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Ken Ribet
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Kenneth Alan Ribet (/ˈrɪbɪt/; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fermat's Last Theorem, as well as for his service as President of the American Mathematical Society from 2017 to 2019. He is currently a professor of mathematics at the University of California, Berkeley.

Key Information

Early life and education

[edit]

Kenneth Ribet was born in Brooklyn, New York to parents David Ribet and Pearl Ribet, both Jewish, on June 28, 1948.[1] As a student at Far Rockaway High School, Ribet was on a competitive mathematics team, but his first field of study was chemistry.[2]

Ribet earned his bachelor's degree and master's degree from Brown University in 1969.[3][4] In 1973, Ribet received his Ph.D. from Harvard University under the supervision of John Tate.[3][5][4]

Career

[edit]

After receiving his doctoral degree, Ribet taught at Princeton University for three years before spending two years doing research in Paris.[3] In 1978, Ribet joined the Department of Mathematics at the University of California, Berkeley, where he served three separate terms as supervisor of the department's graduate program, supervisor of the department's undergraduate program, and supervisor of the department's development.[3]

Ribet has served as an editor for several mathematics journals, a book series editor for the Cambridge University Press, and a book series editor for Springer.[3] He also served on the United States National Committee for Mathematics, representing the United States at the International Mathematical Union, and was the Chair of the Mathematics section of the National Academy of Sciences.[3]

From February 1, 2017 to January 31, 2019, Ribet was President of the American Mathematical Society.[6]

Research

[edit]
Further information: Ribet's theorem

Ribet's contributions in number theory and algebraic geometry were described by Benedict Gross and Barry Mazur as being "key to our understanding of the connections between the theory of modular forms and the ℓ-adic representations of the absolute Galois group of the field of rational numbers."[3]

Ribet is credited with paving the way towards Andrew Wiles's proof of Fermat's Last Theorem. In 1986, Ribet proved that the epsilon conjecture formulated by Jean-Pierre Serre was true, and thereby proved that Fermat's Last Theorem would follow from the Taniyama–Shimura conjecture. Crucially it also followed that the full conjecture was not needed, but a special case, that of semistable elliptic curves, sufficed.[7] An earlier theorem of Ribet's, the Herbrand–Ribet theorem, is the converse to Herbrand's theorem on the divisibility properties of Bernoulli numbers and is also related to Fermat's Last Theorem.[8][9][10][11]

Awards and honors

[edit]

Ribet received the Fermat Prize in 1989 jointly with Abbas Bahri.[6][3] He was elected to the American Academy of Arts and Sciences in 1997 and the National Academy of Sciences in 2000.[6][3][4] In 2012, he became a Fellow of the American Mathematical Society.[12][6] In 2017, Ribet received the Brouwer Medal.[13] For 2025 Ribet was awarded the AMS Leroy P. Steele Prize for Seminal Contribution to Research.[14]

In 1988, Ribet was inducted as a vigneron d'honneur by the Jurade de Saint-Émilion.[4] In 1998, Ribet received an honorary doctorate from Brown University.[4]

Personal life

[edit]

Ribet is married to statistician Lisa Goldberg.[15]

References

[edit]
[edit]
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Ken Ribet

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Kenneth Alan Ribet (born June 28, 1948) is an American mathematician specializing in algebraic number theory and algebraic geometry, best known for his pivotal role in the proof of Fermat's Last Theorem through the Herbrand–Ribet theorem and related results on modular forms. Ribet earned his A.B. and A.M. from in 1969 and his Ph.D. from in 1973 under the supervision of John Tate, with a dissertation on Galois actions on division points of abelian varieties. After serving as a lecturer and assistant professor at from 1973 to 1978, he joined the as an in 1977, advancing to full in 1981 and becoming professor emeritus upon retirement in 2022 while continuing as Distinguished Professor of the Graduate School. His research focuses on the arithmetic of modular forms, elliptic curves, and Galois representations, with seminal contributions including the 1976 refinement of Kummer's criterion using modular forms and the 1990 demonstrating that a to for sufficiently large exponents would violate the Taniyama–Shimura conjecture, thereby guiding ' 1995 proof. The Herbrand–Ribet , extending Jacques Herbrand's 1930s work, links the non-triviality of ideal class groups in cyclotomic fields to the existence of irregular primes and has profoundly influenced modern . Ribet has supervised 24 Ph.D. students, including notable mathematicians like , and has held visiting positions in and . Beyond research, Ribet has been a leader in the mathematical community, serving as president of the from 2017 to 2018 and as an editor for prestigious journals including the Proceedings of the . His honors include the 1989 (shared), the 2017 Brouwer Medal, the 2025 Leroy P. Steele Prize for Seminal Contribution to Research from the AMS, election to the (2000) and the American Academy of Arts and Sciences (1997), and an honorary doctorate from (1998).

Early life and education

Childhood and family background

Kenneth Alan Ribet was born on June 28, 1948, in . He grew up in the Rockaway neighborhood of , where his family resided. His parents, and Pearl Ribet, were devoted to fostering academic excellence among their children, creating a supportive home environment that encouraged intellectual pursuits despite the business-oriented circle of adults around them. Ribet attended , where he participated in a selective "maximum learning programme" designed for high-achieving students, offering advanced preparation in , , English, and . Initially, his primary interest lay in chemistry, inspired by an exceptional high who ignited his passion for the subject. During adolescence, Ribet's focus shifted decisively to following a formative summer experience where he encountered advanced concepts such as epsilons and deltas, marking the beginning of his lifelong dedication to the field. This transition propelled him toward higher education in , including his subsequent studies at .

Academic training

Ribet earned his Bachelor of Arts and Master of Arts degrees in mathematics from Brown University in 1969. Although he had initially shown interest in chemistry during high school, his undergraduate focus shifted decisively to mathematics. In the fall of 1969, Ribet began his graduate studies at Harvard University, where he spent the first two years completing coursework in advanced topics, including classes taught by his advisor John Tate and by John Coates. He also attended lectures by visiting scholars such as Bryan Birch and H.P.F. Swinnerton-Dyer on subjects like modular forms and the Birch–Swinnerton-Dyer conjecture. Early in his program, Ribet worked on a project exploring elliptic curves with complex multiplication in relation to the Birch–Swinnerton-Dyer conjecture, before transitioning to research on abelian varieties of GL(2)-type, inspired by Jean-Pierre Serre's recent publications. Under Tate's supervision, Ribet received pivotal guidance that directed his initial research toward ; Tate suggested specific problems, encouraged deep engagement with Serre's work, and even advocated for Ribet to pass his qualifying exams after an initial setback. This mentorship culminated in Ribet's PhD in 1973, with a titled Galois Action on Division Points of Abelian Varieties with Many Real Multiplications.

Academic career

Early positions

Following the completion of his PhD in 1973 at under the supervision of John Tate, Kenneth Ribet began his academic career with positions as a and then at from 1973 to 1978. During this period, Ribet experienced the challenges typical of a young transitioning from graduate student to independent researcher, including initial feelings of insecurity about his readiness for the role amid Princeton's esteemed faculty. He taught undergraduate and graduate courses while developing his research on Galois representations and modular forms, laying the groundwork for his future contributions to . In 1975–1976, Ribet interrupted his Princeton appointment for a research fellowship at the (IHÉS) in , where he collaborated with on foundational work in arithmetic geometry. This year abroad proved intellectually demanding, as Ribet spent much of it deciphering Deligne's dense manuscripts with minimal direct guidance, fostering his expertise in topics like irreducibility of representations in characteristic p. He extended his stay in for research until 1977, producing seminal papers such as "On l-adic representations attached to modular forms" (1975, Inventiones Mathematicae) and "Galois action on division points of Abelian varieties with real multiplications" (1976, American Journal of Mathematics), which explored connections between modular forms and Galois groups. These early positions marked Ribet's emergence as a promising number theorist, with his Paris-based work yielding collaborations and publications that influenced subsequent developments in the field, including aspects of the . Notable among them was a joint paper with Deligne (published 1980 in Inventiones Mathematicae), addressing powers of 2 in p-adic L-functions and building on Ribet's prior investigations into abelian varieties. The period honed his ability to navigate international research environments and solidified his focus on modular representations, despite the personal adjustments required in moving between institutions.

Faculty role at UC Berkeley

In 1977, Kenneth Ribet joined the faculty of the Department of Mathematics at the University of California, Berkeley, as an associate professor, following positions at Princeton University and research time in Paris. He advanced to full professor in 1981 and held that position until his retirement in 2022, after which he became Professor Emeritus of Mathematics while continuing as a Professor in the Graduate School. Throughout his Berkeley career, Ribet contributed significantly to the department's academic environment, balancing research with teaching and service roles. Ribet earned acclaim for his teaching, receiving the UC Berkeley Department of Mathematics' Distinguished Teaching Award in 1985 and again in 2013 for his engaging and effective instruction in advanced undergraduate and graduate courses, such as linear algebra and . His pedagogical approach emphasized clarity and accessibility, making complex topics in approachable for students at various levels. As a mentor, Ribet guided numerous graduate students, advising dozens of early-career learners and supervising PhD dissertations, including that of San Ling on the arithmetic of modular curves in 1990. His mentorship extended to fostering research skills and , often through one-on-one guidance and collaborative projects within the department. Ribet also participated actively in departmental activities, contributing to seminars that advanced discussions in . During his tenure at Berkeley, Ribet served on editorial boards for several prominent mathematics journals and book series, including the Bulletin of the American Mathematical Society and the Proceedings of the National Academy of Sciences, where he acted as a Member Editor in mathematics. These roles involved reviewing manuscripts and shaping publications in algebraic geometry and number theory, enhancing the quality of scholarly output in his field.

Leadership and administrative roles

Ribet served as president of the () from February 1, 2017, to January 31, 2019, following his term as president-elect in 2016–2017. In this role, he led the organization in advancing mathematical research, , and , including initiatives to promote diversity and international collaboration within the field. At the (), Ribet held significant leadership positions in the . He chaired the NAS Section 11 () from 2009 to 2012, overseeing nominations and activities for this section. Later, he served as secretary of Class I (Physical Sciences) from 2015 to 2018 and as chair of Class I from 2018 to 2021, contributing to the governance of physical and within the academy. Additionally, Ribet was a member of the committee that produced the influential 2013 NAS report The Mathematical Sciences in 2025, which outlined strategic recommendations for the growth and impact of in research, education, and national priorities such as and cybersecurity. Ribet has also contributed to mathematical publishing through editorial roles. He serves as a series editor for Springer Monographs in Mathematics, a prestigious book series focused on advanced topics in pure and applied mathematics. Earlier in his career, he edited for several mathematics journals, supporting the dissemination of research in number theory and related areas.

Mathematical research

Foundations in number theory

Ken Ribet's research foundations lie in and , where he explored the interplay between Galois groups, abelian varieties, and arithmetic structures. His doctoral work at , supervised by John Tate, focused on the Galois action on division points of abelian varieties over the rationals, establishing key results on the endomorphism algebras and real multiplications of such varieties. This thesis, published in 1976, connected the geometry of abelian varieties—including elliptic curves as one-dimensional cases—to Galois representations, providing tools for understanding torsion points and their arithmetic implications. A cornerstone of Ribet's early contributions is the Herbrand–Ribet theorem, which serves as the converse to Herbrand's 1932 theorem on the class group of the Q(μp)\mathbb{Q}(\mu_p) for odd primes p>3p > 3. Herbrand showed that the pp-Sylow subgroup AA of the class group decomposes into eigenspaces under the action of (Z/pZ)×(\mathbb{Z}/p\mathbb{Z})^\times, with the eigenspace A(ω1−k)A(\omega^{1-k}) (for even k=2,4,…,p−3k = 2, 4, \dots, p-3) annihilated by the pp-adic L-value L(0,ωk−1)L(0, \omega^{k-1}), where ω\omega is the Teichmüller character. Ribet proved the converse in by constructing unramified pp-extensions of Q(μp)\mathbb{Q}(\mu_p) using modular forms, showing that if L(0,ωk−1)L(0, \omega^{k-1}) is divisible by pp (equivalently, if pp divides the numerator of the Bk/kB_k/k), then the corresponding eigenspace A(ω1−k)A(\omega^{1-k}) is non-trivial. This result refines Kummer's criterion for the regularity of primes, linking the pp-part of the class number h+h^+ of Q(μp)\mathbb{Q}(\mu_p) to the product of these L-values: #A∼∏k=2,4,…,p−3∣L(0,ωk−1)∣p−1\# A \sim \prod_{k=2,4,\dots,p-3} |L(0, \omega^{k-1})|_p^{-1}. The implications extend to bounding class numbers and identifying irregular primes, such as p=37p=37, where non-vanishing eigenspaces arise from specific Bernoulli numerators. During the and , Ribet advanced the study of Galois representations attached to modular forms, building on his modular construction techniques to explore  mod p\bmod p representations of Gal(Q‾/Q)\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}). His 1976 work demonstrated how and cuspidal Hecke eigenforms yield explicit Galois modules corresponding to unramified extensions, influencing subsequent developments in the . These efforts connected modular forms to elliptic curves via the division fields of their torsion points, as seen in his analysis of abelian varieties with complex multiplication, where Galois images are constrained by endomorphism rings. For instance, in examining elliptic curves over Q\mathbb{Q} with real multiplications, Ribet showed that the Galois action on nn-torsion points stabilizes for large nn, providing effective bounds on conductor and rank. This foundational interplay between and underscored the arithmetic of elliptic curves, setting the stage for broader applications in .

Key contributions to Fermat's Last Theorem

Ken Ribet's key contributions to centered on establishing a profound connection between potential counterexamples to the theorem and the Taniyama–Shimura , now known as the . In , inspired by Gerhard Frey's proposal that a solution to aℓ+bℓ=cℓa^\ell + b^\ell = c^\ell for prime ℓ>2\ell > 2 would yield a semistable elliptic curve Ea,b,c:y2=x(x−aℓ)(x+bℓ)E_{a,b,c}: y^2 = x(x - a^\ell)(x + b^\ell) with unusual properties, and Jean-Pierre Serre's subsequent formulation of the epsilon , Ribet proved the . This result demonstrated that the Galois representation associated with such a Frey curve could not arise from a modular form unless the Taniyama–Shimura held, effectively reducing the proof of to verifying the modularity of certain elliptic curves. The epsilon conjecture, as proved by Ribet, posits that if an irreducible, two-dimensional Galois representation ρ:\Gal(Q‾/Q)→\GL2(Fℓ)\rho: \Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \GL_2(\mathbb{F}_\ell) attached to a of level NℓN\ell (with ℓ∤N\ell \nmid N) is unramified at some prime or exhibits specific ramification behavior, then it is isomorphic to the representation attached to a of level NN. In the context of , Ribet showed that the Frey curve Ea,b,cE_{a,b,c} has conductor 2⋅\rad(abc)2 \cdot \rad(abc) and a Galois representation that, if modular, would require a nonexistent of level 2 and weight 2, leading to a contradiction with the Taniyama–Shimura conjecture. This theorem, often called Ribet's theorem, implied that no such counterexample to could exist without violating the modularity of elliptic curves over Q\mathbb{Q}. Ribet's work culminated in his 1990 paper, which provided a rigorous framework linking Frey curves directly to modular forms via Galois representations. By constructing explicit isomorphisms between these representations and those arising from cusp forms, Ribet refined the epsilon conjecture's implications, showing that the modularity of the Frey curve would force an impossible reduction in the level of the associated modular form. This refinement was instrumental in paving the way for Andrew Wiles' 1994 proof of Fermat's Last Theorem, as it isolated the problem to proving the Taniyama–Shimura conjecture for semistable elliptic curves, a task Wiles ultimately accomplished with Richard Taylor. The historical synergy—Frey's geometric insight, Serre's conjectural bridge, and Ribet's proof—transformed Fermat's centuries-old problem into a question of modularity in the Langlands program.

Later work on modular forms and Galois representations

Following the proof of , Ribet continued to advance the understanding of connections between s and Galois representations, particularly through his work on Serre's conjecture. This conjecture posits that every irreducible, odd, two-dimensional representation of the of the rationals over a of characteristic p arises from a of weight two. Ribet's contributions in the included proving key cases of level optimization, showing that such representations can be associated with s of minimal level determined by the representation's local at primes. For instance, if the representation is unramified at a prime p dividing the level, it arises from a form with level reduced by p, under suitable conditions like non-scalar Frobenius traces. In collaboration with William A. Stein, Ribet delivered comprehensive lectures in 2001 that synthesized progress on Serre's , emphasizing residual and level-lowering techniques post-Wiles. These lectures detailed Ribet's pivot method for level optimization, which uses a fixed prime q (such as q=2) to eliminate unnecessary prime factors from the level of the associated , relying on multiplicity one theorems and deformation theory. This work extended residual results, confirming that modular Galois representations p could be lifted to characteristic zero while preserving key arithmetic properties. Ribet's level-lowering theorems, refined in this period, played a crucial role in verifying the of specific representations arising in arithmetic geometry, such as those attached to abelian varieties over number fields. Ribet's later research influenced ongoing developments in the , particularly the arithmetic aspects linking automorphic forms to Galois representations. His techniques for controlling the image of Galois representations attached to modular forms have informed studies of torsion points on Jacobians and semistable representations, with applications to Diophantine equations and elliptic curves over global fields. For example, work on the Manin constant in 2006, joint with Agashe and Stein, explored congruences that refine level-lowering in the context of optimal quotients of modular Jacobians. As of 2023, Ribet reflected on these themes in his lecture "Fermat's Last Theorem: Looking Back after 30 Years" at the University of Science and Technology's , discussing the enduring impact of modularity theorems on contemporary arithmetic geometry. He highlighted how post-Wiles advancements in Serre's conjecture have broadened the Langlands correspondence, enabling proofs of modularity for representations over totally real fields and influencing conjectures on L-functions. These reflections underscore Ribet's role in bridging classical with modern Galois-theoretic methods.

Awards and honors

Major prizes

In 1989, Kenneth Ribet jointly received the with Abbas Bahri, a biennial established by the Institut de Mathématiques de to recognize outstanding mathematical research in fields influenced by , such as , probability, and variational principles; the prize, valued at €20,000, is selected by an international committee of mathematicians based on the profound impact and originality of the work. Ribet's was specifically for his contributions to , particularly his work linking modular forms to elliptic curves, which advanced the path toward proving . In 2017, Ribet was awarded the Brouwer Medal by the Royal Dutch Mathematical Society (Koninklijk Wiskundig Genootschap), a triennial honor given to a prominent for lifetime achievements in the field, with selection emphasizing groundbreaking contributions and the ability to deliver an accessible lecture to a broad mathematical audience during the Netherlands Mathematical Congress. The medal recognizes Ribet's enduring impact on , including his pivotal role in establishing connections between Galois representations and modular forms that facilitated major breakthroughs in . In 2025, Ribet earned the Leroy P. Steele Prize for Seminal Contribution to Research from the (AMS), an annual award of $5,000 granted for a single paper—recent or historical—that demonstrates fundamental and lasting importance to mathematical research, selected by an AMS committee reviewing nominations for depth, influence, and transformative effects on the discipline. The prize honors Ribet's 1976 paper "A modular construction of unramified p-extensions of Q(μp)\mathbb{Q}(\mu_p)," published in Inventiones Mathematicae, which provided a converse to Herbrand's theorem on irregular primes and introduced modular methods for constructing unramified extensions of cyclotomic fields, influencing and related areas.

Professional memberships and recognitions

In recognition of his contributions to , Kenneth A. Ribet was elected to the American Academy of Arts and Sciences in 1997. This honor reflects peer acknowledgment of his work on modular forms and Galois representations. The following year, Ribet received an honorary degree from in 1998, where he had earlier earned his A.M. in 1969. This distinction highlights his early academic ties and ongoing influence in and research. Ribet's election to the in 2000 further underscored his stature among leading scientists. As a member, he has contributed to advisory roles, including serving as Secretary of Class I from 2014 to 2017. In 2013, Ribet was named a of the as part of its inaugural class, an honor bestowed for exceptional contributions to the field and service to the profession. He later served as AMS President from 2017 to 2018, advancing initiatives in mathematical research and outreach.

Personal life

Ribet is married to Lisa R. Goldberg, a and financial economist at the University of California, Berkeley. They have two daughters.

References

  1. https://math.berkeley.edu/~ribet/ribet_cv.pdf
  2. https://www.nasonline.org/directory-entry/kenneth-a-ribet-htzclx/
  3. https://news.berkeley.edu/2024/12/16/ken-ribet-awarded-math-prize-for-influential-proof/
  4. https://math.berkeley.edu/people/faculty/kenneth-ribet
  5. https://legacy.slmath.org/people/176
  6. https://www.ams.org/about-us/presidents/64-ribet
  7. https://www.rockawave.com/articles/rockaways-math-superstar-2/
  8. https://www.rockawave.com/articles/chatting-with-chapey-50/
  9. https://www.nieuwarchief.nl/serie5/pdf/naw5-2017-18-2-115.pdf
  10. https://math.berkeley.edu/~ribet/biosketch.html
  11. https://static1.squarespace.com/static/5b2cf993266c07b1ca7a812f/t/619ba607c0a14a4fdb5aa05a/1637590535303/Fermat%E2%80%99s+Last+Theorem+-+with+Ken+Ribet.pdf
  12. https://math.berkeley.edu/~ribet/Articles/
  13. https://math.berkeley.edu/~ribet/courses.html
  14. https://www.ams.org/journals/notices/201908/rnoti-p1303.pdf
  15. https://math.berkeley.edu/people/past-department-members/past-phd-students
  16. https://www.ams.org/journals/bull/1999-36-01/bull-36-1-print-matter.pdf
  17. https://link.springer.com/series/3733
  18. https://scholar.google.com/citations?user=U63vv5wAAAAJ&hl=en
  19. https://math.berkeley.edu/~ribet/herbrand.pdf
  20. https://math.berkeley.edu/~ribet/Articles/toulousela.pdf
  21. https://eudml.org/doc/143793
  22. https://arxiv.org/pdf/math/9503219
  23. https://wstein.org/papers/serre/ribet-stein.pdf
  24. https://www.youtube.com/watch?v=wYw4H6JIaxA
  25. https://www.math.univ-toulouse.fr/en/activites/prix-fermat/
  26. https://www.nieuwarchief.nl/serie5/pdf/naw5-2017-18-2-113.pdf
  27. https://math.berkeley.edu/news/kenneth-ribet-awarded-2017-brouwer-medal-congratulations
  28. https://www.ams.org/news?news_id=7391
  29. https://www.ams.org/steele-research
  30. https://corporation.brown.edu/honorary-degrees
  31. https://vcresearch.berkeley.edu/news/ken-ribet-awarded-math-prize-influential-proof
  32. https://kpknudson.com/my-favorite-theorem/2018/7/10/episode-22-ken-ribet
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