 1955 international symposium on ] at ] and ], 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 ]. During the 1955 international symposium on ] at ] and ], 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 ].<ref name="Mazur"/>


The most influential Taniyama's problems led to the formulation of the ] (now known as the ]), which states that every elliptic curve over the rational numbers is ]. This conjecture became central to modern number theory and played a crucial role in ]' ] of ] in 1995. The most influential Taniyama's problems led to the formulation of the ] (now known as the ]), which states that every elliptic curve over the rational numbers is ]. This conjecture became central to modern number theory and played a crucial role in ]' ] of ] in 1995.<ref name="Mazur"/>


Taniyama's problems influenced the development of modern ] and ], including the ], the theory of ]s, and the study of ]s. Taniyama's problems influenced the development of modern ] and ], including the ], the theory of ]s, and the study of ]s.<ref name="Mazur"/>


== Problems == == Problems ==
Line 20: Line 20:
{{Math proof|title=Taniyama's twelfth problem (translated)|proof=Let <math>C</math> be an ] defined over an ] <math>k</math>, and <math>L_C(s)</math> the ] 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 ] is true for <math>\zeta_C(s)</math>, then the ] obtained from <math>L_C(s)</math> by the inverse ] must be an ] of dimension -2 of a special type (see ]{{efn|The reference to ] 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 ]s not ] with it.}}). If so, it is very plausible that this form is an ellipic ] 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>}} {{Math proof|title=Taniyama's twelfth problem (translated)|proof=Let <math>C</math> be an ] defined over an ] <math>k</math>, and <math>L_C(s)</math> the ] 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 ] is true for <math>\zeta_C(s)</math>, then the ] obtained from <math>L_C(s)</math> by the inverse ] must be an ] of dimension -2 of a special type (see ]{{efn|The reference to ] 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 ]s not ] with it.}}). If so, it is very plausible that this form is an ellipic ] 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>}}


In particular, fellow Japanese mathematician ] noted that Taniyama's formulation in his twelfth problem was unclear: the proposed ] method would only work for elliptic curves over ].<ref name="Shimura"/> For curves over ]s, the situation is substantially more complex and remains unclear even at a conjectural level today. In particular, fellow Japanese mathematician ] noted that Taniyama's formulation in his twelfth problem was unclear: the proposed ] method would only work for elliptic curves over ].<ref name="Shimura"/> For curves over ]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 ''Stufe''{{clarify|date=December 2024}} <math>N</math>, and especially to decompose the ] <math>J</math> of this ] into simple factors up to ]. Also it is well known that if <math>N = q</math>, a ], 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>?}} {{Math proof|title=Taniyama's thirteenth problem (translated)|proof=To characterize the field of elliptic modular functions of ''Stufe''{{clarify|date=December 2024}} <math>N</math>, and especially to decompose the ] <math>J</math> of this ] into simple factors up to ]. Also it is well known that if <math>N = q</math>, a ], 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 == == See also ==
Line 34: Line 34:
== References == == References ==
{{reflist}} {{reflist}}
*{{citation | last=Mazur|first=B.|title=Number Theory as Gadfly|journal=The American Mathematical Monthly|volume=98|number=7|pages=593–610|year=1991}}


] ]

Revision as of 16:27, 27 December 2024

Set of 36 mathematics problems posed by Yutaka Taniyama
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Taniyama's problems" – news · newspapers · books · scholar · JSTOR (December 2024) (Learn how and when to remove this message)
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (December 2024) (Learn how and when to remove this message)
(Learn how and when to remove this message)

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

This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2024) (Learn how and when to remove this message)

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

Taniyama's twelfth problem (translated)

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

Taniyama's thirteenth problem (translated)

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

See also

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

References

  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
  2. ^ Mazur, B. (1991), "Number Theory as Gadfly", The American Mathematical Monthly, 98 (7): 593–610
  3. Taniyama, Yutaka (1956), "Problem 12", Sugaku (in Japanese), 7: 269
Categories: