Celebratio Mathematica

[1]B. Mazur: “Orthotopy and spherical knots,” Publ. Math., Inst. Hautes Étud. Sci. 3 : 1 (December 1959), pp. 121–140. MR 0116348 Zbl 0119.38704 article

[2]B. Mazur: “On embeddings of spheres,” Bull. Am. Math. Soc. 65 : 2 (1959), pp. 59–65. Research announcement for the author’s PhD thesis (1959). MR 0117693 Zbl 0086.37004 article

[3]B. Mazur: “On the structure of certain semi-groups of spherical knot classes,” Publ. Math., Inst. Hautes Étud. Sci. 3 : 1 (December 1959), pp. 111–119. MR 0116347 Zbl 0119.38703 article

[4]B. Mazur: “The definition of equivalence of combinatorial imbeddings,” Publ. Math., Inst. Hautes Étud. Sci. 3 : 1 (1959), pp. 97–109. MR 0116346 Zbl 0119.38702 article

In treating the intriguing problem of equivalences of imbeddings of complexes \( K \) in \( E^r \), a brutish technical question arises: The question of whether or not one can extend an isotopy from \( K \) to \( K^{\prime} \) to an isotopy of the entire ambient space \( E^r \) onto itself (which would bring of course, \( K \) to \( K \)).

For if one could not, a multiplicity of possible definitions of knot-equivalence would arise. And what is worse, an isotopy from \( K \) to \( K^{\prime} \) would tell little about the relationship between the respective complementary spaces.

So, the purpose of this paper is to prove just that: Any isotropy of a subcomplex \( K\subseteq E^r \) to \( K^{\prime} \) can be extended to an isotopy of \( E^r \) to \( E^r \).

The construction of the extended isotopy is done in two stages. The first stage is to reduce the problem to one of moderately local considerations. The second is to solve the local problem remaining. The technical machinations involved in solving the local problem consists in attempting to restate it as a common extension problem.

[5]B. C. Mazur: On embedings of spheres. Ph.D. thesis, Princeton University, 1959. Advised by R. H. Fox and R. H. Bing. A research announcement based on this thesis was published in Bull. Am. Math. Soc. 65:2 (1959). MR 2613038 phdthesis

Ralph Hartzler Fox	Related
R H Bing	Related	Volume

[6]B. Mazur: “A note on some contractible 4-manifolds,” Ann. Math. (2) 73 : 1 (January 1961), pp. 221–228. MR 0125574 Zbl 0127.13604 article

[7]B. C. Mazur: “On embeddings of spheres,” Acta Math. 105 : 1–2 (1961), pp. 1–17. MR 0125570 Zbl 0096.37904 article

[8]B. Mazur: “Stable equivalence of differentiable manifolds,” Bull. Am. Math. Soc. 67 : 4 (1961), pp. 377–384. MR 0130697 Zbl 0107.17002 article

[9]B. Mazur: “Simple neighborhoods,” Bull. Am. Math. Soc. 68 : 2 (1962), pp. 87–92. MR 0137125 Zbl 0105.17002 article

[10]B. Mazur: “Symmetric homology spheres,” Ill. J. Math. 6 : 2 (1962), pp. 245–250. Corrections were published in Ill. J. Math. 8:1 (1962). MR 0140102 Zbl 0103.39503 article

[11]B. Mazur: Differential topology from the point of view of simple homotopy theory, published as Publ. Math. Inst. Hautes Étud. Sci. 15 (December 1963). Corrections and further remarks were published in Publ. Math. Inst. Hautes Étud. Sci. 22 (1964). MR 0161342 Zbl 0173.51203 book

[12]B. Mazur: “Relative neighborhoods and the theorems of Smale,” Ann. Math. (2) 77 : 2 (March 1963), pp. 232–249. MR 0150786 Zbl 0112.38301 article

[13]B. Mazur: “Les théorèmes généraux de lissage” [General smoothing theorems] in Séminaire de topologie combinatoire et différentielle de l’Institut des Hautes Etudes Scientifiques: 1962/63 [IHÉS seminar on combinatorial and differential topology: 1962/63]. IHÉS (Paris), 1963. Exposé no. 3, 9 pages. incollection

[14]B. Mazur: “Les théorèmes d’unicité de lissage” [Uniqueness theorems for smoothing] in Séminaire de topologie combinatoire et différentielle de l’Institut des Hautes Études Scientifiques: 1962/63 [IHÉS seminar on combinatorial and differential topology: 1962/63]. IHÉS (Paris), 1963. Exposé no. 4, 6 pages. incollection

[15]B. Mazur: “Corrections to my paper, ‘Symmetric homology spheres’,” Ill. J. Math. 8 : 1 (1964), pp. 175. Corrections to an article published in Ill. J. Math. 6:2 (1962). MR 0157379 Zbl 0116.14901 article

[16]B. Mazur: “Combinatorial equivalence versus topological equivalence,” Trans. Am. Math. Soc. 111 : 2 (1964), pp. 288–316. MR 0202124 Zbl 0124.16601 article

In [1961], Milnor exhibited an example of two finite complexes \( X_1 \), \( X_2 \) which are homeomorphic, but combinatorially distinct. That was the first example of the disparity between combinatorial notions versus topological ones on finite complexes. Previously the only results were positive in nature. Papakyriakopoulos had proved the Hauptvermutung for 2-dimensional complexes [1943], and Moise [1952] (and later Bing) had proved the Hauptvermutung for 3-dimensional manifolds. The object of this paper is to provide an example of a finite complex \( K \) and two simplicial imbeddings, \[ \alpha,\beta:K\to \mathbb{S}^m \] which are combinatorially inequivalent, yet topologically equivalent. The nature of the argument is such that the minimum \( m \) it yields is \( m = 23 \).

[17]B. Mazur: “The method of infinite repetition in pure topology, I,” Ann. Math. (2) 80 : 2 (September 1964), pp. 201–226. MR 0169224 Zbl 0133.16506 article

[18]B. Mazur: “Corrections to: ‘Differential topology from the point of view of simple homotopy theory’ and further remarks,” Publ. Math. Inst. Hautes Étud. Sci. 22 (December 1964), pp. 81–91. Corrections and further remarks for an article in Publ. Math. Inst. Hautes Étud. Sci. 15 (1963). MR 0185612 Zbl 0173.51301 article

[19]B. Mazur: Remarks on the Alexander polynomial, 1964?. Never published, but recently a scanned version was posted online by Mazur himself. misc

[20]M. Artin and B. Mazur: “On periodic points,” Ann. Math. (2) 81 : 1 (January 1965), pp. 82–99. MR 0176482 Zbl 0127.13401 article

[21]B. Mazur: “Morse theory,” pp. 145–165 in Differential and combinatorial topology: A symposium in honor of Marston Morse (Princeton, NJ, 1964). Edited by S. S. Cairns. Princeton Mathematical Series 27. Princeton University Press, 1965. MR 0185613 Zbl 0158.42202 incollection

Harold Calvin Marston Morse	Related
Stewart Scott Cairns	Related

[22]B. Mazur: “The method of infinite repetition in pure topology, II: Stable applications,” Ann. Math. (2) 83 : 3 (May 1966), pp. 387–401. MR 0196758 Zbl 0141.40501 article

[23]M. Artin and B. Mazur: “On the van Kampen theorem,” Topology 5 (1966), pp. 179–189. MR 0192495 Zbl 0138.18301 article

[24]M. Artin and B. Mazur: “Homotopy of varieties in the etale topology,” pp. 1–15 in Proceedings of the conference on local fields (Driebergen, Netherlands, 1966). Edited by T. A. Springer. Springer (Berlin), 1967. MR 0230730 Zbl 0188.53402 incollection

This paper presents an outline of some recent work; proofs of the results we announce will be published elsewhere.

Our purpose is to study the analogues of homotopy invariants which can be obtained from varieties by using the étale topology of Grothendieck . (We refer to [Artin et al. 1973] for the definitions and properties of this topology.) In Section 6, we give a complete description of the relation between the étale topology and the classical topology for a normal variety \( X \) over the field of complex numbers, as far as homotopy invariants are concerned. We show that the homotopy type of \( X \) in the étale topology is a certain profinite completion of the classical homotype type. This statement includes the classical Riemann existence theorem (cf. [Artin et al. 1973, XI (4.3)] for instance) which asserts that every finite topological covering space of \( X \) has a unique algebraic structure (i.e., that the algebraic fundamental group is the profinite completion of the topological one), and the comparison theorem for cohomology, ([Artin et al. 1973, XVI (4.1)]), which asserts that the classical and étale cohomologies of \( X \) with values in twisted finite coefficient groups are equal. In fact, our result is derived from these two theorems.

Michael Artin	Related
Tonny A. Springer	Related

[25]M. Artin and B. Mazur: Etale homotopy (Harvard University, 1965–1966). Lecture Notes in Mathematics 100. Springer (Berlin), 1969. A second edition was published in 1986. MR 0245577 Zbl 0182.26001 book

[26]B. Mazur and L. Roberts: “Local Euler characteristics,” Invent. Math. 9 : 3 (1969–1970), pp. 201–234. MR 0258844 Zbl 0191.19202 article

[27]B. Mazur: “Local flat duality,” Am. J. Math. 92 : 2 (April 1970), pp. 343–361. MR 0271119 Zbl 0199.24501 article

[28]B. Mazur: “Finite flat structures,” pp. 219–225 in Applications of categorical algebra (New York, 10–11 April 1968). Edited by A. Heller. Proceedings of Symposia in Pure Mathematics 17. American Mathematical Society (Providence, RI), 1970. MR 0257182 Zbl 0221.14025 incollection

Fix a rational prime \( p \), an algebraic closure of the \( p \)-adic numbers, \( \bar{Q}_p \), and a subfield \( K\subset\bar{Q}_p \), of finite degree over \( Q_p \). Consider the category \( \mathcal{g} \) of finite flat commutative group schemes over \( R \), the ring of integers in \( K \) (call objects of \( \mathcal{g} \) group schemes, for short). We take morphisms of \( \mathcal{g} \) to be arbitrary group scheme homomorphisms over \( R \). The category \( \mathcal{g} \) is not an abelian category. One remedy for this is to imbed \( \mathcal{g} \) as a full subcategory of the abelian category of sheaves of abelian groups for the \( fppf \) site over \( \operatorname{Spec}(R) \). One thus contains the \( fppf = \) flat cohomology groups over \( \operatorname{Spec}(R) \) with coefficients in objects of \( \mathcal{g} \). One has a fairly good computational understanding of this cohomology theory, and it is the purpose of this paper to show how, conversely, the knowledge of flat cohomology determines the object of \( \mathcal{g} \).

[29]B. Mazur: “Frobenius and the Hodge filtration,” Bull. Am. Math. Soc. 78 : 5 (1972), pp. 653–667. See also related article in Ann. Math. 98:1 (1973). MR 0330169 Zbl 0258.14006 article

The problem of \( p \)-adically estimating the number of solutions of an algebraic variety over a finite field of characteristic \( p \) was initiated by the classical result of Chevalley–Warning [1936].

Let \( G(x_1,\dots,x_n) \) be a polynomial with integral coefficients, of degree less than \( n \). The number of solutions of \[ G(x_1,\dots,x_n) \equiv 0 \mod p\] is divisible by \( p \).

My aim in this paper, is to explain a conjecture of N. Katz [1970], which provides quite sharp informatin concerning this general problem, and to indicate some theorems I have obtained which affirm his conjecture, under a mild hypothesis. Proofs of these theorems will be contained in [Mazur 1973].

[30]B. Mazur and J. Vélu: “Courbes de Weil de conducteur 26” [Weil curves of conductor 26], C. R. Acad. Sci., Paris, Sér. A 275 (1972), pp. A743–A745. MR 0320010 Zbl 0241.14015 article

[31]B. Mazur: “Rational points of abelian varieties with values in towers of number fields,” Invent. Math. 18 : 3–4 (1972), pp. 183–266. A Russian translation was published in two parts in Matematika 17:2 (1973) and Matematika 17:3 (1973). MR 0444670 Zbl 0245.14015 article

[32]B. Mazur: “Frobenius and the Hodge filtration (estimates),” Ann. Math. (2) 98 : 1 (July 1973), pp. 58–95. See also related article in Bull. Am. Math. Soc. 78:5 (1972). MR 0321932 Zbl 0261.14005 article

[33]B. Mazur: “Courbes elliptiques et symboles modulaires” [Elliptic curves and modular symbols], pp. 277–294 in Séminaire Bourbaki: 1971/72, exposés 400–417. Lecture Notes in Mathematics 317. Springer (Berlin), 1973. Exposé no. 414. In French. MR 0429921 Zbl 0276.14012 incollection

[34]B. Mazur: “Notes on étale cohomology of number fields,” Ann. Sci. Éc. Norm. Supér. (4) 6 : 4 (1973), pp. 521–552. MR 0344254 Zbl 0282.14004 article

In the setting of étale cohomology, M. Artin and J.-L. Verdier have proved a duality theorem for constructible abelian sheaves over the scheme \( \operatorname{Spec} D \) where \( D \) is the ring of integers in a number field (see [Artin and Verdier 1964]). This duality theorem contains within it Tate duality for finite Galois modules over local and global fields, and is indeed the natural extension of Tate duality to the context of such schemes.

In what follows, I would like to poke at this result from various angles, and make what I hope to be enlightening remarks and computations, so as to convey in a concrete way a sense of the information contained in this theorem. Rather than give its proof completely, I shall emphasize those aspects of the proof that related closely to number theory, and I shall try to suppress much of the general sheaf-theoretic techniques that go into it. I will assume some familiarity with étale cohomology, but whenever something may be done more explicitly (i.e. in the category of Galois modules over a field, or more generally over a discrete valuation ring) I shall do so and try to indicate the appropriate translation.

[35]B. Mazur: “Rational points of Abelian varieties with values in towers of number fields,” Matematika 17 : 2 (1973), pp. 3–57. Russian translation of part of an article in Invent. Math. 18:3–4 (1972). Zbl 0251.14015 article

[36]B. Mazur: “Book review: Stephen S. Shatz, ‘Profinite groups, arithmetic, and geometry’,” Am. Sci. 61 : 3 (May–June 1973), pp. 377. article

[37]B. Mazur: “Rational points of Abelian varieties with values in towers of number fields,” Matematika 17 : 3 (1973), pp. 114–136. Russian translation of part of an article in Invent. Math. 18:3–4 (1972). Zbl 0258.14009 article

[38]B. Mazur and J. Tate: “Points of order 13 on elliptic curves,” Invent. Math. 22 : 1 (1973–1974), pp. 41–49. MR 0347826 Zbl 0268.14009 article

[39]M. W. Hirsch and B. Mazur: Smoothings of piecewise linear manifolds. Annals of Mathematics Studies 80. Princeton University Press, 1974. MR 0415630 Zbl 0298.57007 book

[40]B. Mazur and P. Swinnerton-Dyer: “Arithmetic of Weil curves,” Invent. Math. 25 : 1 (1974), pp. 1–61. MR 0354674 Zbl 0281.14016 article

[41]B. Mazur and W. Messing: Universal extensions and one-dimensional crystalline cohomology. Lecture Notes in Mathematics 370. Springer (Berlin), 1974. MR 0374150 Zbl 0301.14016 book

[42]B. Mazur: “\( p \)-adic analytic number theory of elliptic curves and Abelian varieties over \( \mathbb{Q} \),” pp. 369–377 in Proceedings of the International Congress of Mathematicians (Vancouver, BC, 21–29 August 1974), vol. 1. Edited by R. D. James. Canadian Mathematical Congress (Montreal), 1975. MR 0439844 Zbl 0351.14011 incollection

[43]B. Mazur: “Eigenvalues of Frobenius acting on algebraic varieties over finite fields,” pp. 231–261 in Algebraic geometry (Arcata, CA, 29 July–16 August 1974). Edited by R. Hartshorne. Proceedings of Symposia in Pure Mathematics XXIX. American Mathematical Society (Providence, RI), 1975. MR 0371896 Zbl 0306.14011 incollection

[44]B. Mazur: Arithmetic in the geometry of symmetric spaces. A letter written to John Millson in the mid-1970s. misc

[45]B. Mazur and J.-P. Serre: “Points rationnels des courbes modulaires \( X_{0}(N) \) (d’après A. Ogg)” [Rational points of modular curves \( X_{0}(N) \) (after A. Ogg)], pp. 238–255 in Séminaire Bourbaki: 1974/75, exposés 453–470. Lecture Notes in Mathematics 514. Springer (Berlin), 1976. Exposé no. 469. In French. MR 0485882 Zbl 0346.14013 incollection

Jean-Pierre Serre	Related
Andrew Pollard Ogg	Related	Volume

[46]B. Mazur: “Book review: S. Lefschetz, ‘Applications of algebraic topology’,” Am. Sci. 64 : 4 (July–August 1976), pp. 460. article

[47]B. Mazur: “Modular curves and the Eisenstein ideal,” Publ. Math. Inst. Hautes Étud. Sci. 47 : 1 (1977), pp. 33–186. MR 488287 Zbl 0394.14008 article

[48]B. Mazur: “Rational points on modular curves,” pp. 107–148 in Modular functions of one variable, V: Proceedings of the second international conference (Bonn, 2–14 July 1976). Edited by J.-P. Serre and D. B. Zagier. Lecture Notes in Mathematics 601. Springer (Berlin), 1977. MR 0450283 Zbl 0357.14005 incollection

Don Bernard Zagier	Related
Jean-Pierre Serre	Related

[49]M. Artin and B. Mazur: “Formal groups arising from algebraic varieties,” Ann. Sci. Éc. Norm. Supér. (4) 10 : 1 (1977), pp. 87–131. MR 0457458 Zbl 0351.14023 article

[50]B. Mazur: “Book review: Ernst Edward Kummer, ‘Collected papers’,” Bull. Am. Math. Soc. 83 : 5 (1977), pp. 976–988. MR 1566995 article

[51]B. Mazur: “Rational isogenies of prime degree,” Invent. Math. 44 : 2 (1978), pp. 129–162. With an appendix by D. Goldfeld. MR 482230 Zbl 0386.14009 article

[52]O. Zariski: Collected papers, vol. III: Topology of curves and surfaces, and special topics in the theory of algebraic varieties. Edited by M. Artin and B. Mazur. Mathematicians of Our Time 12. MIT Press (Cambridge, MA), 1978. MR 0505104 Zbl 0446.14001 book

Michael Artin	Related
Oscar Zariski	Related

[53]B. Mazur: “On the arithmetic of special values of \( L \) functions,” Invent. Math. 55 : 3 (1979), pp. 207–240. MR 553997 Zbl 0426.14009 article

[54]J.-P. Serre: “Points rationnels des courbes modulaires \( X_{0}(N) \) [d’après Barry Mazur],” pp. 89–100 in Séminaire Bourbaki, 30e année (1977/78). Lecture Notes in Math. 710. Springer (Berlin), 1979. Exposé no. 511. MR 554216

[55] B. H. Gross: Arithmetic on elliptic curves with complex multiplication. Lecture Notes in Mathematics 776. Springer (Cham, Switzerland), 1980. With an appendix by B. Mazur. MR 563921 Zbl 0433.14032 book

[56]B. Mazur: “Book review: C. Herbert Clemens, ‘A scrapbook of complex curve theory’,” Am. Sci. 70 : 2 (March–April 1982), pp. 228. article

[57]B. Mazur and A. Wiles: “Analogies between function fields and number fields,” Am. J. Math. 105 : 2 (April 1983), pp. 507–521. Dedicated to André Weil for his 77th birthday. MR 701567 Zbl 0531.12015 article

Andrew John Wiles	Related
André Weil	Related

[58]B. Mazur and J. Tate: “Canonical height pairings via biextensions,” pp. 195–237 in Arithmetic and geometry: Papers dedicated to I. R. Sharfarevich on the occasion of his sixtieth birthday, vol. 1. Edited by M. Artin and J. Tate. Progress in Mathematics 35. Birkhäuser (Boston, MA), 1983. MR 717595 Zbl 0574.14036 incollection

Michael Artin	Related
John Torrence Tate, Jr.	Related
Igor Rostislavovich Shafarevich	Related

[59]B. Mazur: “Solution of math problem,” Science 222 : 4623 (4 November 1983), pp. 456. A letter to Science. article

[60]B. Mazur: “Modular curves and arithmetic,” pp. 185–211 in Proceedings of the International Congress of Mathematicians (Warsaw, 16–24 August 1983), vol. 1. Edited by Z. Ciesielski and C. Olech. PWN (Warsaw), 1984. MR 804682 Zbl 0597.14023 incollection

Czesław Olech	Related
Zbigniew Ciesielski	Related

[61]B. Mazur and A. Wiles: “Class fields of abelian extensions of \( \mathbb{Q} \),” Invent. Math. 76 : 2 (1984), pp. 179–330. To K. Iwasawa. MR 742853 Zbl 0545.12005 article

Andrew John Wiles	Related
Kenkichi Iwasawa	Related

[62]B. Mazur: On the arithmetic of curves, 1984. Lecture notes distributed in conjunction with the colloquium lectures given at the 90th annual meeting of the AMS, Louisville, KY, 25–28 January 1984. misc

[63]N. M. Katz and B. Mazur: Arithmetic moduli of elliptic curves. Annals of Mathematics Studies 108. Princeton University Press, 1985. MR 772569 Zbl 0576.14026 book

[64]B. Mazur and A. Wiles: “On \( p \)-adic analytic families of Galois representations,” Compos. Math. 59 : 2 (1986), pp. 231–264. With an appendix by Nigel Boston. MR 860140 Zbl 0654.12008 article

Andrew John Wiles	Related
Nigel Boston	Related

[65]B. Mazur, J. Tate, and J. Teitelbaum: “On \( p \)-adic analogues of the conjectures of Birch and Swinnerton-Dyer,” Invent. Math. 84 : 1 (1986), pp. 1–48. MR 830037 Zbl 0699.14028 article

The conjectures of Birch and Swinnerton-Dyer connect arithmetic invariants of an elliptic curve \( E \) over \( \mathbb{Q} \) (or more generally of an abelian variety over a global field) with the order of zero and the leading coefficient of the Taylor expansion of its Hasse–Weil zeta function at the “central point”. One of the arithmetic invariants entering into this conjecture is the “regulator of \( E \)”, i.e., the discriminant of the quadratic form on \( E(\mathbb{Q}) \) defined by the “canonical height pairing”.

If \( E \) is an elliptic curve over \( \mathbb{Q} \) parametrized by modular functions (a Weil curve) then the \( p \)-adic analogue of its Hasse–Weil \( L \)-function has been defined, and recently \( p \)-adic theories analogous to the theory of canonical height have been developed. It seemed to us, then, to be an appropriate time to embark on the project of formulating a \( p \)-adic analogue of the conjecture of Birch and Swinnerton-Dyer, and gathering numerical data in its support. It also seemed, at the outset, that this would be a relatively routine project.

The project has proved to be anything but routine, and this article is an attempt to report on our findings so far.

Jeremy Thau Teitelbaum	Related
John Torrence Tate, Jr.	Related

[66]B. Mazur: “Arithmetic on curves,” Bull. Am. Math. Soc., New Ser. 14 : 2 (April 1986), pp. 207–259. MR 828821 Zbl 0593.14021 article

[67]M. Artin and B. Mazur: Etale homotopy (Harvard University, 1965–1966), 2nd edition. Lecture Notes in Mathematics 100. Springer (Berlin), 1986. Republication of the 1969 original. MR 883959 book

[68]J. Moser, V. Strassen, J. W. Milnor, M. F. Atiyah, and B. Mazur: Talks on the medal and prize winners, 1986. 78 minute VHS videocassette. American Mathematical Society (Providence, RI). A group of four filmed lectures on the work of four prize-winning scholars in the fields of mathematics and computer science. Mazur lectures on the work of Fields medalist Gerd Faltings. Filmed at the International Congress of Mathematicians, Berkeley, CA, 3–11 August 1986. misc

Leslie G. Valiant	Related
M. F. Atiyah	Related	Volume
M. H. Freedman	Related	Volume
Gerd Faltings	Related
Volker Strassen	Related
John W. Milnor	Related
Jürgen K. Moser	Related
S. K. Donaldson	Related

[69]B. Mazur: “On some of the mathematical contributions of Gerd Faltings,” pp. 7–12 in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 3–11 August 1986), vol. 1. Edited by A. M. Gleason. American Mathematical Society (Providence, RI), 1987. MR 934210 Zbl 0663.01002 incollection

Andrew Mattei Gleason	Related	Volume
Gerd Faltings	Related

[70]B. Mazur and J. Tate: “Refined conjectures of the ‘Birch and Swinnerton-Dyer type’,” Duke Math. J. 54 : 2 (1987), pp. 711–750. To Yuri Manin on the occasion of his fiftieth birthday. MR 899413 Zbl 0636.14004 article

The idea behind the present article is that, in certain instances, arithmetic conjectures concerning the special values of derivatives of \( p \)-adic \( L \) functions can be “refined” to obtain formulations of stronger conjectures. These stronger conjectures avoid any mention of \( p \)-adic \( L \) functions and therefore obviate the necessity of constructing the \( p \)-adic \( L \) functions for the statement of the conjectures. Moreover, they avoid any reference to a prime number \( p \), and require no \( p \)-adic limiting process; they should ultimately be phrased, perhaps, in adelic language. In this paper, however, we state our conjectures “at a finite layer \( M \),” where \( M \) is a possibly composite number (somewhat restricted). Even when \( M = p \), however, our conjecture “at layer \( p \)” is not implied by the analogous conjecture for the \( p \)-adic \( L \) function. Indeed, our conjecture predicts congruence formulas modulo divisors of \( p - 1 \), in this case. When \( M \) is a product of distinct primes, our conjectured congruence formulas involve what seems to us to be a thoroughgoing mixture of phenomena related to those prime divisors.

John Torrence Tate, Jr.	Related
Yuri Ivanovich Manin	Related

[71]Algebraic number theory: In honor of K. Iwasawa (Berkeley, CA, 20–24 January 1987). Edited by J. Coates, R. Greenberg, B. Mazur, and I. Satake. Advanced Studies in Pure Mathematics 17. Academic Press (Boston), 1989. Collected papers dedicated to K. Iwasawa on his 70th birthday and papers based on the lectures delivered at the workshop on Iwasawa theory and special values of \( L \)-functions. MR 1097604 Zbl 0721.00006 book

Ichirō Satake	Related
Ralph Greenberg	Related
Kenkichi Iwasawa	Related
John Henry Coates	Related

[72]N. Boston and B. Mazur: “Explicit universal deformations of Galois representations,” pp. 1–21 in Algebraic number theory: In honor of K. Iwasawa (Berkeley, CA, 20–24 January 1987). Edited by J. Coates, R. Greenberg, B. Mazur, and I. Satake. Advanced Studies in Pure Mathematics 17. Academic Press (Boston, MA), 1989. MR 1097607 Zbl 0739.11021 incollection

Ichirō Satake	Related
Ralph Greenberg	Related
Kenkichi Iwasawa	Related
John Henry Coates	Related
Nigel Boston	Related

[73]B. Mazur: “Deforming Galois representations,” pp. 385–437 in Galois groups over \( \mathbb{Q} \) (Berkeley, CA, 23–27 March 1987). Edited by Y. Ihara, K. Ribet, and J.-P. Serre. Mathematical Sciences Research Institute Publications 16. Springer (New York), 1989. For A. Mazur. Available open access here. MR 1012172 Zbl 0714.11076 incollection

A. Mazur	Related
Yasutaka Ihara	Related
Jean-Pierre Serre	Related
Kenneth A. Ribet	Related

[74]A. M. Gleason, A. Jaffe, B. Mazur, R. H. Herman, C. H. Clemens, J. Kollár, K. Gawędzki, C. Soulé, and M. Sipser: “ICM-90,” Notices Am. Math. Soc. 37 : 9 (1990), pp. 1209–1216. Report on the International Congress of Mathematicians held in Kyoto, 21–29 August 1990. MR 1076560 Zbl 1194.01041 article

János Kollár	Related
Arthur Jaffe	Related
Andrew Mattei Gleason	Related	Volume
C. Herbert Clemens	Related
Krzysztof Gawędzki	Related
Richard Howard Herman	Related
Michael Fredric Sipser	Related
Christophe Soulé	Related

[75]B. Mazur and J. Tilouine: “Représentations galoisiennes, différentielles de Kähler et ‘conjectures principales’” [Galois representations, Kähler differentials and the ‘main conjectures’], Publ. Math. Inst. Hautes Étud. Sci. 71 (1990), pp. 65–103. In French. MR 1079644 Zbl 0744.11053 article

[76]B. Mazur: “Two-dimensional \( p \)-adic Galois representations unramified away from \( p \),” Compos. Math. 74 : 2 (1990), pp. 115–133. MR 1047735 Zbl 0773.11036 article

[77]B. Mazur: “Number theory as gadfly,” pp. 15–33 in Number theory: Proceedings of a symposium (Washington, DC, 4 May 1989). National Research Council (Washington, DC), 1990. From this presentation evolved an article with the same title which was published in Am. Math. Mon. 98:7 (1991). Zbl 0764.11022 incollection

[78]B. Mazur, A. Beilinson, and B. Feigin: Notes on conformal field theory. Notes, Harvard University. techreport

Boris L. Feigin	Related
Alexander A. Beilinson	Related

[79]H. Gillet, B. Mazur, and C. Soulé: “A note on a classical theorem of Blichfeldt,” Bull. London Math. Soc. 23 : 2 (1991), pp. 131–132. MR 1122898 Zbl 0746.52021 article

Christophe Soulé	Related
Henri Antoine Gillet	Related

[80]B. Mazur and K. A. Ribet: “Two-dimensional representations in the arithmetic of modular curves,” pp. 215–255 in Courbes modulaires et courbes de Shimura [Modular curves and Shimura curves] (Orsay, France, 1987–1988). Astérisque 196–197. Société Mathématique de France (Paris), 1991. MR 1141460 Zbl 0780.14015 incollection

[81]F. Gouvêa and B. Mazur: “The square-free sieve and the rank of elliptic curves,” J. Am. Math. Soc. 4 : 1 (January 1991), pp. 1–23. MR 1080648 Zbl 0725.11027 article

[82]B. Mazur and J. Tate: “The \( p \)-adic sigma function,” Duke Math. J. 62 : 3 (1991), pp. 663–688. MR 1104813 Zbl 0735.14020 article

[83]B. Mazur: “Number theory as gadfly,” Am. Math. Mon. 98 : 7 (August–September 1991), pp. 593–610. This evolved from the presentation with the same title which was published in Number theory: Proceedings of a symposium (1990). MR 1121312 Zbl 0764.11021 article

[84]F. Gouvêa and B. Mazur: “Families of modular eigenforms,” Math. Comput. 58 : 198 (April 1992), pp. 793–805. MR 1122070 Zbl 0773.11030 article

[85]B. Mazur: “The topology of rational points,” Exp. Math. 1 : 1 (1992), pp. 35–45. MR 1181085 Zbl 0784.14012 article

[86]B. Mazur: Galois deformations and Hecke curves, 1993. Harvard course notes, Fall 1993. misc

[87]F. Q. Gouvêa and B. Mazur: “On the characteristic power series of the \( U \) operator,” Ann. Inst. Fourier 43 : 2 (1993), pp. 301–312. MR 1220270 Zbl 0779.11022 article

[88]B. Mazur: “On the passage from local to global in number theory,” Bull. Am. Math. Soc., New Ser. 29 : 1 (July 1993), pp. 14–50. MR 1202293 Zbl 0813.14016 article

[89]B. Mazur: “On monodromy invariants occurring in global arithmetic, and Fontaine’s theory,” pp. 1–20 in \( p \)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 12–16 August 1991). Edited by B. Mazur and G. Stevens. Contemporary Mathematics 165. American Mathematical Society (Providence, RI), 1994. MR 1279599 Zbl 0846.11039 incollection

[90]\( p \)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 12–16 August 1991). Edited by B. Mazur and G. Stevens. Contemporary Mathematics 165. American Mathematical Society (Providence, RI), 1994. MR 1279598 Zbl 0794.00016 book

[91]B. Mazur: “Questions of decidability and undecidability in number theory,” J. Symb. Log. 59 : 2 (June 1994), pp. 353–371. In memory of David Lachterman. MR 1276620 Zbl 0814.11059 article

[92]E. M. Friedlander and B. Mazur: “Correspondence homomorphisms for singular varieties,” Ann. Inst. Fourier 44 : 3 (1994), pp. 703–727. MR 1303882 Zbl 0811.14007 article

[93]E. M. Friedlander and B. Mazur: Filtrations on the homology of algebraic varieties. Memoirs of the American Mathematical Society 529. July 1994. With an appendix by Daniel Quillen. MR 1211371 Zbl 0841.14019 book

Daniel Gray Quillen	Related
Eric Mark Friedlander	Related

[94]J.-M. Fontaine and B. Mazur: “Geometric Galois representations,” pp. 41–78 in Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 18–21 December 1993). Edited by J. Coates and S.-T. Yau. Series in Number Theory 1. International Press (Somerville, MA), 1995. MR 1363495 Zbl 0839.14011 incollection

The main point of this paper is to present in detail some of our conjectures concerning \( p \)-adic representations of the Galois groups of number fields (cf. [Fontaine 1988] for a brief synopsis). The motivating idea behind our conjectures is simply the hope that, given an irreducible \( p \)-adic representation \( \rho \) of \( G_K \), the Galois group of a number field \( K \) (assumed unramified except at a finite number of places), one can find necessary and sufficient local conditions (on the restriction of the representation to the decomposition grpous \( G_{K_{\nu}} \) at the finite set of places \( \nu \) of \( K \) of residual characteristic \( p \)) for the representation \( \rho \) to “come from algebraic geometry”. The local condition we have in mind is that \( \rho \) restricted to \( G_{K_{\nu}} \) be potentially semi-stable. This will be made precise in Conjecture 1 of §1 below.

John Henry Coates	Related
Jean-Marc Fontaine	Related
Shing-Tung Yau	Related

[95]B. Mazur: Fermat’s last theorem, 1995. 60 minute videocassette. American Mathematical Society (Providence, RI). A lecture presented in Vancouver, BC, 15 August 1993. MR 1419093 Zbl 0974.11500 misc

[96]F. Q. Gouvêa and B. Mazur: “Searching for \( p \)-adic eigenfunctions,” Math. Res. Lett. 2 : 5 (1995), pp. 515–536. To A. O. L. Atkin, on his retirement. MR 1359960 Zbl 0851.11031 article

Arthur Oliver Lonsdale Atkin	Related
Fernando Quadros Gouvêa	Related

[97]L. Caporaso, J. Harris, and B. Mazur: “How many rational points can a curve have?,” pp. 13–31 in The moduli space of curves (Texel Island, Netherlands, April 1994). Edited by R. H. Dijkgraaf, C. Faber, and G. B. M. van der Geer. Progress in Mathematics 129. Birkhäuser (Boston, MA), 1995. MR 1363052 Zbl 0862.14012 incollection

Carel Filip Faber	Related
Gerard B. M. van der Geer	Related
Joseph Daniel Harris	Related
Lucia Caporaso	Related
Robbert Dijkgraaf	Related

[98]B. Mazur: “Speculations about the topology of rational points: An up-date,” pp. 165–182 in Columbia University number theory seminar (New York, 1992). Astérisque 228. 1995. MR 1330932 Zbl 0851.14009 incollection

[99]S. Kamienny and B. Mazur: “Rational torsion of prime order in elliptic curves over number fields,” pp. 81–100 in Columbia University number theory seminar (New York, 1992). Astérisque 228. Société Mathématique de France (Paris), 1995. With an appendix by A. Granville. MR 1330929 Zbl 0846.14012 incollection

Andrew James Granville	Related
Sheldon Kamienny	Related

[100]D. Abramovich: “Formal finiteness and the torsion conjecture on elliptic curves: A footnote to a paper: ‘Rational torsion of prime order in elliptic curves over number fields’ by S. Kamienny and B. Mazur,” pp. 3, 5–17 in Columbia University Number Theory Seminar (New York, 1992). Astérisque 228. 1995. MR 1330925

[101]B. Mazur: Questions about number. Lecture notes, Harvard University. An expository lecture on the ABC conjecture, written for a volume entitled New Directions in Mathematics. techreport

[102]B. Mazur: “An ‘infinite fern’ in the universal deformation space of Galois representations,” pp. 155–193 in Journées arithmétiques [Arithmetic days] (Barcelona, July 1995), published as Collect. Math. 48 : 1–2. Springer (Berlin), 1997. In memory of R. H. Bing. MR 1464022 Zbl 0865.11046 incollection

I hope this article will be helpful to people who might want a quick overview of how modular representations fit into the theory of deformations of Galois representations.

There is also a more specific aim: to sketch a construction of a “point-set topological” configuration (the image of an “infinite fern”) which emerges from consideration of modular representations in the universal deformation space of all Galois representations. This is a configuration hinted at in [Gouvêa and Mazur 1992], but now, thanks to some recent important work of Coleman [1996, 1997], it is something one can actually produce! The “infinite fern” is joint work with F. Q. Gouvêa, and will be the subject of slightly more systematic study in a future paper in which some consequences of its existence will be discussed.

[103]B. Mazur: “An introduction to the deformation theory of Galois representations,” pp. 243–311 in Modular forms and Fermat’s last theorem (Boston, MA, 9–18 August 1995). Edited by G. Cornell, J. H. Silverman, and G. Stevens. Springer (New York), 1997. MR 1638481 Zbl 0901.11015 incollection

Gary Cornell	Related
Glenn Howard Stevens	Related
Joseph Hillel Silverman	Related

[104]Elliptic curves and modular forms: Papers from the NAS Colloquium (Washington, DC, 15–17 March 1996), published as Proc. Natl. Acad. Sci. USA 94 : 21. Issue edited by B. Mazur and K. Rubin. National Academy of Sciences (Washington, DC), 1997. MR 1491966 Zbl 0882.00034 booklet

[105]B. Mazur: “Conjecture,” pp. 197–210 in Proof and progress in mathematics (Boston, MA, 1996), published as Synthese 111 : 2. Springer (Dordrecht), 1997. MR 1465271 Zbl 1052.00513 incollection

[106]D. Eisenbud and B. Mazur: “Evolutions, symbolic squares, and Fitting ideals,” J. Reine Angew. Math. 488 (1997), pp. 189–201. MR 1465370 Zbl 0912.13010 ArXiv alg-geom/9511008 article

Given a reduced local algebra \( T \) over a suitable ring or field \( k \) we study the question of whether there are nontrivial algebra surjections \( R\to T \) which induce isomorphism \[ \Omega_{R/k}\otimes T\to \Omega_{T/K} .\] Such maps, called evolutions, arise naturally in the study of Hecke algebras, as in the recent work of Wiles, Taylor–Wiles, and Flach. We show that the existence of non-trivial evolutions of an algebra \( T \) can be characterized in terms of the symbolic square of an ideal \( I \) defining \( T \), and under broad circumstances our question amounts to asking when the symbolic square of \( I \) is in the maximal ideal times \( I \). We give a characterization of the symbolic square in terms of Fitting ideals. Using this and other techniques we show that certain classes of reduced algebras — codimension-2 Cohen–Macaulay, codimension-3 Gorenstein, licci algebras in general, and some others — admit no nontrivial evolutions. On the other hand we give examples showing that non-trivial evolutions of reduced Cohen–Macaulay algebras of codimension 3 do exist in every positive characteristic.

[107]L. Caporaso, J. Harris, and B. Mazur: “Uniformity of rational points,” J. Am. Math. Soc. 10 : 1 (January 1997), pp. 1–35. MR 1325796 Zbl 0872.14017 article

Lucia Caporaso	Related
Joseph Daniel Harris	Related

[108]B. Mazur: “Book review: Alice Fulton, ‘Sensual math’,” Harvard Review 12 (1997), pp. 170–172. article

[109]J. Harris, B. Mazur, and R. Pandharipande: “Hypersurfaces of low degree,” Duke Math. J. 95 : 1 (1998), pp. 125–160. MR 1646558 Zbl 0991.14018 article

Rahul Pandharipande	Related
Joseph Daniel Harris	Related

[110]B. Mazur: “Open problems regarding rational points on curves and varieties,” pp. 239–265 in Galois representations in arithmetic algebraic geometry (Durham, UK, 9–18 July 1996). Edited by A. J. Scholl and R. L. Taylor. London Mathematical Society Lecture Note Series 254. Cambridge University Press, 1998. MR 1696485 Zbl 0943.14009 incollection

Robert L. Taylor	Related
Anthony James Scholl	Related

[111]F. Q. Gouvêa and B. Mazur: “On the density of modular representations,” pp. 127–142 in Computational perspectives on number theory: Proceedings of a conference in honor of A. O. L. Atkin (Chicago, September 1995). Edited by D. A. Buell and J. T. Teitelbaum. AMS/IP Studies in Advanced Mathematics 7. American Mathematical Society (Providence, RI), 1998. MR 1486834 Zbl 1134.11324 incollection

Fernando Quadros Gouvêa	Related
Arthur Oliver Lonsdale Atkin	Related
Duncan Alan Buell	Related
Jeremy Thau Teitelbaum	Related

[112]R. Coleman and B. Mazur: “The eigencurve,” pp. 1–113 in Galois representations in arithmetic algebraic geometry (Durham, UK, 9–18 July 1996). Edited by A. J. Scholl and R. L. Taylor. London Mathematical Society Lecture Note Series 254. Cambridge University Press, 1998. MR 1696469 Zbl 0932.11030 incollection

Robert Frederick Coleman	Related
Robert L. Taylor	Related
Anthony James Scholl	Related

[113]B. Mazur: “Book review: Rochelle Owens, ‘New and selected poems, 1961–1996’,” Harvard Review 14 (1998), pp. 135–137. article

[114]B. Mazur: “Book review: Martha Ronk, ‘Eyetrouble’,” Harvard Review 15 (1998), pp. 171–173. article

[115] Current developments in mathematics 1997 (Cambridge, MA, 1997). Edited by R. Bott, A. Jaffe, S.-T. Yau, D. Jerison, G. Lusztig, and I. Singer. Current Developments in Mathematics. International Press (Boston), 1999. A preliminary version was published in 1997. MR 1698851 Zbl 0927.00019 book

George Lusztig	Related
Arthur Jaffe	Related
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume
David Saul Jerison	Related

[116] Current developments in mathematics 1998 (Cambridge, MA, 1998). Edited by B. Mazur, W. Schmid, S.-T. Yau, D. Jerison, I. M. Singer, and D. Stroock. Current Developments in Mathematics. International Press (Somerville, MA), 1999. MR 1772320 Zbl 0963.00022 book

Daniel W. Stroock	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume
David Saul Jerison	Related

[117] Current developments in mathematics 1999 (Cambridge, MA, 1999). Edited by B. Mazur, W. Schmid, S.-T. Yau, D. Jerison, I. M. Singer, and D. Stroock. Current Developments in Mathematics. International Press (Somerville, MA), 1999. MR 1990245 Zbl 1051.00008 book

Daniel W. Stroock	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume
David Saul Jerison	Related

[118]B. Mazur: “Visualizing elements of order three in the Shafarevich–Tate group,” Asian J. Math. 3 : 1 (March 1999), pp. 221–232. MR 1701928 Zbl 0958.11043 article

[119]B. Mazur: “Open problems in number theory,” pp. 199–203 in Current developments in mathematics, 1997 (Sommerville, MA). Edited by R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau. International Press, 1999. MR 1700300 incollection

George Lusztig	Related
Arthur Jaffe	Related
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume
David Saul Jerison	Related

[120]B. Mazur: “Book review: Kurt Brown (ed.), ‘Verse & universe: Poems about science and mathematics’,” Harvard Review 16 (1999), pp. 141–146. article

[121]Mathematics: Frontiers and perspectives. Edited by V. Arnold, M. Atiyah, P. Lax, and B. Mazur. American Mathematical Society (Providence, RI), 2000. MR 1754762 Zbl 1047.00015 book

M. F. Atiyah	Related	Volume
Peter D. Lax	Related
Vladimir Igorevich Arnold	Related

[122] “2000 Steele Prizes,” Notices Amer. Math. Soc. 47 : 4 (April 2000), pp. 477–480. MR 1745621 Zbl 1040.01503 article

Isadore M. Singer	Related	Volume
John Horton Conway	Related

[123]B. Mazur: “Abelian varieties and the Mordell–Lang conjecture,” pp. 199–227 in Model theory, algebra, and geometry. Edited by D. Haskell, A. Pillay, and C. Steinhorn. Mathematical Sciences Research Institute Publications 39. Cambridge University Press, 2000. MR 1773708 Zbl 0990.14006 incollection

Deirdre Haskell	Related
Anand Pillay	Related
Charles Ira Steinhorn	Related

[124]B. Mazur: “The theme of \( p \)-adic variation,” pp. 433–459 in Mathematics: Frontiers and perspectives. Edited by V. Arnold, M. Atiyah, P. Lax, and B. Mazur. American Mathematical Society (Providence, RI), 2000. MR 1754790 Zbl 0959.14008 incollection

M. F. Atiyah	Related	Volume
Peter D. Lax	Related
Vladimir Igorevich Arnold	Related

[125]J. E. Cremona and B. Mazur: “Visualizing elements in the Shafarevich–Tate group,” Exp. Math. 9 : 1 (2000), pp. 13–28. To Bryan Birch. MR 1758797 Zbl 0972.11049 article

We review a number of ways of “visualizing” the elements of the Shafarevich–Tate group of an elliptic curve \( E \) over a number field \( K \). We are specifically interested in cases where the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of \( X_0(N) \), where \( N \) is the conductor of the elliptic curve). For a given such \( E \) with nontrivial Shafarevich–Tate group, we pose the question:

Are all the curves of genus one representing elements of the Shafarevich–Tate group of \( E \) isomorphic (over the rationals) to curves contained in a (single) abelian surface \( A \), itself defined over the rationals, containing \( E \) as a subelliptic curve, and contained in turn in the new part of the jacobian of a modular curve \( X_0(N) \)?

At first view, one might imagine that there are few \( E \) with nontrivial Shafarevich–Tate group for which the answer is yes. Indeed we have a small number of examples where the answer is no, and it is very likely that the answer will be no if the order of the Shafarevich–Tate group is large enough. Nonetheless, among all (modular) elliptic curves \( E \) as above, with conductors up to 5500 and with no rational point of order 2, we have found the answer to the question to be yes in the vast majority of cases. We are puzzled by this and wonder whether there is some conceptual reason for it. We present a substantial amount of data relating to the curves investigated.

Bryan John Birch	Related
John Edward Cremona	Related

[126]B. Mazur: “Questions about powers of numbers,” Notices Am. Math. Soc. 47 : 2 (2000), pp. 195–202. MR 1740351 Zbl 1041.11065 article

[127]D. Kazhdan, B. Mazur, and C.-G. Schmidt: “Relative modular symbols and Rankin–Selberg convolutions,” J. Reine Angew. Math. 519 (2000), pp. 97–141. MR 1739728 Zbl 0941.11020 article

David A. Kazhdan	Related
Claus-Günther Schmidt	Related

[128]B. Mazur: “Open wide,” Harvard Review 14 (2001), pp. 67. A poem. article

[129]B. Mazur: “LIBR,” Harvard Review 20 (2001), pp. 24–25. A poem. article

[130]Current developments in mathematics, 2001 (Cambridge, MA, 2001). Edited by A. J. de Jong, D. Jerison, G. Lusztig, B. Mazur, W. Schmid, and S.-T. Yau. International Press (Somerville, MA), 2001. MR 1971375 Zbl 1013.00015 book

George Lusztig	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
Aise Johan de Jong	Related
David Saul Jerison	Related

[131]Current developments in mathematics, 2000 (Cambridge, MA, 2000). Edited by B. Mazur, W. Schmid, S.-T. Yau, A. J. de Jong, D. Jerison, and G. Lusztig. International Press (Somerville, MA), 2001. MR 1882532 Zbl 0990.00013 book

George Lusztig	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
Aise Johan de Jong	Related
David Saul Jerison	Related

[132]B. Mazur: “Qu’est-ce qu’un probleme? Une hypothese? Une conjecture?” [What is a problem? A hypothesis? A conjecture?], La Recherche 32 : 346 (2001), pp. 18. article

[133]B. Mazur: “Book review: Laure-Anne Bosselaar, ‘Urban nature: Poems about wildlife in the city’,” Harvard Review 16 (2001), pp. 141–144. article

[134]B. Mazur and K. Rubin: “Elliptic curves and class field theory,” pp. 185–195 in Proceedings of the International Congress of Mathematicians (Beijing, 20–28 August 2002), vol. 2: Invited lectures. Edited by L. Tatsien, C. Zhijie, X. Mi, and Z. Chunlian. Higher Education Press (Beijing), 2002. MR 1957032 Zbl 1036.11023 ArXiv math/0304235 incollection

Suppose \( E \) is an elliptic curve defined over \( \mathbb{Q} \). At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjecturally) produces almost all of the rational points on \( E \) over those fields. Those conjectures are to a large extent settled by recent work of Vatsal and of Cornut, building on work of Kolyvagin and others. In this paper we describe a collection of interrelated conjectures still open regarding the variation of Mordell–Weil groups of \( E \) over abelian extensions of imaginary quadratic fields, and suggest a possible algebraic framework to organize them.

Cai Zhijie	Related
Xue Mi	Related
Karl Cooper Rubin	Related
Zhou Chunlian	Related
Li Tatsien	Related

[135]F. La Nave and B. Mazur: “Reading Bombelli,” Math. Intell. 24 : 1 (2002), pp. 12–21. MR 1889932 Zbl 1023.01005 article

Rafael Bombelli	Related
Federica La Nave	Related

[136]B. Mazur: “Math and art without the filters,” LA Times (May 18 2003). article

[137]Current developments in mathematics, 2003. Edited by B. Mazur, W. Schmid, S.-T. Yau, A. J. de Jong, D. Jerison, and G. Lusztig. International Press (Somerville, MA), 2003. MR 2132322 Zbl 1090.00008 book

George Lusztig	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
Aise Johan de Jong	Related
David Saul Jerison	Related

[138]B. Mazur and K. Rubin: “Studying the growth of Mordell–Weil,” Doc. Math. Extra volume (2003), pp. 585–607. To Kazuya Kato, with admiration. MR 2046609 Zbl 1142.11339 article

Kazuya Kato	Related
Karl Cooper Rubin	Related

[139]Current developments in mathematics, 2002 (Cambridge, MA, 15–17 November 2002). Edited by D. Jerison, G. Lusztig, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2003. Proceedings of a seminar in honor of Wilfried Schmid and George Lusztig. MR 2051781 Zbl 1033.00012 book

Tomasz Stanislaw Mrowka	Related
George Lusztig	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[140]B. Mazur: Imagining numbers (particularly the square root of minus fifteen). Farrar, Straus and Giroux (New York), 2003. MR 1950850 Zbl 1049.00502 book

[141]B. Mazur: History of mathematics as a tool. Lecture notes, Harvard University, 2003. Lecture notes for seminar organized by Anthony Liu and Lucia Mocz. techreport

[142]P. Diaconis: “The problem of thinking too much,” Bull. Am. Acad. Arts Sci. 56 : 3 (2003), pp. 26–38. With an introduction by Barry Mazur. article

[143]B. Mazur: “Plus symétrique que la sphère” [More symmetric than the sphere], Pour la Science. Dossier 41 (October–December 2003), pp. 78–85. article

[144]T. Graber, J. Harris, B. Mazur, and J. Starr: “Arithmetic questions related to rationally connected varieties,” pp. 531–542 in The legacy of Niels Henrik Abel: The Abel bicentennial (Oslo, 3–8 June 2002). Edited by O. A. Laudal and R. Piene. Springer (Berlin), 2004. MR 2077583 Zbl 1071.14053 incollection

Ragni Piene	Related
Thomas Benjamin Graber	Related
Niels Henrik Abel	Related
Olav Arnfinn Laudal	Related
Joseph Daniel Harris	Related
Jason Michael Starr	Related

[145]T. Graber, J. Harris, B. Mazur, and J. Starr: “Jumps in Mordell–Weil rank and arithmetic surjectivity,” pp. 141–147 in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 11–20 December 2002). Edited by B. Poonen and Y. Tschinkel. Progress in Mathematics 226. Birkhäuser (Boston, MA), 2004. MR 2029867 Zbl 1075.14021 incollection

Thomas Benjamin Graber	Related
Bjorn Mikhail Poonen	Related
Joseph Daniel Harris	Related
Jason Michael Starr	Related
Yuri Tschinkel	Related

[146]B. Mazur: “What is … a motive?,” Notices Am. Math. Soc. 51 : 10 (2004), pp. 1214–1216. MR 2104916 Zbl 1054.14511 article

[147]B. Mazur and K. Rubin: “Kolyvagin systems,” Mem. Am. Math. Soc. 168 : 799 (2004), pp. viii+96. Available open access here. MR 2031496 Zbl 1055.11041 article

[148]B. Mazur: “Perturbations, deformations, and variations (and ‘near-misses’) in geometry, physics, and number theory,” Bull. Am. Math. Soc., New Ser. 41 : 3 (2004), pp. 307–336. In memory of René Thom. MR 2058289 Zbl 1057.11033 article

[149]B. Mazur and K. Rubin: “Introduction to Kolyvagin systems,” pp. 207–221 in Stark’s conjectures: Recent work and new directions (Baltimore, MD, 5–9 August 2002). Edited by D. Burns, C. Popescu, J. Sands, and D. Solomon. Contemporary Mathematics 358. American Mathematical Society (Providence, RI), 2004. Available open access here. MR 2088718 Zbl 1081.11069 incollection

Cristian Dumitru Popescu	Related
Karl Cooper Rubin	Related
Jonathan Winslow Sands	Related
Daniel Matthew Burns, Jr.	Related
David Robert Solomon	Related

[150]B. Mazur and K. Rubin: “Pairings in the arithmetic of elliptic curves,” pp. 151–163 in Modular curves and abelian varieties (Barcelona, 15–18 July 2002). Edited by J. Cremona, J.-C. Lario, J. Quer, and K. Ribet. Progress in Mathematics 224. Birkhäuser (Basel), 2004. Available open access here. MR 2058649 Zbl 1071.11028 incollection

Jordi Quer Bosor	Related
Karl Cooper Rubin	Related
Kenneth A. Ribet	Related
John Edward Cremona	Related
Joan-Carles Lario	Related

[151]B. Mazur: “On the absence of time in mathematics,” Learn. Math. 24 : 3 (November 2004), pp. 18–20. article

[152]A. Agashe and W. Stein: “Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero,” Math. Comput. 74 : 249 (2005), pp. 455–484. With an appendix by J. Cremona and B. Mazur. MR 2085902 Zbl 1084.11033 article

William Arthur Stein	Related
Amod Sadanand Agashe	Related
John Edward Cremona	Related

[153]B. Mazur and K. Rubin: “Finding large Selmer groups,” J. Differ. Geom. 70 : 1 (2005), pp. 1–22. Available open access here. MR 2192059 Zbl 1211.11068 article

[154]T. Graber, J. Harris, B. Mazur, and J. Starr: “Rational connectivity and sections of families over curves,” Ann. Sci. Éc. Norm. Supér. (4) 38 : 5 (September–October 2005), pp. 671–692. MR 2195256 Zbl 1092.14062 article

A “pseudosection” of the total space \( X \) of a family of varieties over a base variety \( B \) is a subvariety of \( X \) whose general fiber over \( B \) is rationally connected. We prove a theorem which is a converse, in some sense, of the main result of [Graber et al. 2003]: a family of varieties over \( B \) has a “pseudosection” if its restriction to each one-parameter subfamily has a “pseudosection” (which, due to [Graber et al. 2003], holds if and only if each one-parameter subfamily has a section). This is used to give a negative answer to a question posed by Serre to Grothendieck: There exists a family of \( O \)-acyclic varieties (a family of Enriques surfaces) parametrized by \( P^1 \) with no section.

Jason Michael Starr	Related
Thomas Benjamin Graber	Related
Joseph Daniel Harris	Related

[155]B. Mazur and K. Rubin: “Organizing the arithmetic of elliptic curves,” Adv. Math. 198 : 2 (2005), pp. 504–546. MR 2183387 Zbl 1122.11038 article

Suppose that \( E \) is an elliptic curve defined over a number field \( K \), \( p \) is a rational prime, and \( K_{\infty} \) is the maximal \( \mathbb{Z}_p \)-power extension of \( K \). In previous work [Mazur and Rubin 2002, 2004] we discussed the possibility that much of the arithmetic of \( E \) over \( K_{\infty} \) (i.e., the Mordell–Weil groups and their \( p \)-adic height pairings, the Shafarevich–Tate groups and their Cassels pairings, over all finite extensions of \( K \) in \( K_{\infty} \)) can be described efficiently in terms of a single skew-Hermitian matrix with entries drawn from the Iwasawa algebra of \( K_{\infty}/K \).

In this paper, using work of Nekovář [2006], we show that under not-too-stringent conditions such an “organizing” matrix does in fact exist. We also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly.

[156]B. Mazur: “Pourquoi les nombres premiers?” [Why the prime numbers?], Les Dossiers de la Recherche 20 (August–October 2005), pp. 20–24. article

[157]B. Mazur and P. Pesic: “On mathematics, imagination & the beauty of numbers,” Daedalus 134 : 2 (2005), pp. 124–130. article

[158]A. Ash and R. Gross: Fearless symmetry: Exposing the hidden patterns of numbers. Princeton University Press, 2006. With a foreword by Barry Mazur. MR 2229945 Zbl 1111.11001 book

Avner Dolnick Ash	Related
Robert Howard Gross	Related

[159]Current developments in mathematics, 2004 (Cambridge, MA, 2005). Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2006. MR 2459295 Zbl 1161.00012 book

Tomasz Stanislaw Mrowka	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[160]B. Mazur, W. Stein, and J. Tate: “Computation of \( p \)-adic heights and log convergence,” pp. 577–614 in John H. Coates’ sixtieth birthday, published as Doc. Math. Extra volume. Deutsche Mathematiker-Vereinigung (Bielefeld), 2006. MR 2290599 Zbl 1135.11034 incollection

This paper is about computational and theoretical questions regarding \( p \)-adic height pairings on elliptic curves over a global field \( K \). The main stumbling block to computing them efficiently is in calculating, for each of the completions \( K_v \) at the places \( v \) of \( K \) dividing \( p \), a single quantity: the value of the \( p \)-adic modular form \( \mathbf{E}_2 \) associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky–Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of \( \mathbf{E}_2 \) of an elliptic curve. We also discuss the \( p \)-adic convergence rate of canonical expansions of the \( p \)-adic modular form \( \mathbf{E}_2 \) on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that \( \mathbf{E}_2 \) is log convergent.

John Henry Coates	Related
William Arthur Stein	Related
John Torrence Tate, Jr.	Related

[161]Current developments in mathematics, 2004. Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2006. MR 2459289

[162]B. Mazur: A “Basic Notions” lecture about sign intuitions. Lecture notes, Harvard University, February 2006. Based on joint work with Karl Rubin. Notes to Basic Notions Seminar talk, given 6 February 2006. techreport

[163]B. Mazur: “Mathematics: Controlling our errors,” Nature 443 : 7107 (7 September 2006), pp. 38–40. article

[164]B. Mazur: “About the cover: Diophantus’s ‘Arithmetica’,” Bull. Amer. Math. Soc., New Ser. 43 : 3 (July 2006), pp. 399–401. article

[165]B. Mazur: Ruminations on my very minor running injury, 2007. A poetic response to Rumi’s “Unmarked boxes”. misc

Harold Schechter	Related
Kurt Patrick Brown	Related

[166]B. Bektemirov, B. Mazur, W. Stein, and M. Watkins: “Average ranks of elliptic curves: Tension between data and conjecture,” Bull. Am. Math. Soc., New Ser. 44 : 2 (April 2007), pp. 233–254. MR 2291676 Zbl 1190.11032 article

Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control — thanks to ideas of Kolyvagin — the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today.

We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

Baur Bektemirov	Related
Mark James Watkins	Related
William Arthur Stein	Related

[167]B. Mazur, K. Rubin, and A. Silverberg: “Twisting commutative algebraic groups,” J. Algebra 314 : 1 (August 2007), pp. 419–438. MR 2331769 Zbl 1128.14034 ArXiv math/0609066 article

If \( V \) is a commutative algebraic group over a field \( k \), \( \mathcal{O} \) is a commutative ring that acts on \( V \), and \( \mathcal{I} \) is a finitely generated free \( \mathcal{O} \)-module with a right action of the absolute Galois group of \( k \), then there is a commutative algebraic group \( \mathcal{O}\otimes_{\mathcal{O}} V \) over \( k \), which is a twist of a power of \( V \). These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties.

Alice Silverberg	Related
Karl Cooper Rubin	Related

[168]B. Mazur and K. Rubin: “Finding large Selmer rank via an arithmetic theory of local constants,” Ann. Math. (2) 166 : 2 (2007), pp. 579–612. MR 2373150 Zbl 1219.11084 ArXiv math/0512085 article

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields.

Suppose \( K/k \) is a quadratic extension of number fields, \( E \) is an elliptic curve defined over \( k \), and \( p \) is an odd prime. Let \( \mathcal{K}^- \) denote the maximal abelian \( p \)-extension of \( K \) that is unramified at all primes where \( E \) has bad reduction and that is Galois over \( k \) with dihedral Galois group (i.e., the generator \( c \) of \( \mathrm{Gal}(K/k) \) acts on \( \mathrm{Gal}(\mathcal{K}^-/K) \) by inversion). We prove (under mild hypotheses on \( p \)) that if the \( \mathbb{Z}_p \)-rank of the pro-\( p \) Selmer group \( \mathcal{S}_p(E/K) \) is odd, then \[ \mathrm{rank}_{\mathbb{Z}_p}\mathcal{S}_p(E/F)\geq [F:K] \] for every finite extension \( F \) of \( K \) in \( \mathcal{K}^- \).

[169]B. Mazur and K. Rubin: Growth of Selmer rank in nonabelian extensions of number fields. Preprint, 2007. ArXiv 0703363 techreport

[170]T. Dantzig: Number: The language of science, 4th reprinted edition. Edited by J. Mazur. Plume (New York), 2007. With a new foreword by Barry Mazur. book

Joseph Conrad Mazur	Related
Tobias Dantzig	Related

[171]L. R. Lieber: The education of T. C. Mits: What modern mathematics means to you, reprint edition. Paul Dry Books (Philadelphia), 2007. With a foreword by Barry Mazur. book

[172]L. R. Lieber: Infinty: Beyond the beyond the beyond, 2nd edition. Edited by B. Mazur. Paul Dry Books (Philadelphia), 2007. Edited and with a new foreword by Barry Mazur. book

[173]E. Galvin: Do the math. Tupelo Press (North Adams, MA), 2008. With a foreword by Barry Mazur. book

[174]B. Mazur: “When is one thing equal to some other thing?,” pp. 221–241 in Proof and other dilemmas: Mathematics and philosophy. Edited by B. Gold and R. A. Simons. Spectrum. Mathematical Association of America (Washington, DC), 2008. In memory of Saunders Mac Lane. MR 2452082 incollection

Bonnie Gold	Related
Roger Alan Simons	Related
S. Mac Lane	Related	Volume

[175]B. Mazur and K. Rubin: “Growth of Selmer rank in nonabelian extensions of number fields,” Duke Math. J. 143 : 3 (2008), pp. 437–461. MR 2423759 Zbl 1151.11023 article

Let \( p \) be an odd prime number, let \( E \) be an elliptic curve over a number field \( k \), and let \( F/k \) be a Galois extension of degree twice a power of \( p \). We study the \( \mathbb{Z}_p \)-corank \( \operatorname{rk}_p(E/F) \) of the \( p \)-power Selmer group of \( E \) over \( F \). We obtain lower bounds for \( \operatorname{rk}_p(E/F) \), generalizing the results in [Mazur and Rubin 2007], which applied to dihedral extensions.

If \( K \) is the (unique) quadratic extension of \( k \) in \( F \), if \[ G = \mathrm{Gal}(F/K) ,\] if \( G^+ \) is the subgroup of elements of \( G \) commuting with a choice of involution of \( F \) over \( k \), and if \( \operatorname{rk}_p(E/K) \) is odd, then we show that (under mild hypotheses) \[ \operatorname{rk}_p(E/F) \geq [G:G^+] .\]

As a very specific example of this, suppose that \( A \) is an elliptic curve over \( \mathbb{Q} \) with a rational torsion point of order \( p \) and without complex multiplication. If \( E \) is an elliptic curve over \( \mathbb{Q} \) with good ordinary reduction at \( p \) such that every prime where both \( E \) and \( A \) have bad reduction has odd order in \( \mathbb{F}^{\times}_p \) and such that the negative of the conductor of \( E \) is not a square modulo \( p \), then there is a positive constant \( B \) depending on \( A \) but not on \( E \) or \( n \) such that \[ \operatorname{rk}_p(E/\mathbb{Q}(A[p^n])) \geq Bp^{2n} \] for every \( n \).

[176]Current developments in mathematics, 2006 (Cambridge, MA, 2006). Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2008. MR 2459303 Zbl 1149.00018 book

Tomasz Stanislaw Mrowka	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[177]B. Mazur: “Finding meaning in error terms,” Bull. Am. Math. Soc., New Ser. 45 : 2 (April 2008), pp. 185–228. In memory of Serge Lang. MR 2383303 Zbl 1229.11001 article

[178]B. Mazur: “Mathematical Platonism and its opposites,” Eur. Math. Soc. Newsl. 68 (2008), pp. 19–21. Zbl 1154.00310 article

[179]B. Mazur: “Algebraic numbers,” pp. 315–332 in The Princeton companion for mathematics. Edited by T. Gowers, J. Barrow-Green, and I. Leader. American Mathematical Society (Providence, RI), 2008. Section IV.1. incollection

William Timothy Gowers	Related
June Elizabeth Barrow-Green	Related
Imre Bennett Leader	Related

[180]B. Mazur: “How did Theaetetus prove his theorem?,” pp. 227–250 in The envisioned life: Essays in honor of Eva Brann. Edited by P. Kalkavage and E. Salem. Paul Dry Books (Philadelphia), 2008. incollection

Eric Salem	Related
Eva Brann	Related
Peter Kalkavage	Related

[181]B. Mazur: Some notes about imagination and mathematics related to a conversation with Eva Brann. Lecture notes, Harvard University, 2008. A conversation with Eva Brann. techreport

[182]F. Calegari and B. Mazur: “Nearly ordinary Galois deformations over arbitrary number fields,” J. Inst. Math. Jussieu 8 : 1 (January 2009), pp. 99–177. MR 2461903 Zbl 1211.11065 ArXiv 0708.2451 article

Let \( K \) be an arbitrary number field, and let \[ \rho:\mathrm{Gal}(\bar{K}/K)\to \mathrm{GL}_2(E) \] be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of \( \rho \). When \( K \) is totally real and \( \rho \) is modular, results of Hida imply that the nearly ordinary deformation space associated to \( \rho \) contains a Zariski dense set of points corresponding to “automorphic” Galois representations. We conjecture that if \( K \) is not totally real, then this is never the case, except in three exceptional cases, corresponding to

“base change”,
“CM” forms, and
“even” representations.

The latter case conjecturally can only occur if the image of \( \rho \) is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when \( K \) is an imaginary quadratic field, we prove an unconditional result that implies the existence of “many” positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about “\( p \)-adic functoriality,” as well as some remarks on how our methods should apply to \( n \)-dimensional representations of \( \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \) when \( n > 2 \).

[183]Current developments in mathematics, 2008 (Cambridge, MA, 2008). Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2009. MR 2555925 Zbl 1173.00021 book

Tomasz Stanislaw Mrowka	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[184]Current developments in mathematics, 2007 (Cambridge, MA, 2007). Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. Somerville, MA: International Press, 2009. MR 2531714 Zbl 1166.00310 book

Tomasz Stanislaw Mrowka	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[185]B. Mazur: Arithmetic and some of its aspirations. Lecture notes, Harvard University, May 2009. Brief talk given to the “Friends of the Harvard Mathematics Department” on 5 May 2009. techreport

[186]B. Mazur: What should a professional mathematician know?. Technical report, Harvard University, 2009. An essay written in response to a question posed by Phil Davis and David Mumford. techreport

David Bryant Mumford	Related
Philip J. Davis	Related

[187]B. Mazur: The language of explanation. Lecture notes, Harvard University, November 2009. An essay written for the University of Utah Symposium in Science and Literature (November, 2009) and in anticipation of a panel discussion on Mathematics, Language, and Imagination. techreport

[188]B. Mazur: Problems connnecting logic and number theory. Lecture notes, Harvard University, May 2009. Notes for a panel on Number Theory and Logic, at the MAMLS@Harvard conference, 9 May 2009. techreport

[189]B. Mazur: On time (in mathematics and literature). Lecture notes, Harvard University, March 2009. Notes for a talk given at a conference in comparative literature, Harvard, March 2009. techreport

[190] P. D. Lax, F. Hirzebruch, B. Mazur, L. Conlon, E. B. Curtis, H. M. Edwards, J. Huebschmann, and H. Shulman: “Raoul Bott as we knew him,” pp. 43–49 in A celebration of the mathematical legacy of Raoul Bott (Montreal, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648884 Zbl 1195.01029 incollection

F. Hirzebruch	Related	Volume
Johannes Huebschmann	Related
Lawrence William Conlon	Related
Raoul H. Bott	Related	Volume
Edward Baldwin Curtis	Related
Peter Robert Kotiuga	Related
Peter D. Lax	Related
Harold Mortimer Edwards, Jr.	Related
Herbert Shulman	Related

[191]Current developments in mathematics, 2009 (Cambridge, MA, November 2009). Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2010. MR 2757356 Zbl 1205.00075 book

Tomasz Stanislaw Mrowka	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[192]B. Mazur and K. Rubin: “Ranks of twists of elliptic curves and Hilbert’s tenth problem,” Invent. Math. 181 : 3 (September 2010), pp. 541–575. MR 2660452 Zbl 1227.11075 ArXiv 0904.3709 article

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell–Weil group, and (assuming the Shafarevich–Tate conjecture) many others with infinite cyclic Mordell–Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich–Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.

[193]B. Mazur, B. Gross, and D. Mumford: “Andrew Gleason, 4 November 1921–17 October 2008,” Proc. Am. Phil. Soc. 154 : 4 (December 2010), pp. 471–476. A slightly expanded version of the obituary published in the Harvard University Gazette (1 April 2010). article

Andrew Mattei Gleason	Related	Volume
Benedict H. Gross	Related	Volume
David Bryant Mumford	Related

[194]B. Mazur: Hilbert’s tenth problem and elliptic curves. Lecture notes, Harvard University, February 2010. “Basic Notions” talk given on 1 February 2010. techreport

[195]B. Mazur: Ranks of twists of elliptic curves. Lecture notes, Harvard University, February 2010. Number Theory Seminar lecture on 3 February 2010. techreport

[196]B. Mazur, B. Gross, and D. Mumford: “Andrew Mattei Gleason,” Harvard Gazette (1 April 2010). A slightly expanded version of this obituary was published in Proc. Am. Phil. Soc. 154:4 (2010). article

Andrew Mattei Gleason	Related	Volume
Benedict H. Gross	Related	Volume
David Bryant Mumford	Related

[197]B. Mazur: “How can we construct abelian Galois extensions of basic number fields?,” Bull. Am. Math. Soc., New Ser. 48 : 2 (2011), pp. 155–209. MR 2774089 Zbl 1228.11163 article

Irregular primes — 37 being the first such prime — have played a great role in number theory. This article discusses Ken Ribet’s construction — for all irregular primes \( p \) — of specific abelian, unramified, degree-\( p \) extensions of the number fields \( \mathbb{Q}(e^{2\pi i/p}) \). These extensions with explicit information about their Galois groups (they are Galois over \( \mathbb{Q} \)) were predicted to exist ever since the work of Herbrand in the 1930s. Ribet’s method involves a tour through the theory of modular forms; it demonstrates the usefulness of congruences between cuspforms and Eisenstein series, a fact that has inspired, and continues to inspire, much work in number theory.

[198]B. Mazur and K. Rubin: “Refined class number formulas and Kolyvagin systems,” Compos. Math. 147 : 1 (2011), pp. 56–74. MR 2771126 Zbl 1225.11144 ArXiv 0909.3916 article

[199]B. Mazur: Disparity in the statistics for quadratic twist families. Lecture notes, Harvard University, April 2011. Based on joint work with Zev Klagsbrun and Karl Rubin. Notes for the 13 April 2011 workshop of the program “Arithmetic Statistics” at MSRI. techreport

Zev Klagsbrun	Related
Karl Cooper Rubin	Related

[200]B. Mazur: Elliptic curves, their companions, and their statistics. Lecture notes, Harvard University, 2011. Lecture notes for a joint MSRI-UC Berkeley colloquium for graduate students entitled “Elliptic curves, their companions, and their statistics” given in the winter of 2011 in conjunction with the MSRI program “Arithmetic Statistics.”. techreport

[201]B. Mazur: What is the surface area of a hedgehog?. Lecture notes, Harvard University, September 2011. Lecture notes for a Steiner Lecture given at St. Johns College, Annapolis, MD, 30 September 2011. techreport

[202]Current developments in mathematics, 2010 (Cambridge, MA, 2010). Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2011. MR 2906369 Zbl 1245.00031 book

Tomasz Stanislaw Mrowka	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[203]B. Mazur: Primes, knots and Po. Lecture notes, Harvard University, July 2012. techreport

[204]B. Mazur: Mathematics and story, 20 April 2012. Guest contribution to literary website berfrois. misc

[205]Current developments in mathematics, 2011 (Cambridge, MA, November 2011). Edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S.-T. Yau. International Press (Somerville, MA), 2012. Zbl 1256.00015 book

Tomasz Stanislaw Mrowka	Related
Richard Peter Stanley	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[206]J. Hoffman: “Q&A: The maths raconteur,” Nature 483 : 7390 (22 March 2012), pp. 405. An interview with Mazur connected to the publication of his book Circles disturbed (2012). article

[207]B. Mazur: “About Hermann Weyl’s ‘Ramifications, old and new, of the eigenvalue problem’,” Bull. Am. Math. Soc., New Ser. 49 : 2 (2012), pp. 325–326. MR 2888169 Zbl 1236.01013 article

[208]B. Mazur: Is it plausible?. Lecture notes, Harvard University, January 2012. Lecture notes for the joint AMS-MAA conference, 5 January 2012. techreport

[209]B. Mazur: Elliptic curves and their statistics. Lecture notes, Harvard University, February 2012. Notes for a seminar given 28 February 2012. techreport

[210]Circles disturbed: The interplay of mathematics and narrative. Edited by A. Doxiadis and B. Mazur. Princeton University Press, 2012. MR 2906079 Zbl 1237.00022 book

[211]B. Mazur: “Visions, dreams, and mathematics,” pp. 183–210 in Circles disturbed: The interplay of mathematics and narrative. Edited by A. Doxiadis and B. Mazur. Princeton University Press, 2012. An interview with Mazur connected to the publication of this book was published in Nature 483:7390 (2012). MR 2918069 incollection

[212]B. Mazur: A brief introduction to the work of Haruzo Hida. Lecture notes, Harvard University, June 2012. Lecture notes for the conference in celebration of the 60th birthday of Haruzo Hida, UCLA, 18–23 June 2012. techreport

[213]B. Mazur and K. Rubin: Refined class number formulas for \( \mathbb{G}_m \). Preprint, December 2013. ArXiv 1312.4053 techreport

[214]J. B. Friedlander, H. Iwaniec, B. Mazur, and K. Rubin: “The spin of prime ideals,” Invent. Math. 193 : 3 (September 2013), pp. 697–749. MR 3091978 Zbl pre06222256 article

Henryk Iwaniec	Related
John Benjamin Friedlander	Related
Karl Cooper Rubin	Related

[215]B. Mazur: Some comments on elliptic curves over general number fields and Brill–Noether modular varieties. Unpublished lecture notes, Harvard University, 2013. Notes for a lecture given at the Quebec/Maine Number Theory Conference, 2013. techreport

[216]Z. Klagsbrun, B. Mazur, and K. Rubin: “Disparity in Selmer ranks of quadratic twists of elliptic curves,” Ann. Math. (2) 178 : 1 (2013), pp. 287–320. MR 3043582 Zbl pre06190561 article

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve \( E \) over an arbitrary number field \( K \). We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve \( E \) such that as \( K \) varies, these fractions are dense in \( [0,1] \). More generally, our results also apply to \( p \)-Selmer ranks of twists of 2-dimensional self-dual \( \mathbb{F}_p \)-representations of the absolute Galois group of \( K \) by characters of order \( p \).

Zev Klagsbrun	Related
Karl Cooper Rubin	Related

[217]B. Mazur: Arithmetic statistics: Elliptic curves and other things. Unpublished lecture notes, Harvard University, 2013. Notes for the Erdős Memorial Lecture, Temple University, 12 October 2013. techreport

[218] B. Mazur: “Is it plausible?,” Math. Intelligencer 36 : 1 (2014), pp. 24–33. MR 3166990 Zbl 1307.11003 article

[219] Z. Klagsbrun, B. Mazur, and K. Rubin: “A Markov model for Selmer ranks in families of twists,” Compos. Math. 150 : 7 (2014), pp. 1077–1106. MR 3230846 Zbl 1316.11045 ArXiv 1303.6507 article

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve \( E \) over an arbitrary number field \( K \). Under the assumption that \[ \operatorname{Gal}\bigl(K(E[2])/K\bigr) \cong S_3 ,\] we show that the density (counted in a nonstandard way) of twists with Selmer rank \( r \) exists for all positive integers \( r \), and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of \( E \) and \( K \). More generally, our results also apply to \( p \)-Selmer ranks of twists of two-dimensional self-dual \( \mathbf{F}_p \)-representations of the absolute Galois group of \( K \) by characters of order \( p \).

Zev Klagsbrun	Related
Karl Cooper Rubin	Related

[220] B. Mazur and K. Rubin: “Selmer companion curves,” Trans. Amer. Math. Soc. 367 : 1 (2015), pp. 401–421. MR 3271266 Zbl 1316.11047 ArXiv 1203.0620 article

[221] M. Baker, B. Mazur, and K. Ribet: “Robert F. Coleman 1954–2014,” Res. Math. Sci. 2 (2015), pp. Art. 21, 15. MR 3394751 Zbl 1338.01025 ArXiv arXiv:1411.3655 article

Kenneth A. Ribet	Related
Matthew Baker	Related

[222] J. B. Friedlander, H. Iwaniec, B. Mazur, and K. Rubin: “Erratum to: The spin of prime ideals,” Invent. Math. 202 : 2 (2015), pp. 923–925. Correction to Invent. Math. 193 (2013), 697–749. MR 3418248 Zbl 1318.57009 article

Henryk Iwaniec	Related
John Benjamin Friedlander	Related
Karl Cooper Rubin	Related

[223] B. Mazur and K. Rubin: “Controlling Selmer groups in the higher core rank case,” J. Théor. Nombres Bordeaux 28 : 1 (2016), pp. 145–183. MR 3464616 Zbl 1411.11065 ArXiv 1312.4052 article

[224] B. Mazur and K. Rubin: “Refined class number formulas for \( \mathbb G_m \),” J. Théor. Nombres Bordeaux 28 : 1 (2016), pp. 185–211. MR 3464617 Zbl 1414.11155 ArXiv arXiv:1312.4053 article

[225] B. Mazur: “A celebration of the mathematical work of Glenn Stevens,” Ann. Math. Qué. 40 : 1 (2016), pp. 1–16. MR 3512520 Zbl 1354.11004 article

[226] B. Mazur and W. Stein: Prime numbers and the Riemann hypothesis. Cambridge University Press (Cambridge, UK), 2016. MR 3616260 Zbl 1356.11001 book

Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann hypothesis, which remains one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann hypothesis. Students with a minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible explanation of the key ideas. The exposition of these ideas is generously illuminated by computational graphics that exhibit the key concepts and phenomena in enticing detail. Readers with more mathematical experience will then go deeper into the structure of primes and see how the Riemann hypothesis relates to Fourier analysis using the vocabulary of spectra. Readers with a strong mathematical background will be able to connect these ideas to historical formulations of the Riemann hypothesis.

[227] B. Mazur: “To the edge of the map,” pp. 389–401 in The map and the territory. Edited by S. Wuppuluri and F. A. Doria. Frontiers Collection. Springer, 2018. MR 3751949 Zbl 1079.74574 incollection

[228] M. Derickx, B. Mazur, and S. Kamienny: “Rational families of 17-torsion points of elliptic curves over number fields,” pp. 81–104 in Number theory related to modular curves: Momose memorial volume. Edited by J.-C. Lario and V. K. Murty. Contemp. Math. 701. Amer. Math. Soc. (Providence, RI), 2018. MR 3755909 Zbl 1440.11092 incollection

Sheldon Kamienny	Related
Joan-Carles Lario	Related
Maarten Derickx	Related
Vijaya Kumar Murty	Related

[229] B. Mazur: “Afterword to the article ‘Arithmetic on curves’,” Bull. Amer. Math. Soc. (N.S.) 55 : 3 (2018), pp. 353–358. Discussion of Bulletin of the American Mathematical Society 14:2 (1986), 207–259. MR 3803160 Zbl 1390.11005 article

[230] B. Mazur and K. Rubin: “Diophantine stability,” Amer. J. Math. 140 : 3 (2018), pp. 571–616. With an appendix by Michael Larsen. MR 3805014 Zbl 1491.14036 ArXiv 1503.04642 article

Michael J. Larsen	Related
Karl Cooper Rubin	Related

[231] B. Mazur: “Reprint: Arithmetic on curves,” Bull. Am. Math. Soc., New Ser. 55 : 3 (2018), pp. 207–259 (original pagination). Reprint of Bull. Am. Math. Soc. 14, 207–259 (1986) in the print issue only. Zbl 1398.14034 article

[232] B. Mazur: “Grand unity,” ICCM Not. 7 : 1 (2019), pp. 76. MR 3960563 article

[233] B. Mazur and K. Rubin: “Big fields that are not large,” Proc. Amer. Math. Soc. Ser. B 7 (2020), pp. 159–169. MR 4173816 Zbl 1468.11218 article

[234] B. Mazur: “Math in the time of plague,” Math. Intelligencer 42 : 4 (2020), pp. 1–6. MR 4182034 Zbl 1466.00006 article

[235] K. Acquista, J. Buhler, D. Clausen, J. H. Coates, B. H. Gross, S. Lichtenbaum, J. Lubin, B. Mazur, V. K. Murty, B. Perrin-Riou, C. D. Popescu, K. A. Ribet, J. H. Silverman, K. Uhlenbeck, and J. F. Voloch: “Memorial article for John Tate.” Edited by B. Mazur and K. Ribet. Notices Am. Math. Soc. 68 : 5 (2021), pp. 768–781. MR 4249436 Zbl 1470.01012 article

Bernadette Perrin-Riou	Related
Karen Acquista	Related
Dustin Clausen	Related
Benedict H. Gross	Related	Volume
Karen Uhlenbeck	Related	Volume
Cristian Dumitru Popescu	Related
Joe Peter Buhler	Related
José Felipe Voloch	Related
John Torrence Tate, Jr.	Related
Joseph Hillel Silverman	Related
John Henry Coates	Related
Kenneth A. Ribet	Related
Stephen Lichtenbaum	Related
Jonathan Darby Lubin	Related
Vijaya Kumar Murty	Related

[236] L. Caporaso, J. Harris, and B. Mazur: “Uniformity of rational points: an up-date and corrections,” Tunis. J. Math. 4 : 1 (2022), pp. 183–201. Corrects and enlarges on J. Amer. Math. Soc. 10:1 (1997), 1–35. MR 4401791 Zbl 1493.14039 ArXiv 2012.14461 article

Lucia Caporaso	Related
Joseph Daniel Harris	Related

[237] B. Mazur: “Notes for a seminar in axiomatic reasoning,” pp. 105–144 in Axiomatic thinking, I. Edited by F. Ferreira, R. Kahle, and G. Sommaruga. Springer, 2022. MR 4510892 Zbl 07632999 incollection

[238] B. Mazur: “Happy birthday, Dennis,” EMS Surv. Math. Sci. 9 : 2 (2022), pp. 275–277. MR 4659684 Zbl 07800510 article

[239] B. Mazur, K. Rubin, and A. Shlapentokh: Defining \( \mathbb{Z} \) using unit groups. Preprint, 2023. ArXiv 2303.02521 techreport

Alexandra Shlapentokh	Related
Karl Cooper Rubin	Related

[240] T. Feng, M. Harris, and B. Mazur: Derived class field theory. Preprint, 2023. In memory of John Coates. ArXiv 2304.14161 techreport

Michael Howard Harris	Related
Tony Feng	Related

[241] J. S. Balakrishnan and B. Mazur: Ogg’s Torsion conjecture: Fifty years later. Preprint, 2023. ArXiv 2307.04752 techreport

[242] Z. Klagsbrun, B. Mazur, and K. Rubin: “Erratum to ‘Disparity in Selmer ranks of quadratic twists of elliptic curves’,” Ann. of Math. (2) 198 : 2 (2023), pp. 879–880. Correction to Ann. Math. 178:1 (2013), 287–320. MR 4635307 Zbl 1523.11099 article

Zev Klagsbrun	Related
Karl Cooper Rubin	Related

[243] B. Mazur and K. Rubin: “Arithmetic conjectures suggested by the statistical behavior of modular symbols,” Exp. Math. 32 : 4 (2023), pp. 657–672. MR 4669286 Zbl 07772718 article

Suppose \( E \) is an elliptic curve over \( \mathbb{Q} \) and \( \chi \) is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that \[ L(E,\chi,1)=0 .\] Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell–Weil rank grows in abelian extensions of \( \mathbb{Q} \). Using this heuristic, we find a large class of infinite abelian extensions \( F \) where we expect \( E(F) \) to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, and Kisilevsky. Where our predictions and theirs overlap, the predictions are consistent.

[244] B. Mazur, K. Rubin, and A. Shlapentokh: “Existential definability and diophantine stability,” J. Number Theory 254 (2024), pp. 1–64. MR 4633727 Zbl 07748216 ArXiv 2208.09963 article

Alexandra Shlapentokh	Related
Karl Cooper Rubin	Related

[245] B. Mazur, S. Lichtenbaum, A. Wiles, L. Schneps, Sujatha, and M. Kakde: “John Coates (1945–2022): Celebration of a life,” Notices Amer. Math. Soc. 71 : 1 (2024), pp. 46–56. MR 4693600 Zbl 1546.01120 article

[246] B. Mazur, K. Rubin, and A. Shlapentokh: “Corrigendum to ‘Existential definability and diophantine stability’ \( [ \)J. Number Theory 254 (2024) 1–64\( ] \),” J. Number Theory 262 (2024), pp. 539–540. Corrections to a paper previously published in J. Number Theorey 254 (2024). MR 4748764 Zbl 1541.11105 article

[247] B. Mazur, K. Rubin, and A. Shlapentokh: “Defining \( \mathbb{Z} \) using unit groups,” Acta Arith. 214 (2024), pp. 235–255. MR 4772285 Zbl 1555.11196 article

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of \( \mathbb{Q} \), paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of \( \mathbb{Z} \). Namely, we prove that for a large collection of algebraic extensions \( K/\mathbb{Q} \), \[ \{x \in \mathcal{O}_K \mid \forall \varepsilon \in \mathcal{O}^{\times}_K \, \exists \partial\in \mathcal{O}^{\times}_K: \partial - 1 \equiv (\varepsilon -1) x \mod{(\varepsilon-1)^2}\} = \mathbb{Z}, \] where \( \mathcal{O}_K \) denotes the ring of integers of \( K \). One of the corollaries of our results is undecidability of the field of constructible numbers, a question posed by Tarski in 1948.

[248] J. S. Balakrishnan and B. Mazur: “Ogg’s torsion conjecture: Fifty years later,” Bull. Amer. Math. Soc. (N.S.) 62 : 2 (2025), pp. 235–268. with an appendix by Netan Dogra. MR 4885858 Zbl 08036525 article

Andrew Ogg’s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg’s celebrated torsion conjecture — as it relates to modular curves — can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg’s torsion conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years.

Barry C. Mazur

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