Emeritus Sadleirian Professor of Mathematics  FRS

Research Interests

Number theory, arithmetical algebraic geometry, Iwasawa theory

Publications

• Classical Iwasawa theory and infinite descent on a family of abelian varieties. Preprint. (With J. Li and Y. Li) [ArXiv:2008.10310]

• A brief account of the history of exact formulae in arithmetic geometry. Preprint.

• Non-vanishing theorems for central L-values of some elliptic curves with complex multiplication. Proc. Lond. Math. Soc. (3) 121 (2020), no. 6, 1531–1578. (With Y. Li) 

• Peter Swinnerton-Dyer (1927–2018). Notices Amer. Math. Soc. 66 (2019), no. 7, 1058–1067. (With B. Birch, J-L. Colliot-Thélène, A. Skorobogatov)

• Shing-Tung Yau and the rebirth of Chinese number theory. ICCM Not. 7 (2019), no. 1, 12–13.

• Iwasawa theory of quadratic twists of X₀(49). Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 1, 19–28. (With J. Choi)

• The oldest problem. ICCM Not. 5 (2017), no. 2, 8–13.

• The conjecture of Birch and Swinnerton-Dyer. Open problems in mathematics, 207–223, Springer, [Cham], 2016.

• Values of the Riemann zeta function at the odd positive integers and Iwasawa theory. The Bloch-Kato conjecture for the Riemann zeta function, 45–64, London Math. Soc. Lecture Note Ser., 418, Cambridge Univ. Press, Cambridge, 2015.

• Quadratic twists of elliptic curves. Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 357–394. (With Y. Li, Y. Tian, S. Zhai)

• On the 2-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. Münster J. Math. 7 (2014), no. 1, 83–103. (With M. Kim, Z. Liang, C. Zhao)

• Congruent numbers. Acta Math. Vietnam. 39 (2014), no. 1, 3–10.

• Lectures on the Birch-Swinnerton-Dyer conjecture. ICCM Not. 1 (2013), no. 2, 29–46.

• Non-commutative Iwasawa theory for modular forms. Proc. Lond. Math. Soc. (3) 107 (2013), no. 3, 481–516. (With T. Dokchitser,  Z. Liang, W. Stein, R. Sujatha)

• Introduction to the work of M. Kakde on the non-commutative main conjectures for totally real fields. Noncommutative Iwasawa main conjectures over totally real fields, 1–22, Springer Proc. Math. Stat., 29, Springer, Heidelberg, 2013. (With D. Kim)

• Congruent numbers. Proc. Natl. Acad. Sci. USA 109 (2012), no. 52, 21182-21183.

• The enigmatic Tate-Shafarevich group. Fifth International Congress of Chinese Mathematicians. Part 1, 2, 43–50, AMS/IP Stud. Adv. Math., 51, pt. 1, 2, Amer. Math. Soc., Providence, RI, 2012.

• On the ?_H(G)-conjecture. Non-abelian fundamental groups and Iwasawa theory, 132–161, London Math. Soc. Lecture Note Ser., 393, Cambridge Univ. Press, Cambridge, 2012. (With R. Sujatha)

• Galois cohomology of elliptic curves. Second edition. Published by Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2010. xii+98 pp. (With R. Sujatha)

• The Tate-Shafarevich group for elliptic curves with complex multiplication II. Milan J. Math. 78 (2010), no. 2, 395–416. (With Z. Liang, R. Sujatha)

• The main conjecture. Guwahati Workshop on Iwasawa Theory of Totally Real Fields, 113–140, Ramanujan Math. Soc. Lect. Notes Ser., 12, Ramanujan Math. Soc., Mysore, 2010. (With R. Sujatha)

• Number theory, ancient and modern. Fourth International Congress of Chinese Mathematicians, 3–12, AMS/IP Stud. Adv. Math., 48, Amer. Math. Soc., Providence, RI, 2010. 

• Selmer varieties for curves with CM Jacobians. Kyoto J. Math. 50 (2010), no. 4, 827–852. (With M. Kim)

• Root numbers, Selmer groups, and non-commutative Iwasawa theory. J. Algebraic Geom. 19 (2010), no. 1, 19–97. (With T. Fukaya, K. Kato, R. Sujatha)

• Elliptic curves and Iwasawa theory. Colloquium De Giorgi 2007 and 2008, 47–55, Colloquia, 2, Ed. Norm., Pisa, 2009.

• The Tate-Shafarevich group for elliptic curves with complex multiplication. J. Algebra 322 (2009), no. 3, 657–674. (With Z. Liang, R. Sujatha)

• Euler's work on zeta and L-functions and their special values. BSHM Bull. 23 (2008), no. 1, 37–41. 

• Cyclotomic fields and zeta values. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2006. x+113 pp. (With R. Sujatha) 

• The GL₂ main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Études Sci. No. 101 (2005), 163–208. (With T. Fukaya, K. Kato, R. Sujatha, O. Venjakob)

• Fine Selmer groups for elliptic curves with complex multiplication. Algebra and number theory, 327–337, Hindustan Book Agency, Delhi, 2005. (With R. Sujatha)

• Congruent number problem. Q. J. Pure Appl. Math. 1 (2005), no. 1, 14–27.

• Fine Selmer groups of elliptic curves over p-adic Lie extensions. Math. Ann. 331 (2005), no. 4, 809–839. (With R. Sujatha)

• Preface [Special issue dedicated to the memory of Professor Armand Borel, 1923–2003]. Asian J. Math. 8 (2004), no. 4, iii. (With L. Ji, G. Prasad, Y-T. Siu, S-T. Yau)

• Courbes elliptiques. Leçons de mathématiques d'aujourd'hui. Vol. 1. (French) [Lectures on mathematics today. Vol. 1] With a preface by Éric Charpentier and Nicolaï Nikolski. Second edition. Le Sel et le Fer [Salt and Iron], 4. Cassini, Paris, 2003. xvi+332 pp. 

• Iwasawa algebras and arithmetic. Séminaire Bourbaki. Vol. 2001/2002. Astérisque No. 290 (2003), Exp. No. 896, vii, 37–52. 

• Links between cyclotomic and GL₂ Iwasawa theory. Kazuya Kato's fiftieth birthday. Doc. Math. 2003, Extra Vol., 187–215. (With P. Schneider, R. Sujatha)

• Modules over Iwasawa algebras. J. Inst. Math. Jussieu 2 (2003), no. 1, 73–108. (With P. Schneider, R. Sujatha)

• Elliptic curves — the crossroads of theory and computation. Algorithmic number theory (Sydney, 2002), 9–19, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002. 

• On the Euler-Poincaré characteristics of finite dimensional p-adic Galois representations. Publ. Math. Inst. Hautes Études Sci. No. 93 (2001), 107–143. (With R. Sujatha, J-P. Wintenberger)

• The work of Shouwu Zhang. First International Congress of Chinese Mathematicians (Beijing, 1998), xliii–xlv, AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, 2001. 

• Euler characteristics and elliptic curves. II. J. Math. Soc. Japan 53 (2001), no. 1, 175–235. (With S. Howson)

• Galois cohomology of elliptic curves. Tata Institute of Fundamental Research Lectures on Mathematics, 88. Published by Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2000. x+100 pp. (With R. Sujatha) 

• Fragments of the GL₂ Iwasawa theory of elliptic curves without complex multiplication. Arithmetic theory of elliptic curves (Cetraro, 1997), 1–50, Lecture Notes in Math., 1716, Springer, Berlin, 1999. 

• Kenkichi Iwasawa (1917–1998). Notices Amer. Math. Soc. 46 (1999), no. 10, 1221–1225. 

• Euler-Poincaré characteristics of abelian varieties. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 4, 309–313. (With R. Sujatha)

• Euler characteristics and elliptic curves. Elliptic curves and modular forms (Washington, DC, 1996). Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11115–11117. (With S. Howson) 

• Wiles receives NAS award in mathematics. Notices Amer. Math. Soc. 43 (1996), no. 7, 760–763.

• Kummer theory for abelian varieties over local fields. Invent. Math. 124 (1996), no. 1-3, 129–174. (With R. Greenberg)

• On the symmetric square of a modular elliptic curve. Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), 2–21, Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995. (With A. Sydenhamh)

• Iwasawa theory of modular elliptic curves of analytic rank at most 1. J. London Math. Soc. (2) 50 (1994), no. 2, 243–264. (With G. McConnell)

• Elliptic curves with complex multiplication and Iwasawa theory. Bull. London Math. Soc. 23 (1991), no. 4, 321–350. 

• Motivic p-adic L-functions. L-functions and arithmetic (Durham, 1989), 141–172, London Math. Soc. Lecture Note Ser., 153, Cambridge Univ. Press, Cambridge, 1991.

• On p-adic L-functions attached to motives over Q. II. Bol. Soc. Brasil. Mat. (N.S.) 20 (1989), no. 1, 101–112. 

• On p-adic L-functions attached to motives over Q. Algebraic number theory, 23–54, Adv. Stud. Pure Math., 17, Academic Press, Boston, MA, 1989. (With B. Perrin-Riou)

• On p-adic L-functions. Séminaire Bourbaki, Vol. 1988/89. Astérisque No. 177-178 (1989), Exp. No. 701, 33–59.

• Iwasawa theory for the symmetric square of an elliptic curve. J. Reine Angew. Math. 375/376 (1987), 104–156. (With C-G.Schmidt)

• The work of Gross and Zagier on Heegner points and the derivatives of L-series. Seminar Bourbaki, Vol. 1984/85. Astérisque No. 133-134 (1986), 57–72. 

• Elliptic curves and Iwasawa theory. Modular forms (Durham, 1983), 51–73, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984. 

• Infinite descent on elliptic curves with complex multiplication. Arithmetic and geometry, Vol. I, 107–137, Progr. Math., 35, Birkhäuser Boston, Boston, MA, 1983.

• Some remarks on the main conjecture for elliptic curves with complex multiplication. Amer. J. Math. 105 (1983), no. 2, 337–366. (With C. Goldstein)

• The work of Mazur and Wiles on cyclotomic fields. Bourbaki Seminar, Vol. 1980/81, pp. 220–242, Lecture Notes in Math., 901, Springer, Berlin-New York, 1981.

• Elliptic curves of conductor 11. Math. Comp. 35 (1980), no. 151, 991–1002. (With M. Agrawal, D. Hunt, A. van der Poorten) 

• The arithmetic of elliptic curves with complex multiplication. Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 351–355, Acad. Sci. Fennica, Helsinki, 1980.

• On p-adic L-functions and elliptic units. J. Austral. Math. Soc. Ser. A 26 (1978), no. 1, 1–25. (With A. Wiles)

• On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39 (1977), no. 3, 223–251. (With A. Wiles)

• Explicit reciprocity laws. Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 7–17. Astérisque No. 41-42, Soc. Math. France, Paris, 1977. (With A. Wiles)

• p-adic L-functions and Iwasawa's theory. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 269–353. Academic Press, London, 1977.

• Kummer's criterion for Hurwitz numbers. Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), pp. 9–23. Japan Soc. Promotion Sci., Tokyo, 1977. (With A. Wiles)

• Integrality properties of the values of partial zeta functions. Proc. London Math. Soc. (3) 34 (1977), no. 2, 365–384. (With W. Sinnott)

• Diophantine approximation on Abelian varieties with complex multiplication. Invent. Math. 34 (1976), no. 2, 129–133. (With S. Lang)

• Fonctions zêta partielles d'un corps de nombres totalement réel. (French) Séminaire Delange-Pisot-Poitou (16e année: 1974/75), Théorie des nombres, Fasc. 1, Exp. No. 1, 9 pp. Secrétariat Mathématique, Paris, 1975.

• Fractional parts of powers of rationals. Math. Proc. Cambridge Philos. Soc. 77 (1975), 269–279. (With A. Baker)

• An analogue of Stickelberger's theorem for the higher K-groups. Invent. Math. 24 (1974), 149–161. (With W. Sinnott)

• On p-adic L-functions over real quadratic fields. Invent. Math. 25 (1974), 253–279. (With W. Sinnott)

• K-theory and Iwasawa's analogue of the Jacobian. Algebraic K-theory, II: "Classical'' algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 502–520. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973.

• Research problems: Arithmetic questions in K-theory. Algebraic K-theory, II: "Classical'' algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 521–523. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973.

• Linear relations between 2πi and the periods of two elliptic curves. Diophantine approximation and its applications (Proc. Conf., Washington, D.C., 1972), pp. 77–99. Academic Press, New York, 1973.

• On Iwasawa's analogue of the Jacobian for totally real number fields. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 51–61. Amer. Math. Soc., Providence, R.I., 1973.

• On l-adic zeta functions. Ann. of Math. (2) 98 (1973), 498–550. (With S. Lichtenbaum)

• Verification of Weil's conjecture on elliptic curves over Q in some special cases. Proceedings of the 1972 Number Theory Conference (Univ. Colorado, Boulder, Colo.), pp. 43–48. Univ. Colorado, Boulder, Colo., 1972.

• On K₂ and some classical conjectures in algebraic number theory. Ann. of Math. (2) 95 (1972), 99–116.

• Linear forms in the periods of the exponential and elliptic functions. Invent. Math. 12 (1971), 290–299.

• The transcendence of linear forms in ω₁, ω₂, η₁, η₂, 2πi. Amer. J. Math. 93 (1971), 385–397. 

• An application of the Thue-Siegel-Roth theorem to elliptic function. Proc. Cambridge Philos. Soc. 69 (1971), 157–161.

• An application of the division theory of elliptic functions to diophantine approximation. Invent. Math. 11 (1970), 167–182.

• Construction of rational functions on a curve. Proc. Cambridge Philos. Soc. 68 (1970), 105–123.

• Integer points on curves of genus 1. Proc. Cambridge Philos. Soc. 67 (1970), 595–602. (With A. Baker)

• An effective p-adic analogue of a theorem of Thue. III. The diophantine equation y²=x³+k. Acta Arith. 16 (1969/70), 425–435.

• An effective p-adic analogue of a theorem of Thue. II. The greatest prime factor of a binary form. Acta Arith. 16 (1969/70), 399–412.

• An effective p-adic analogue of a theorem of Thue. Acta Arith. 15 (1968/69), 279–305. 

• Approximation in algebraic function fields of one variable. J. Austral. Math. Soc. 7 1967 341–355.

• On the algebraic approximation of functions. IV. Nederl. Akad. Wetensch. Proc. Ser. A 70 = Indag. Math. 29 1967 205–212.

• On the algebraic approximation of functions. III. Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 1966 449–461.

• On the algebraic approximation of functions. II. Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 1966 435–448.

• On the algebraic approximation of functions. I. Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 1966 421–434.

Address

Emmanuel College

St Andrew's Street

Cambridge CB2 3AP

United Kingdom