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Goro Shimura
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Gorō Shimura (志村 五郎, Shimura Gorō; 23 February 1930 – 3 May 2019) was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry.[1] He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.

Key Information

Biography

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Gorō Shimura was born in Hamamatsu, Japan, on 23 February 1930.[2] Shimura graduated with a B.A. in mathematics and a D.Sc. in mathematics from the University of Tokyo in 1952 and 1958, respectively.[3][2]

After graduating, Shimura became a lecturer at the University of Tokyo, then worked abroad — including ten months in Paris and a seven-month stint at Princeton's Institute for Advanced Study — before returning to Tokyo, where he married Chikako Ishiguro.[4][2] He then moved from Tokyo to join the faculty of Osaka University, but growing unhappy with his funding situation, he decided to seek employment in the United States.[4][2] Through André Weil he obtained a position at Princeton University.[4] Shimura joined the Princeton faculty in 1964 and retired in 1999, during which time he advised over 28 doctoral students and received the Guggenheim Fellowship in 1970, the Cole Prize for number theory in 1977, the Asahi Prize in 1991, and the Steele Prize for lifetime achievement in 1996.[1][5]

Shimura described his approach to mathematics as "phenomenological": his interest was in finding new types of interesting behavior in the theory of automorphic forms. He also argued for a "romantic" approach, something he found lacking in the younger generation of mathematicians.[6] Shimura used a two-part process for research, using one desk in his home dedicated to working on new research in the mornings and a second desk for perfecting papers in the afternoon.[2]

Shimura had two children, Tomoko and Haru, with his wife Chikako.[2] Shimura died on 3 May 2019 in Princeton, New Jersey at the age of 89.[1][2]

Research

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Shimura was a colleague and a friend of Yutaka Taniyama, with whom he wrote the first book on the complex multiplication of abelian varieties and formulated the Taniyama–Shimura conjecture.[7] Shimura then wrote a long series of major papers, extending the phenomena found in the theory of complex multiplication of elliptic curves and the theory of modular forms to higher dimensions (e.g. Shimura varieties). This work provided examples for which the equivalence between motivic and automorphic L-functions postulated in the Langlands program could be tested: automorphic forms realized in the cohomology of a Shimura variety have a construction that attaches Galois representations to them.[8]

In 1958, Shimura generalized the initial work of Martin Eichler on the Eichler–Shimura congruence relation between the local L-function of a modular curve and the eigenvalues of Hecke operators.[9][10] In 1959, Shimura extended the work of Eichler on the Eichler–Shimura isomorphism between Eichler cohomology groups and spaces of cusp forms which would be used in Pierre Deligne's proof of the Weil conjectures.[11][12]

In 1971, Shimura's work on explicit class field theory in the spirit of Kronecker's Jugendtraum resulted in his proof of Shimura's reciprocity law.[13] In 1973, Shimura established the Shimura correspondence between modular forms of half integral weight k+1/2, and modular forms of even weight 2k.[14]

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.[15] Shimura dryly commented that his first reaction on hearing of Andrew Wiles's proof of the semistable case was 'I told you so'.[16]

Other interests

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His hobbies were shogi problems of extreme length and collecting Imari porcelain. The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelain is a non-fiction work about the Imari porcelain that he collected over 30 years that was published by Ten Speed Press in 2008.[2][17]

Works

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Mathematical books

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  • Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Tokyo: The Mathematical Society of Japan, MR 0125113 Later expanded and published as Shimura (1997)
  • Shimura, Goro (1968). Automorphic Functions and Number Theory. Lecture Notes in Mathematics, Vol. 54 (Paperback ed.). Springer. ISBN 978-3-540-04224-2.
  • Shimura, Goro (1 August 1971). Introduction to the Arithmetic Theory of Automorphic Functions (Paperback ed.). Princeton University Press. ISBN 978-0-691-08092-5. - It is published from Iwanami Shoten in Japan.[18]
  • Shimura, Goro (1 July 1997). Euler Products and Eisenstein Series. CBMS Regional Conference Series in Mathematics (Paperback ed.). American Mathematical Society. ISBN 978-0-8218-0574-9.
  • Shimura, Goro (1997). Abelian Varieties with Complex Multiplication and Modular Functions (Hardcover ed.). Princeton University Press. ISBN 978-0-691-01656-6.[19] An expanded version of Shimura & Taniyama (1961).
  • Shimura, Goro (22 August 2000). Arithmeticity in the Theory of Automorphic Forms. Mathematical Surveys and Monographs (Paperback ed.). American Mathematical Society. ISBN 978-0-8218-2671-3.[20]
  • Shimura, Goro (1 March 2004). Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups. Mathematical Surveys and Monographs (Hardcover ed.). American Mathematical Society. ISBN 978-0-8218-3573-9.
  • Shimura, Goro (2007). Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Hardcover ed.). Springer. ISBN 978-0-387-72473-7.
    • Shimura, Goro (28 December 2009). Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Paperback ed.). Springer New York. ISBN 978-1-4419-2478-0.
  • Shimura, Goro (15 July 2010). Arithmetic of Quadratic Forms. Springer Monographs in Mathematics (Hardcover ed.). Springer. ISBN 978-1-4419-1731-7.

Non-fiction

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Collected papers

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References

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Goro Shimura

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Gōrō Shimura (February 23, 1930 – May 3, 2019) was a Japanese mathematician renowned for his foundational contributions to modern , particularly in the fields of automorphic forms, modular forms, and elliptic curves. His pioneering work, including the formulation of the Shimura–Taniyama–Weil conjecture in 1955 (later refined), established deep connections between elliptic curves and modular forms, providing the theoretical basis for Andrew Wiles's 1994 proof of . Over a career spanning more than six decades, Shimura authored over 100 scholarly papers and several influential monographs, shaping arithmetic geometry and influencing applications in . Born in , , as the youngest of five children, Shimura grew up in a family that relocated frequently due to his father's banking career, settling in by 1933. His early education was interrupted by , including the destruction of his family home in a bombing and postwar food shortages, but he excelled academically, entering the in 1949 after attending elite preparatory schools. He earned a B.A. in 1952 and a D.Sc. in 1958 from the , where he studied under influential mentors such as and was exposed to . Shimura's career began as an assistant at the in 1952, progressing to lecturer in 1954 and associate professor in 1957; he then served as a full professor at starting in 1961. In 1962, he joined , where he became the Michael Henry Strater University Professor of and remained until his retirement in 1999, continuing as emeritus professor thereafter. During stays in in the early 1950s, influenced by figures like , , and , he advanced theories on modular function fields, Fuchsian groups, and complex multiplication of abelian varieties. His seminal 1961 book with , Complex Multiplication of Abelian Varieties and Its Applications to , and the 1971 Introduction to the Arithmetic Theory of Automorphic Functions remain cornerstone texts in the field. Among his major achievements, Shimura developed the theory of Shimura varieties—higher-dimensional analogs of modular curves that unify arithmetic and geometric structures—and explored L-functions associated with automorphic forms. He received prestigious honors, including the Frank Nelson Cole Prize in Number Theory from the in 1977, the Asahi Prize in 1991, and the Leroy P. Steele Prize for Lifetime Achievement in 1996. In his 2008 autobiography, The Map of My Life, Shimura reflected on his trajectory from and in his early research to transformative insights in arithmetic theory. He was survived by his wife Chikako (who died in 2023), daughter Tomoko, and son Haru.

Early Life and Education

Childhood and Family

Goro Shimura was born on February 23, 1930, in , , . His family moved frequently due to his father's employment at a bank, which involved transfers between branches. In March 1933, when Shimura was three years old, the family relocated to , settling in the Ward near ancestral lands from the , where his forebears had served as retainers. His mother hailed from as the third of four daughters, and the family maintained a modest, stable home environment despite the economic and political turbulence of pre-war ; Shimura later described his early years from 1933 to 1938 as happy ones in an old-fashioned one-story house. He was the youngest of five children, including at least one brother who worked during air raids and several sisters, contributing to a close-knit where Shimura received an indulgent upbringing that encouraged his . Shimura's early education began in April 1936 at a local elementary school near a in the Ushigome area of , where he enjoyed reading aloud but struggled with and manual arts. He graduated from elementary school in March 1942 amid Japan's deepening involvement in , which soon disrupted his studies. That April, he enrolled at the Fourth Prefectural , facing wartime challenges including labor mobilizations from to August 1945, factory work producing fighter plane parts, and frequent air raids that forced the family into shelters. These experiences, including witnessing destruction and deprivation over more than a decade, marked his adolescence; the war's end on August 15, 1945, brought relief but left lasting impressions of hardship. Initially disinterested in the standard curriculum, which he found routine, Shimura's curiosity was sparked through self-study during the war years, as he independently explored advanced topics by reading books borrowed from libraries. Family influences played a subtle role in nurturing Shimura's intellectual growth, with his parents providing stability and encouragement despite concerns over his slight build and reserved nature. There were no direct familial ties to mathematics—his father's banking career and mother's homemaking offered practical rather than academic inspiration—but the home environment fostered a love for learning through access to books and a supportive atmosphere amid adversity. This early self-directed engagement laid the groundwork for his later academic pursuits, transitioning into formal university studies after the war.

Academic Training in Japan

Goro Shimura enrolled at the in 1949, beginning his undergraduate studies in the Department of Mathematics shortly after the end of . There, he studied under prominent mathematicians including Shokichi Iyanaga, whose lectures on axiomatic geometry profoundly shaped Shimura's early mathematical perspective, introducing him to rigorous approaches in and related fields, and , whose course on he particularly enjoyed. Shimura found the standard curriculum somewhat repetitive but persisted, completing his bachelor's degree in mathematics in 1952. Following his undergraduate graduation, Shimura continued his graduate studies at the , focusing on advanced topics in and . The mentorship of Iyanaga and Iwasawa guided Shimura toward deeper explorations in algebraic structures and their arithmetic properties. In 1958, Shimura earned his (D.Sc.) degree. This work reflected the blend of and prevalent in the Japanese mathematical community at the time. During his graduate years, Shimura gained early exposure to through participation in seminars and emerging collaborations within Japan's mathematical community. He attended key events, such as the 1953 conference on and organized by Yasuo Akizuki, where he delivered an impromptu talk that marked his entry into active discourse. These interactions, alongside seminars led by Iyanaga and Iwasawa, introduced him to contemporary problems in the field. Additionally, Shimura began initial collaborative work on abelian varieties, notably with , exploring their arithmetic properties and complex multiplication aspects, which would inform his later research trajectory.

Professional Career

Early Appointments in Japan

After completing his doctoral training at the University of Tokyo, Goro Shimura transitioned to independent research roles within Japan's academic landscape. Upon graduating with a B.A. in 1952, he was appointed as an assistant at the College of General Education of the , where he began teaching and pursuing research in . In 1954, Shimura advanced to the position of lecturer at the , delivering courses in linear algebra and while publishing his initial papers on abelian varieties. In 1957, Shimura was promoted to associate professor at the University of Tokyo, a role that allowed greater focus on advanced research amid the institution's emphasis on and . He completed his D.Sc. degree there in 1958. During this time, he engaged in significant collaborations, notably with , co-authoring Modern Number Theory (in Japanese) in 1957, which synthesized contemporary developments in the field. Their partnership extended to the 1961 publication Complex Multiplication of Abelian Varieties and Its Applications to , a seminal work that built on shared interests in abelian varieties despite the challenges of the era. In 1961, he became a full at . The post-war research environment in posed substantial hurdles for mathematicians like Shimura, characterized by severe resource shortages and infrastructural deficits in universities recovering from wartime destruction. Food and limited access to international literature hindered progress, yet a vibrant focus on persisted, exemplified by the 1955 International Symposium on Algebraic Number Theory held in and Nikko, which fostered key exchanges among Japanese scholars. Shimura's work during these years reflected this resilient intellectual climate, prioritizing theoretical advancements in a setting of constrained material support.

Career at Princeton University

Goro Shimura arrived at in September 1962 as a visiting professor, following his earlier positions at the and , where his emerging work in had garnered international attention and led to an invitation facilitated by . He transitioned to a full professorship in the Department of Mathematics in 1964, marking the beginning of a distinguished tenure that spanned over three decades. During his time at Princeton, Shimura maintained a rigorous schedule, delivering undergraduate and graduate courses known for their precision, detail, and subtle humor, often alternating between introductory lectures and advanced seminars. He organized specialized seminars, such as a private discussion group on zeta functions and abelian varieties, fostering deep engagement with complex topics in . Additionally, he supervised 28 PhD students, many of whom became prominent researchers, and advised numerous senior theses, emphasizing direct interaction and research guidance over formal lecturing. In 1991, Shimura was appointed the Michael Henry Strater University Professor of Mathematics, a prestigious endowed chair he held until his retirement in 1999 at age 69. He continued as professor emeritus thereafter, remaining active in the scholarly community. Shimura resided in , throughout his career and until his death on May 3, 2019, at the age of 89.

Mathematical Contributions

Automorphic Forms and Hecke Theory

Goro Shimura developed a comprehensive framework for on the general linear group GL(2) over number fields, extending classical modular forms to an arithmetic setting. In this theory, an automorphic form on GL(2) over a number field F is a smooth function φ on the adelic group GL(2, 𝔸_F), where 𝔸_F is the adele ring of F, satisfying left invariance under GL(2, F), right invariance under a compact open K of GL(2, 𝔸_F), and specific transformation properties: at finite places, it factors through a finite-dimensional representation, and at archimedean places, it transforms under a finite-dimensional of GL(2, ℝ) (or more generally over ℂ). Cusp forms are those automorphic forms that vanish at the cusps, meaning their constant term in the Fourier expansion along the unipotent radical is zero. These forms exhibit key properties such as having Whittaker models, admitting Fourier expansions with coefficients tied to additive characters, and forming Hecke-invariant spaces under operator actions. Shimura extended Hecke theory from classical holomorphic modular forms to these automorphic representations by defining Hecke operators in the adelic context. For an unramified place v of F, the Hecke operator T_v acts on the space of automorphic forms by integration over double cosets GL(2, F_v) g GL(2, F_v) ∩ K, where g runs over matrices with the uniformizer at v. The ℋ, generated by these operators for all unramified places, is commutative and acts on the space of cusp forms 𝒜_0(K), with eigenforms diagonalizing this action. Normalized eigen-cusp forms, or newforms, have eigenvalues λ_v that are algebraic integers, and the algebra decomposes into products over irreducible representations. This extension preserves multiplicativity, allowing the construction of Euler products for associated L-functions. A central result in Shimura's framework is the reciprocity law, often referred to as the Shimura-Deligne reciprocity, which links the arithmetic of these automorphic forms to Galois representations. Specifically, for a cuspidal automorphic representation π on GL(2, 𝔸_F) that is a Hecke eigenform with eigenvalues λ_p for unramified primes p of F, there exists a continuous, semisimple, 2-dimensional representation ρ_π,λ : Gal(\bar{F}/F) → GL_2(\bar{ℚ}_ℓ) (for ℓ not dividing the conductor), such that the central character of π corresponds to the determinant of ρ_π,λ, and for unramified p, trace(ρ_π,λ(Frob_p)) = λ_p. This establishes a precise reciprocity between the Hecke eigenvalues and the traces of Frobenius elements in the Galois group. The proof sketch proceeds in two parts: first, Shimura's construction of the automorphic side ensures the eigenvalues satisfy the necessary algebraic and analytic properties; second, Deligne's realization via motives or étale cohomology attaches the Galois representation, generalizing the Eichler-Shimura isomorphism for weight 2 forms—where the representation arises from H^1 of the modular curve with coefficients in a local system—to higher weights by embedding the form into a motive whose cohomology yields ρ_π,λ, with compatibility under base change. Shimura applied this theory to associated with automorphic forms, proving their analytic properties. The L(s, π) for a cuspidal automorphic representation π on GL(2, 𝔸_F) is defined as an Euler product ∏_v L(s, π_v)^{-1} over places v of F, where local factors L(s, π_v) are degree-2 polynomials determined by the Satake parameters at unramified finite places and Gamma factors at archimedean places. Shimura established that L(s, π) admits meromorphic continuation to the entire and satisfies a L(s, π) = ε π(s) L(1-s, \tilde{π}), where \tilde{π} is the contragredient and ε is a root number of 1, using representations involving Poincaré series and unfolding techniques to relate it to zeta functions. These results provide arithmetic insights, such as bounds on eigenvalues via the Ramanujan conjecture (conjectured by Shimura and proved in special cases). This framework briefly connects to elliptic curves, where weight-2 cusp forms correspond to motives of elliptic curves over F via the .

Shimura Varieties and Complex Multiplication

Shimura varieties are algebraic varieties that arise as quotients of Hermitian symmetric domains by arithmetic subgroups, serving as moduli spaces for polarized abelian varieties with additional structures such as level-KK structures. These domains are associated with the symmetric space attached to a reductive algebraic group GG over Q\mathbb{Q}, where the quotient encodes arithmetic data from automorphic forms. A is formally defined via a Shimura datum (G,X,D)(G, X, D), where GG is a reductive algebraic group over Q\mathbb{Q}, XX is a G(R)G(\mathbb{R})-conjugacy class of homomorphisms h:S→GRh: S \to G_{\mathbb{R}} with S=\ResC/RGmS = \Res_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m the Deligne torus satisfying certain axioms (SV1–SV4), and DD is a discrete subgroup of G(Af)G(\mathbb{A}_f) (often a neat arithmetic subgroup). The variety \ShK(G,X)\Sh_K(G, X) for a compact open subgroup K⊂G(Af)K \subset G(\mathbb{A}_f) is the quotient Γ\H\Gamma \backslash \mathcal{H}, where H\mathcal{H} is the Hermitian symmetric domain parametrized by XX, and Γ=D∩G(Q)\Gamma = D \cap G(\mathbb{Q}). This construction ensures the varieties have canonical models over the reflex field E(G,X)E(G, X), the field of definition of the conjugacy class c(X)c(X). Complex multiplication (CM) on abelian varieties, as developed by Shimura and Taniyama, refers to the action of a CM field EE—a totally imaginary quadratic extension of a totally real field—on the endomorphism algebra of an abelian variety AA. The endomorphism ring \End0(A)\End^0(A) is a CM field of degree 2dim⁡A2 \dim A over Q\mathbb{Q}, with the Rosati involution induced by complex conjugation on EE. For AA of CM type (E,Φ)(E, \Phi), where Φ\Phi is a subset of embeddings E↪CE \hookrightarrow \mathbb{C} with ∣Φ∣=dim⁡A|\Phi| = \dim A, the ring OE\mathcal{O}_E acts on the tangent space \Tgt0(A)≅CΦ\Tgt_0(A) \cong \mathbb{C}^\Phi, satisfying the trace condition \Tr(i(a)∣\Tgt0(A))=∑ϕ∈Φϕ(a)\Tr(i(a) \mid \Tgt_0(A)) = \sum_{\phi \in \Phi} \phi(a) for a∈OEa \in \mathcal{O}_E. This action extends to cohomology: H1(A,Q)H^1(A, \mathbb{Q}) forms a 1-dimensional EE-vector space, preserved by the Hodge structure, with the CM type defining a cocharacter μΦ:Gm→(Gm)E/Q\mu_\Phi: \mathbb{G}_m \to ( \mathbb{G}_m )_{E/\mathbb{Q}} acting on H1(A,Q)⊗RH^1(A, \mathbb{Q}) \otimes \mathbb{R} by z↦(z,…,z,z‾,…,z‾)z \mapsto (z, \dots, z, \overline{z}, \dots, \overline{z}) on the decomposition according to Φ\Phi and its complement. The absolute Galois group acts via EE-linear isogenies α:A→σA\alpha: A \to {}^\sigma A for σ∈\Aut(C/E∗)\sigma \in \Aut(\mathbb{C}/E^*), satisfying α(NΦ∗(s)⋅x)=σx\alpha(N_\Phi^*(s) \cdot x) = \sigma x for torsion points xx and s∈AE∗,f×s \in A_{E^*,f}^\times, where NΦ∗N_\Phi^* is the norm ideal. Shimura-Taniyama theory expresses the zeta function of such AA as a product of Hecke LL-series, linking CM points on Shimura varieties to special arithmetic loci. A key result is Shimura's theorem on the algebraicity of special values of automorphic LL-functions at CM points: for a CM abelian variety AA over a number field KK with CM type (E,Φ)(E, \Phi), the central value L(A,1)L(A, 1) is algebraic up to a rational multiple involving periods of AA. More generally, for an on a corresponding to a CM point xx, the value at xx is algebraic (if finite) and satisfies an explicit under the Galois action over the reflex field E(x)E(x). This algebraicity ties to the periods of differentials on AA and critical values of associated zeta functions, with the Frobenius at a prime of good reduction satisfying \ordv(π)/\ordv(q)=∣Φ∩Hv∣/∣Hv∣\ord_v(\pi)/\ord_v(q) = |\Phi \cap H_v| / |H_v| for the decomposition HvH_v of embeddings.

Modularity Theorem and Elliptic Curves

Goro Shimura, in collaboration with , formulated the during the 1950s, which posits that every defined over the rational numbers Q\mathbb{Q} is modular. Specifically, the asserts that for any EE over Q\mathbb{Q}, there exists a cusp form ff of weight 2 on the Γ0(N)\Gamma_0(N), where NN is the conductor of EE, such that the of EE coincides with the associated to ff. This association links the arithmetic geometry of to the analytic properties of modular forms, providing a bridge between Diophantine equations and automorphic representations. Shimura's contributions were instrumental in rigorizing the , particularly through his development of the theory of complex multiplication for abelian varieties, which extended the framework to higher dimensions. Shimura established lifting theorems that associate modular forms to abelian varieties of higher dimension, generalizing the correspondence for elliptic curves. In particular, his work shows that a Hecke eigenform of weight 2 and level NN gives rise to a simple abelian variety over Q\mathbb{Q} whose endomorphism algebra is determined by the field's properties, with the dimension of the variety related to the multiplicity of the form. These lifting constructions, building on the Eichler-Shimura relation, allow for the realization of modular forms as cohomology classes on abelian varieties, facilitating the study of their arithmetic invariants. Through these theorems, Shimura provided tools to lift representations from elliptic modular forms to those arising from Jacobians of higher genus curves or abelian surfaces. Shimura's modularity framework played a pivotal role in ' proof of between 1994 and 1995. Wiles demonstrated the for semistable elliptic s over Q\mathbb{Q}, relying on Shimura's earlier associations between elliptic curves and weight-2 cusp forms to reduce the problem to properties of Galois representations and deformation theory. This partial proof sufficed to establish that no nontrivial solutions exist for xn+yn=znx^n + y^n = z^n with n>2n > 2, as the Frey curve attached to such a solution would contradict the modularity conjecture in its semistable case. Shimura's foundational insights into the correspondence ensured the robustness of the link between the elliptic curve's arithmetic and the modular form's . In the arithmetic of s, Shimura advanced the understanding of conductors, j-invariants, and Galois representations via Tate modules. He determined that the conductor of an with complex is tied to the of its endomorphism ring, providing explicit computations for CM cases that influence the level of the associated . Regarding j-invariants, Shimura's for s with complex classifies them as singular moduli, linking their values to ideals in imaginary quadratic fields and facilitating primes. For Galois representations, Shimura showed that the action of Gal(Q‾/Q)\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) on the Tate module Tℓ(E)T_\ell(E) of an EE yields a 2-dimensional representation whose trace matches the coefficients of the corresponding , establishing a deep arithmetic connection.

Recognition and Honors

Major Awards and Prizes

Shimura received the Frank Nelson Cole Prize in Number Theory in 1977 from the , recognizing his fundamental contributions to automorphic forms. The prize cited his two papers: "Class fields over real quadratic fields and Hecke operators" (1972) and "On modular forms of half integral weight" (1973). In 1991, during his tenure at , Shimura was awarded the Asahi Prize by for his outstanding contributions to and broader impact on . The honored Shimura with the Leroy P. Steele Prize for Lifetime Achievement in 1996, celebrating his extensive body of work in arithmetic algebraic geometry. The award specifically acknowledged the seminal role of his concepts in advancing modern .

Fellowships and Memberships

Shimura was awarded a in 1970 to support his research on automorphic functions. This fellowship enabled him to advance his studies in and during a pivotal period in his career. He was a member of the and the Mathematical Society of Japan, reflecting his international standing in the mathematical community. Shimura was elected to the in 1983. He was an honorary member of the London Mathematical Society and a foreign member of the . Shimura served as a visiting member at the Institute for Advanced Study in Princeton on five occasions, where he collaborated with leading mathematicians and contributed to key developments in automorphic forms. These visits underscored his role in fostering global mathematical exchange.

Personal Life and Interests

Family and Later Years

Goro Shimura married Chikako Ishiguro in 1959 in Tokyo, after knowing her for six years. The couple had two children, daughter Tomoko and son Haru, and maintained a long-term residence in Princeton, New Jersey, where Shimura lived with his family until his death. Following his retirement from in 1999, Shimura continued his scholarly pursuits, authoring numerous books and papers on and related topics. In 2008, he published his , The Map of My Life, which offered humorous and reflective accounts of his personal and professional journey from wartime to his career in the United States. That same year, he also released The Story of Imari, a book on Japanese that drew from his personal collection. His post-retirement routine in Princeton involved methodical writing, often allowing drafts to mature before finalizing them. In his later years, Shimura balanced ongoing mathematical work with personal interests, maintaining an active intellectual life in Princeton. He passed away on May 3, 2019, at his home in Princeton from natural causes at the age of 89.

Hobbies and Non-Mathematical Pursuits

Outside of his mathematical endeavors, Goro Shimura pursued several creative and cultural interests that reflected his Japanese heritage and appreciation for intricate craftsmanship. One of his primary hobbies was , the traditional Japanese variant of chess played on a 9×9 board, which he enjoyed deeply as a strategic pastime. Shimura also dedicated significant time to collecting Imari porcelain, a renowned style of Japanese export ware originating from the region in during the . Over three decades, he amassed a notable collection, focusing on pieces characterized by their blue underglaze and vibrant overglaze enamels depicting motifs such as birds, flowers, and landscapes. His passion extended beyond mere acquisition; Shimura delved into the historical and aesthetic dimensions of , exploring its production techniques, symbolic imagery, and cultural significance in and trade history. This culminated in his 2008 publication, The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelain, a richly illustrated volume that serves as both a personal catalog and an accessible introduction to the artistry of these antiques. In his later years, Shimura occasionally reflected on how these pursuits contributed to his work-life balance, providing respite from rigorous academic demands.

Selected Works

Mathematical Books and Monographs

Goro Shimura authored several influential mathematical monographs that advanced the fields of number theory, automorphic forms, and algebraic geometry. His works are characterized by rigorous expositions that bridge classical and modern techniques, often providing foundational treatments of complex topics. One of Shimura's early collaborative efforts was Complex Multiplication of Abelian Varieties and Its Applications to (1961), co-authored with and published by the Mathematical Society of Japan. This monograph develops the theory of complex multiplication (CM) for abelian varieties over the complex numbers, extending classical results on elliptic curves to higher dimensions, and applies these ideas to , including the construction of ray class fields via CM abelian varieties. The book laid essential groundwork for later advancements in arithmetic geometry, influencing studies of L-functions and reciprocity laws associated with CM varieties. In 1971, Shimura published Introduction to the Arithmetic Theory of Automorphic Functions with Princeton University Press, offering a systematic introduction to automorphic forms on the general linear group GL(2) over the rationals. The text covers the arithmetic aspects of modular forms, Hecke operators, and their connections to L-functions and Eisenstein series, building from basic definitions to advanced topics like the Eichler-Shimura isomorphism. This work became a standard reference for understanding the interplay between automorphic representations and number-theoretic applications, particularly in the context of the Langlands program. Shimura's Abelian Varieties with Complex Multiplication and Modular Functions (, 1998), an expanded edition based on his 1961 collaboration with Taniyama, provides a comprehensive treatment of CM theory in the context of moduli spaces. It explores the moduli interpretations of abelian varieties with CM, the role of functions in constructing these varieties, and generalizations of classical reciprocity laws using modular functions and period matrices. The edition incorporates updates on arithmetic applications, solidifying its status as a key resource for reciprocity in higher dimensions. Earlier, in Automorphic Functions and Number Theory (Springer Lecture Notes in Mathematics, Vol. 10, 1968), Shimura delivered a concise introduction to automorphic functions and their links to , emphasizing transformations and analytic continuations in the context of adelic groups. This short monograph, based on lectures, highlights the beauty of the subject through examples connecting automorphic forms to zeta functions and class numbers. Shimura's later monograph Arithmetic of Quadratic Forms (Springer Monographs in Mathematics, 2010) examines the arithmetic properties of quadratic forms over number fields, including , mass formulas, and connections to orthogonal groups. It draws on Shimura's contributions to exact formulas for genera and classes of forms, providing tools for applications in modular forms and zeta functions. These monographs complement Shimura's collected papers, which compile his research articles on related themes.

Non-Mathematical Writings and Collected Papers

In addition to his extensive mathematical oeuvre, Goro Shimura authored works that delved into personal and cultural realms, reflecting his broader intellectual curiosities. His 2008 autobiography, The Map of My Life, published by Springer, offers an intimate collection of memories spanning his early life in , academic journey, and philosophical musings on mathematics. Shimura recounts influences from mentors such as , with whom he collaborated closely on foundational ideas in , and reflects on the cultural and personal challenges of pursuing in post-war . The book eschews a linear , instead weaving anecdotal stories that illuminate his development as a and his views on the intuitive, almost aesthetic nature of mathematical discovery. Shimura's interest in Japanese art manifested in The Story of Imari: The Symbols and Mysteries of Antique Porcelain, published in 2008 by . This richly illustrated volume explores the historical and cultural significance of Imari porcelain, a renowned style of Japanese ceramics originating from the Arita region in during the . Drawing from his personal hobby of collecting antique ceramics, Shimura examines the symbolic motifs—such as dragons, phoenixes, and floral patterns—that encode , imperial symbolism, and export influences from Dutch traders in the 17th and 18th centuries. The work highlights the technical artistry of overglaze enameling and the socio-economic context of production, connecting it to broader themes of Japanese craftsmanship and global . Shimura also oversaw the compilation of his research into the Collected Papers series, published by Springer across four volumes in the early 2000s, encompassing 103 of his key mathematical contributions from 1954 to 2001. Volume I (1954–1966) gathers early papers on arithmetical geometry, automorphic forms, and zeta functions of algebraic varieties, including seminal works on abelian varieties and cohomology. Subsequent volumes—II (1967–1977), III (1978–1988), and IV (1989–2001)—cover advancements in modular forms, L-functions, and arithmeticity of automorphic representations, with each including editorial notes by Shimura providing historical context and clarifications on evolving ideas. Shimura personally selected the papers and authored the annotations to aid readers in tracing the progression of his research, emphasizing the interconnectedness of his contributions to number theory. These volumes serve as a comprehensive archival resource, distinct from his technical monographs by prioritizing chronological accessibility over thematic exposition.

Legacy and Influence

Impact on Number Theory

Shimura's foundational contributions to the established automorphic forms as a bridge between and Galois groups, providing essential tools for reciprocity laws that connect spectral data in to arithmetic structures. His work on Shimura varieties, developed over decades, linked automorphic representations to Galois representations, particularly for GL(2), and influenced the through the Shimura–Taniyama–Weil conjecture, which associates elliptic curves to modular forms. This framework, crediting Shimura for much of its origins, enabled the global classification of automorphic representations and supported proofs of the local Langlands correspondence for GL(n). In , Shimura's theories underpin construction methods via complex multiplication and modularity, ensuring secure protocols by generating curves with prescribed point counts over finite fields. Shimura reciprocity, integral to class polynomials and j-invariants, facilitates efficient algorithms for ordinary elliptic curves suitable for cryptographic applications, such as those reducing coefficient sizes by factors like 72 using specific modular functions. These techniques, building on Shimura's reciprocity laws, have been implemented in practical tools for creating cryptographically strong curves. Shimura varieties advanced arithmetic by serving as higher-dimensional analogs of modular curves, acting as moduli spaces for abelian varieties with polarizations, endomorphisms, and level structures, thus generalizing classical elliptic modular . Defined through Shimura data involving reductive groups over Q\mathbb{Q} and Hermitian symmetric domains, these varieties enable canonical models over number fields and influence p-adic methods via good reduction at primes, facilitating the study of Galois representations, zeta functions, and point counting over finite fields using p-adic integrals. This structure has become central to the , connecting automorphic forms to motives and arithmetic invariants. Post-2019, Shimura's work continues to receive extensive citations in on special values of L-functions, particularly through applications to cycles on Shimura varieties and twisted Asai L-functions via base change in unitary settings. For instance, recent studies leverage his insights on CM abelian varieties and modular forms to link Weil representations with , informing geometric series and Fourier-Jacobi cycles. Similarly, constructions of p-adic L-functions for unitary groups build directly on Shimura's theories of automorphic forms and , extending earlier frameworks to new differential operators and measures.

Students, Collaborations, and Tributes

Shimura supervised 28 doctoral students at over his career, many of whom went on to become prominent figures in . Notable advisees include Alice Silverberg, Don Blasius, Paul Garrett, and Jacob Sturm, who advanced key areas in and arithmetic geometry through their subsequent work. Jonathan Hanke, Shimura's final graduate student in 1999, credited him with teaching the importance of independently finding research problems. Shimura's most significant collaboration was with , a close friend and colleague from their time in , resulting in joint authorship of foundational texts like Modern (1957, in Japanese) and Complex Multiplication of Abelian Varieties and Its Applications to (1961). Together, they refined the Taniyama-Shimura conjecture in the 1950s, positing a deep connection between elliptic curves and modular forms that later underpinned Andrew Wiles's 1994 proof of . Shimura also maintained extensive correspondence with , whose ideas on influenced his early research, and he drew inspiration from mentors like and during his formative years in the 1950s. Following Shimura's death on May 3, 2019, at age 89, tributes from the mathematical community highlighted his profound influence and personal qualities. Robert Gunning, a colleague at Princeton, described him as "a major research mathematician, creative and original," noting the department's deep sense of loss. , another Princeton mathematician, emphasized the enduring impact of Shimura's contributions, stating that "fundamental mathematical truths have long shelf lives," a trait especially evident in his work. In the New York Times obituary, Shimura was remembered as a foundational figure whose insights extended to modern , with his legacy affirmed by the lasting adoption of Shimura varieties in arithmetic geometry.

References

  1. https://www.princeton.edu/news/2019/05/08/goro-shimura-giant-number-theory-dies-89
  2. https://mathshistory.st-andrews.ac.uk/Biographies/Shimura/
  3. https://www.ias.edu/scholars/goro-shimura
  4. https://www.ams.org/notices/202005/rnoti-p677.pdf
  5. https://www.towntopics.com/2023/02/15/obituaries-2-15-2023/
  6. https://link.springer.com/book/10.1007/978-0-387-79715-1
  7. https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18860
  8. https://www.nytimes.com/2019/05/13/obituaries/goro-shimura-dead.html
  9. https://books.google.com/books/about/Complex_Multiplication_of_Abelian_Variet.html?id=-eHuAAAAMAAJ
  10. https://books.google.com/books/about/The_Map_of_My_Life.html?id=eYuojP7kgvkC
  11. https://www.jmilne.org/math/Documents/TokyoNikko1955.pdf
  12. https://www.math.princeton.edu/people/goro-shimura
  13. https://www.nzdr.ru/data/media/biblio/kolxoz/M/MPop/Shimura%20G.%20The%20map%20of%20my%20life%20%28Springer%2C%202008%29%28ISBN%200387797149%29%28211s%29_MPop_.pdf
  14. https://paw.princeton.edu/article/memoriam-goro-shimura
  15. https://press.princeton.edu/books/paperback/9780691080925/introduction-to-arithmetic-theory-of-automorphic-functions
  16. https://math.bu.edu/people/jsweinst/CEB/BAMS.pdf
  17. https://www.jstor.org/stable/1970507
  18. https://www.jmilne.org/math/xnotes/svi.pdf
  19. https://www.ams.org/journals/bull/2002-39-03/S0273-0979-02-00945-X/S0273-0979-02-00945-X.pdf
  20. https://www.degruyter.com/document/doi/10.1515/9781400883943/html
  21. https://www.ams.org/prizes-awards/pabrowse.cgi?parent_id=15
  22. https://www.asahi.com/corporate/award/asahi/12737983
  23. https://www.ams.org/notices/199207/199207FullIssue.pdf
  24. https://www.ams.org/prizes-awards/pabrowse.cgi?parent_id=25
  25. https://books.google.com/books/about/The_Story_of_Imari.html?id=p9JqaKSNah4C
  26. https://old.maa.org/press/maa-reviews/introduction-to-the-arithmetic-theory-of-automorphic-functions
  27. https://press.princeton.edu/books/hardcover/9780691016566/abelian-varieties-with-complex-multiplication-and-modular-functions
  28. https://www.jstor.org/stable/j.ctt1bpm9xq
  29. https://link.springer.com/book/10.1007/BFb0071097
  30. https://old.maa.org/press/maa-reviews/arithmetic-of-quadratic-forms
  31. https://www.math.columbia.edu/~woit/wordpress/?p=1247
  32. https://www.goodreads.com/book/show/2991641-the-story-of-imari
  33. https://www.abebooks.com/9781580088961/Story-Imari-Symbols-Mysteries-Antique-1580088961/plp
  34. https://link.springer.com/book/9781493918096
  35. https://link.springer.com/book/9781493918324
  36. https://books.google.com/books/about/Collected_Papers_III.html?id=f8gB564cK68C
  37. https://d-nb.info/966423836/04
  38. https://www.math.toronto.edu/arthur/pdf/arthur-the-work-of-robert-langlands.pdf
  39. https://cacr.uwaterloo.ca/conferences/2006/ecc2006/broker.pdf
  40. https://arxiv.org/pdf/2502.16754
  41. https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/padic-lfunctions-for-unitary-groups/E1D22AA60E10A5B5CA5F838EB48467D7
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