# Barry Mazur

#1,101

Most Influential Person Now

American mathematician

## Barry Mazur's AcademicInfluence.com Rankings

Barry Mazurmathematics Degrees

Mathematics

#50

World Rank

#157

Historical Rank

#16

USA Rank

Geometry

#9

World Rank

#20

Historical Rank

#2

USA Rank

Number Theory

#12

World Rank

#26

Historical Rank

#3

USA Rank

Measure Theory

#503

World Rank

#710

Historical Rank

#207

USA Rank

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Mathematics

## Barry Mazur's Degrees

- PhD Mathematics Princeton University

## Why Is Barry Mazur Influential?

(Suggest an Edit or Addition)According to Wikipedia, Barry Charles Mazur is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.

## Barry Mazur's Published Works

### Published Works

- Modular curves and the eisenstein ideal (1977) (1100)
- Arithmetic moduli of elliptic curves (1985) (880)
- Rational isogenies of prime degree (1978) (662)
- Class fields of abelian extensions of Q (1984) (597)
- Onp-adic analogues of the conjectures of Birch and Swinnerton-Dyer (1986) (511)
- Rational points of abelian varieties with values in towers of number fields (1972) (474)
- On Periodic Points (1965) (393)
- Deforming Galois Representations (1989) (350)
- Arithmetic of Weil curves (1974) (327)
- -Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (1994) (221)
- Arithmetic Moduli of Elliptic Curves. (AM-108) (1985) (218)
- Universal Extensions and One Dimensional Crystalline Cohomology (1974) (201)
- Mathematics: Frontiers and Perspectives (2000) (187)
- An Introduction to the Deformation Theory of Galois Representations (1997) (183)
- Uniformity of rational points (1997) (182)
- On $p$-adic analytic families of Galois representations (1986) (171)
- A Note on Some Contractible 4-Manifolds (1961) (160)
- Frobenius and the Hodge filtration (1972) (154)
- Filtrations on the Homology of Algebraic Varieties (1994) (148)
- Formal groups arising from algebraic varieties (1977) (146)
- This is the Story. (1952) (134)
- Ranks of twists of elliptic curves and Hilbert’s tenth problem (2009) (133)
- Notes on étale cohomology of number fields (1973) (127)
- On embeddings of spheres (1959) (125)
- The Topology of Rational Points (1992) (123)
- The square-free sieve and the rank of elliptic curves (1991) (115)
- Algebraic Numbers By (2005) (112)
- Refined conjectures of the “Birch and Swinnerton-Dyer type” (1987) (112)
- Rational points on modular curves (1977) (111)
- Frobenius and the Hodge Filtration (estimates) (1973) (106)
- Arithmetic on curves (1986) (104)
- Smoothings of piecewise linear manifolds (1974) (95)
- On the arithmetic of special values ofL functions (1979) (84)
- Visualizing Elements in the Shafarevich—Tate Group (2000) (83)
- Représentations galoisiennes, différentielles de Kähler et « conjectures principales » (1990) (83)
- Questions of decidability and undecidability in Number Theory (1994) (82)
- Canonical Height Pairings via Biextensions (1983) (81)
- Finding Large Selmer Rank via an Arithmetic Theory of Local Constants (2005) (77)
- Families of modular eigenforms (1992) (77)
- Modular curves and arithmetic (1983) (75)
- Courbes Elliptiques et Symboles Modulaires (1973) (66)
- The $p$-adic sigma function (1991) (62)
- Hypersurfaces of low degree (1998) (61)
- Nearly ordinary Galois deformations over arbitrary number fields (2007) (59)
- Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero (2004) (58)
- Computation of p-Adic Heights and Log Convergence (2006) (58)
- Finding meaning in error terms (2008) (57)
- Number theory as gadfly (1991) (56)
- Rational torsion of prime order in elliptic curves over number fields (1992) (55)
- Relative modular symbols and Rankin-Selberg convolutions (2000) (54)
- Evolutions, symbolic squares, and Fitting ideals. (1995) (53)
- Two-dimensional representations in the arithmetic of modular curves (1991) (53)
- Galois Representations in Arithmetic Algebraic Geometry: The Eigencurve (1998) (52)
- Twisting Commutative Algebraic Groups (2006) (51)
- On the Passage From Local to Global in Number Theory (1993) (49)
- Relative Neighborhoods and the Theorems of Smale (1963) (49)
- Stable equivalence of differentiable manifolds (1961) (47)
- Average Ranks of Elliptic Curves: Tension between Data and Conjecture (2007) (45)
- Rational connectivity and sections of families over curves (2002) (42)
- An "infinite fern" in the universal deformation space of Galois representations (1997) (41)
- Differential topology from the point of view of simple homotopy theory and further remarks (1964) (40)
- Proof and other Dilemmas: When is One Thing Equal to Some Other Thing? (2008) (38)
- How Many Rational Points Can a Curve Have (1995) (37)
- On the van Kampen theorem (1966) (35)
- Analogies Between Function Fields and Number Fields (1983) (34)
- On the density of modular representations (1997) (33)
- Explicit Universal Deformations of Galois Representations (1989) (30)
- Galois Representations in Arithmetic Algebraic Geometry: Open problems regarding rational points on curves and varieties (1998) (30)
- The Method of Infinite Repetition in Pure Topology: I (1964) (29)
- The spin of prime ideals (2011) (28)
- Prime Numbers and the Riemann Hypothesis (2016) (27)
- How can we construct abelian Galois extensions of basic number elds (2011) (26)
- Speculations about the topology of rational points: an update (1995) (26)
- Diophantine stability (2015) (25)
- Perturbations, Deformations, and Variations (and "Near-Misses") in Geometry, Physics, and Number Theory (2004) (25)
- Controlling Selmer groups in the higher core rank case (2013) (24)
- CONJECTURE (2004) (24)
- Elliptic curves and class field theory (2003) (24)
- Refined class number formulas for Gm (2016) (22)
- Points rationnels des courbes modulaires $X_0(N)$ (1976) (22)
- Refined class number formulas and Kolyvagin systems (2009) (22)
- Correspondence homomorphisms for singular varieties (1994) (22)
- Abelian Varieties and the Mordell-Lang Conjecture (2000) (22)
- Organizing the arithmetic of elliptic curves (2005) (22)
- Refined class number formulas for Gm (2016) (22)
- A Markov model for Selmer ranks in families of twists (2013) (21)
- Visualizing elements of order three in the Shafarevich–Tate group (1999) (21)
- On the characteristic power series of the U operator (1993) (20)
- Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols (2019) (20)
- Points of order 13 on elliptic curves (1973) (20)
- Studying the Growth of Mordell-Weil (2003) (18)
- Growth of Selmer rank in nonabelian extensions of number fields (2007) (18)
- Local Flat Duality (1970) (16)
- The definition of equivalence of combinatorial imbeddings (1959) (16)
- Disparity in Selmer ranks of quadratic twists of elliptic curves (2011) (15)
- Topology of curves and surfaces, and special topics in the theory of algebraic varieties (1978) (13)
- Mathematical Platonism and its Opposites (2008) (13)
- Two-dimensional $p$-adic Galois representations unramified away from $p$ (1990) (13)
- Questions about Powers of Numbers (2000) (13)
- Refined class number formulas for $\mathbb{G}_m$ (2013) (12)
- FINDING LARGE SELMER GROUPS (2005) (12)
- Symmetric homology spheres (1962) (12)
- Is it Plausible? (2014) (11)
- Open problems: Descending cohomology, geometrically (2014) (10)
- Current developments in mathematics, 2003 (2001) (10)
- Arithmetic Questions Related to Rationally Connected Varieties (2004) (9)
- "Imagining Numbers" (Particularly the Square Root of Minus Fifteen)@@@"Abel's Proof": An Essay on the Sources and Meaning of Mathematical Unsolvability@@@"The Riemann Hypothesis": The Greatest Unsolved Problem in Mathematics (2003) (9)
- Rational families of 17-torsion points of elliptic curves over number fields (2018) (9)
- RATIONAL CURVES AND POINTS ON K3 SURFACES By FEDOR BOGOMOLOV and YURI TSCHINKEL (2005) (8)
- How Many Primes are There (2016) (7)
- Review: André Weil, Ernst Edward Kummer, Collected Papers (1977) (6)
- Jumps in Mordell-Weil Rank and Arithmetic Surjectivity (2004) (6)
- Smoothings of Piecewise Linear Manifolds. (AM-80), Volume 80 (1975) (6)
- The Method of Infinite Repetition in Pure Topology: II. Stable Applications (1966) (6)
- Homotopy of Varieties in the Etale Topology (1967) (6)
- Selmer companion curves (2012) (6)
- Mathematics: Controlling our errors (2006) (6)
- Searching for $p$-adic eigenfunctions (1995) (6)
- Combinatorial equivalence versus topological equivalence (1964) (6)
- A Note on a Classical Theorem of Blichfeldt (1991) (5)
- Erratum to: The spin of prime ideals (2015) (5)
- Universal extensions and crystals (1974) (5)
- Pairings in the Arithmetic of Elliptic Curves (2004) (5)
- Galois deformations and Hecke curves (1994) (5)
- On the structure of certain semi-groups of spherical knot classes (1959) (4)
- Reading bombelli (2002) (4)
- p-Aàic Analytic Number Theory of Elliptic Curves and Abelian Varieties over Q (1975) (4)
- The Education of T.C. Mits: What modern mathematics means to you (2008) (3)
- A brief introduction to the work of Haruzo Hida (2012) (3)
- Some comments on elliptic curves over general number fields and Brill-Noether modular varieties (2014) (2)
- Big fields that are not large (2020) (2)
- Alexandre Grothendieck, 1928–2014, Part 2 (2016) (2)
- Existential definability and diophantine stability (2022) (2)
- On mathematics, imagination & the beauty of numbers (2005) (2)
- Math in the Time of Plague (2020) (2)
- Corrections to my paper, “Symmetric homology spheres” (1964) (2)
- A question about quadratic points on X 0 ( N ) (2021) (1)
- SMOOTHINGS OF PIECEWISE LINEAR MANIFOLDS II: CLASSIFICATION (1975) (1)
- About Hermann Weyl’s “Ramifications, old and new, of the eigenvalue problem” (2012) (1)
- Orthotopy and spherical knots (1959) (1)
- Robert F. Coleman 1954–2014 (2015) (1)
- What is Riemann's Hypothesis? (2016) (1)
- Primes, Knots and Po (2012) (1)
- Current Developments in Mathematics 1998 (1998) (1)
- About the cover: Diophantus’s Arithmetica (2006) (1)
- CCD Survey for Short Period Variables and sdB/O Stars in 47 TUC (1997) (1)
- Ramsey Geometric and p-adic Modular Forms of Half-Integral Weight (2004) (1)
- Fermat's last theorem : AMS-CMS-MAA joint invited address (1995) (1)
- Corrections to {\it Uniformity of rational points} and further comments (2020) (1)
- A celebration of the mathematical work of Glenn Stevens (2016) (1)
- Questions About Primes (2016) (1)
- Uniformity of rational points: an up-date and corrections (2022) (1)
- To the Edge of the Map (2018) (1)
- BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78 , Number 5 , September 1972 FROBENIUS AND THE HODGE FILTRATION BY (2007) (0)
- Le renard et le hérisson : Grandes et petites énigmes mathématiques (2001) (0)
- The Mystery Moves to the Error Term (2016) (0)
- The definition of equivalence of combinatorial imbeddings ; On the structure of certain semi-groups of spherical knot classes ; Orthotopy and spherical knots (1959) (0)
- Further Questions About Primes (2016) (0)
- To Our Readers of Part I (2016) (0)
- Further Questions About the Riemann Spectrum (2016) (0)
- Chapter 9. MODULI PROBLEMS VIEWED OVER CYCLOTOMIC INTEGER RINGS (1985) (0)
- Thoughts About Numbers (2016) (0)
- Erratum to: The spin of prime ideals (2015) (0)
- Editorial (2020) (0)
- A “BASIC NOTIONS” LECTURE ABOUT SIGN INTUITIONS (2006) (0)
- Hilbert's Tenth Problem and Elliptic Curves (2010) (0)
- Techniques for the Analytic Proof of the Finite Generation of the Canonical Ring (2008) (0)
- In celebration of John Coates’ 60th birthday (2005) (0)
- The Prime Number Theorem (2016) (0)
- AVERAGE RANKS OF ELLIPTIC CURVES (0)
- Spectra and Trigonometric Sums (2016) (0)
- Chapter 1. GENERALITIES ON “ A-STRUCTURES” AND “ A-GENERATORS” (1985) (0)
- P-adic monodromy and the Birch and Swinnerton-Dyer conjecture : a Workshop on p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, August 12-16, 1991 (1994) (0)
- B.: Evolutions, symbolic squares, and Fitting ideals (1997) (0)
- Slopes of Graphs That Have No Slopes (2016) (0)
- Homomorphisms for Singular Varieties (2018) (0)
- Part I The Program for these lectures 1 Around Hilbert ’ s Tenth Problem (2015) (0)
- Is it Plausible? (2013) (0)
- VERY ROUGH NOTES TO ACCOMPANY THE TALKS ABOUT THE METHOD OF CHABAUTY, COLEMAN, KIM (2019) (0)
- How Many θi's are There? (2016) (0)
- Building π(X) from the Spectrum (2016) (0)
- The Verdier functor (1969) (0)
- UNIFORM BOUNDS FOR STABLY INTEGRAL POINTS ON ELLIPTIC CURVES (1999) (0)
- The theory and practice of international market research (1998) (0)
- Thoughts about Andrew Ogg’s (Torsion) Conjecture (2022) (0)
- L-functions and Arithmetic Conference Program 1 Monday , June 13 9 : 15-9 : 30 am (0)
- What is a “Good Approximation”? (2016) (0)
- Thinking about Grothendieck (2016) (0)
- The fundamental group (1969) (0)
- Pure and Applied Mathematics (2016) (0)
- NOTES ADDED IN PROOF (1985) (0)
- Computer Music Files and Prime Numbers (2016) (0)
- Robert F. Coleman 1954–2014 (2014) (0)
- Zariski's Topological and Other Early Papers (1991) (0)
- Chapter 6. CYCLICITY (1985) (0)
- Memorial Article for John Tate (2021) (0)
- What are Prime Numbers (2016) (0)
- Chapter 12. NEW MODULI PROBLEMS IN CHARACTERISTIC p; IGUSA CURVES (1985) (0)
- Prime Numbers Viewed from a Distance (2016) (0)
- Explicit constructions of universal extensions (1974) (0)
- The average elliptic curve has few integral points (2014) (0)
- Explicit Local Class Field Theory (2019) (0)
- Chapter 7. QUOTIENTS BY FINITE GROUPS (1985) (0)
- Preface to the Present Volume (1985) (0)
- Pourquoi les Nombres Premiers (2005) (0)
- Derived class field theory (2023) (0)
- Chapter 6. Visions, Dreams, and Mathematics (2012) (0)
- Afterword to the article “Arithmetic on curves” (2018) (0)
- Chapter 14. APPLICATION TO THEOREMS OF GOOD REDUCTION (1985) (0)
- Completions and fibrations (1969) (0)
- On Losing No Information (2016) (0)
- Why is it plausible (2012) (0)
- Chapter 2. REVIEW OF ELLIPTIC CURVES (1985) (0)
- The Word “Spectrum” (2016) (0)
- A celebration of the mathematical work of Glenn Stevens (2016) (0)
- III · 1 – 3 Elliptic Curves and Class Field Theory (2002) (0)
- Chapter 11. INTERLUDE-EXOTIC MODULAR MORPHISMS AND ISOMORPHISMS (1985) (0)
- Square Root Error and Random Walks (2016) (0)
- Cohomological criteria for ♮-isomorphism (1969) (0)
- Defining $\mathbb Z$ using unit groups (2023) (0)
- Contemporary Mathematics Introduction to Kolyvagin systems (2003) (0)
- Biographical Memoirs: Andrew Gleason (2010) (0)
- Fourier Transforms: Second Visit (2016) (0)
- ANNALES SCIENTIFIQUES DE L'É.N.S. (1973) (0)
- As Riemann Envisioned It (2016) (0)
- A Probabilistic First Guess (2016) (0)
- AN INTRODUCTORY LECTURE ON EULER SYSTEMS (2001) (0)
- Chapter 13. REDUCTIONS mod p OF THE BASIC MODULI PROBLEMS (1985) (0)
- Elliptic Curves and Their Statistics Rough Notes for My Basic Notions Talk, Feb 28 2012 Part I. Densities (0)
- A Sneak Preview of Part III (2016) (0)
- Chapter 3. THE FOUR BASIC MODULI PROBLEMS FOR ELLIPTIC CURVES: SORITES (1985) (0)
- Raoul Bott as we knew him (2010) (0)
- ELLIPTIC CURVES, SERRE’S CONJECTURE AND FERMAT’S LAST THEOREM (2009) (0)
- Open problems : Descending cohomology , geometrically compiled by (2014) (0)
- Chapter 10. THE CALCULUS OF CUSPS AND COMPONENTS VIA THE GROUPS T[N], AND THE GLOBAL STRUCTURE OF THE BASIC MODULI PROBLEMS (1985) (0)
- A View of|Li(X)-π(X)| (2016) (0)
- Homotopy groups of completions (1969) (0)
- A profiniteness theorem (1969) (0)
- Chapter 4. THE FORMALISM OF MODULI PROBLEMS (1985) (0)
- Fourier Transform of Delta (2016) (0)
- The Consolation of Math in Plague Time (2020) (0)
- From Primes to the Riemann Spectrum (2016) (0)
- Chapter 5. REGULARITY THEOREMS (1985) (0)
- The Staircase of Primes (2016) (0)
- This paper was presented at a colloquium entitled “ Elliptic Curves and Modular Forms , ” organized by (1997) (0)
- Topics in Analytic Number Theory (2013) (0)
- The Spectrum and the Staircase of Primes (2016) (0)
- PROBLEMS CONNECTING LOGIC AND NUMBER THEORY (2009) (0)
- A course in Analytic Number Theory Taught (2014) (0)
- L'idéologie, l'art et Franz Kafka (1986) (0)
- Notes on Utility (2017) (0)
- Infinity: Beyond the Beyond the Beyond (1953) (0)
- From the Riemann Spectrum to Primes (2016) (0)
- Pro-objects in the homotopy category (1969) (0)
- The average elliptic curve has few integral (2014) (0)
- Companion to Mathematics Proof 1 Algebraic Numbers By Barry Mazur (2005) (0)
- IV.1 Algebraic Numbers (2010) (0)
- Chapter 8. COARSE MODULI SCHEMES, CUSPS, AND COMPACTIFICATION (1985) (0)
- “Named” Prime Numbers (2016) (0)
- Tinkering with the Staircase of Primes (2016) (0)
- A glossary of the categories in which we shall work, and fibre resolutions (1969) (0)
- The spin of prime ideals (2012) (0)
- Companions to the Zeta Function (2016) (0)

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## What Are Barry Mazur's Academic Contributions?

Barry Mazur is most known for their academic work in the field of mathematics. They are also known for their academic work in the fields of

Barry Mazur has made the following academic contributions: