Editing Goro Shimura (section)

==Research==
Shimura was a colleague and a friend of [[Yutaka Taniyama]], with whom he wrote the first book on the [[abelian variety of CM-type|complex multiplication of abelian varieties]] and formulated the Taniyama–Shimura conjecture.<ref>{{cite journal| last1=Shimura | first1=Goro | title=Yutaka Taniyama and his time. Very personal recollections | doi=10.1112/blms/21.2.186 | mr=976064 | year=1989 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=21 | issue=2 | pages=186–196| doi-access=free }}</ref> Shimura then wrote a long series of major papers, extending the phenomena found in the theory of [[complex multiplication|complex multiplication of elliptic curves]] and the theory of [[modular form]]s to higher dimensions (e.g. Shimura varieties). This work provided examples for which the equivalence between [[Motivic L-function|motivic]] and [[Automorphic L-function|automorphic ''L''-functions]] postulated in the [[Langlands program]] could be tested: [[automorphic form]]s realized in the [[cohomology]] of a Shimura variety have a construction that attaches [[Galois representation]]s to them.<ref name="langlands-shimura">{{cite book|title=Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics|publisher=Chelsea Publishing Company|editor-last1=Borel|editor-first1=Armand|editor-link1=Armand Borel|editor-last2=Casselman|editor-first2=William|editor-link2=Bill Casselman (mathematician)|last=Langlands|first=Robert|authorlink=Robert Langlands|year=1979|volume=XXXIII Part 1
|chapter-url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/autoreps-ps.pdf|chapter=Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen|pages=205–246}}</ref>

In 1958, Shimura generalized the initial work of [[Martin Eichler]] on the [[Eichler–Shimura congruence relation]] between the [[local zeta function|local ''L''-function]] of a [[modular curve]] and the eigenvalues of [[Hecke operators]].<ref>{{cite journal| last1=Shimura | first1=Goro | title=Correspondances modulaires et les fonctions ζ de courbes algébriques | doi=10.2969/JMSJ/01010001 |mr=0095173 | year=1958 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=10 | pages=1–28| doi-access=free }}</ref><ref>{{cite book |first=Ilya |last=Piatetski-Shapiro |authorlink=Ilya Piatetski-Shapiro |chapter=Zeta functions of modular curves |title=Modular functions of one variable II |year=1972 |location=Antwerp |series=Lecture Notes in Mathematics |volume=349 |pages=317–360}}</ref> In 1959, Shimura extended the work of Eichler on the [[Eichler–Shimura isomorphism]] between Eichler cohomology groups and spaces of [[cusp form]]s which would be used in [[Pierre Deligne]]'s proof of the [[Weil conjectures]].<ref>{{cite journal| last1=Shimura | first1=Goro | title=Sur les intégrales attachées aux formes automorphes | doi=10.2969/jmsj/01140291 |mr=0120372 | year=1959 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=11 | issue=4 | pages=291–311| doi-access=free }}</ref><ref>{{cite book| last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Séminaire Bourbaki vol. 1968/69 Exposés 347-363 | url=http://www.numdam.org/item?id=SB_1968-1969__11__139_0 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-05356-9 | doi=10.1007/BFb0058801 | year=1971 | volume=179 | chapter=Formes modulaires et représentations l-adiques }}</ref>

In 1971, Shimura's work on explicit [[class field theory]] in the spirit of [[Hilbert's twelfth problem|Kronecker's Jugendtraum]] resulted in his proof of [[Shimura's reciprocity law]].<ref>{{cite book| last1=Shimura | first1=Goro | authorlink=Goro Shimura | title=Introduction to the arithmetic theory of automorphic functions | publisher=Iwanami Shoten | location=Tokyo | series=Publications of the Mathematical Society of Japan | year=1971 | volume=11 | zbl=0221.10029 }}</ref> In 1973, Shimura established the [[Shimura correspondence]] between modular forms of half integral weight ''k''+1/2, and modular forms of even weight 2''k''.<ref>{{cite journal| last1=Shimura | first1=Goro | title=On modular forms of half integral weight | jstor=1970831 | mr=0332663 | year=1973 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=97 | issue=3 | pages=440–481 | doi=10.2307/1970831}}</ref>

Shimura's formulation of the Taniyama–Shimura conjecture (later known as the modularity theorem) in the 1950s played a key role in the proof of Fermat's Last Theorem by [[Andrew Wiles]] in 1995. In 1990, [[Kenneth Ribet]] proved [[Ribet's theorem]] which demonstrated that Fermat's Last Theorem followed from the semistable case of this conjecture.<ref>{{cite journal|first=Kenneth |last=Ribet |url=http://www.numdam.org/item?id=AFST_1990_5_11_1_116_0 |title=From the Taniyama-Shimura conjecture to Fermat's last theorem |journal=Annales de la Faculté des Sciences de Toulouse |series=Série 5 |volume=11 |issue=1 |year=1990 |pages=116–139 |doi=10.5802/afst.698|doi-access=free }}</ref> Shimura dryly commented that his first reaction on hearing of [[Andrew Wiles]]'s proof of the semistable case was 'I told you so'.<ref>{{cite web|url=https://www.pbs.org/wgbh/nova/proof/ |title=Nova Episode: The Proof|website=[[PBS]]}}</ref>