Skip to main content

Overview of the Work of Kumar Murty

  • Conference paper
  • First Online:
Geometry, Algebra, Number Theory, and Their Information Technology Applications (GANITA 2016)

Abstract

The role of the scholar in society is foundational for the growth of human civilization. In fact, one could argue that without the scholar, civilizations crumble. The transmission of knowledge from generation to generation, to take what is essential from the past, to transform it into a new shape and arrangement relevant to the present and to stimulate future students to add to this knowledge is the primary role of the teacher. Spanning more than four decades, Kumar Murty has been the model teacher and researcher, working in diverse areas of number theory and arithmetic geometry, expanding his contributions to meet the challenges of the digital age and training an army of students and postdoctoral fellows who will teach the future generations. On top of this, he has also given serious attention to how mathematics and mathematical thought can be applied to dealing with large-scale economic problems and the emergence of “smart villages”. We will not discuss this latter work here, nor his other work in the field of Indian philosophy. We will only focus on giving a synoptic overview of his major contributions to mathematics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A. Akbary, D. Ghioca, and V. Kumar Murty, Reductions of points on elliptic curves, Math. Annalen, 347(2010), no. 2, 365–394.

    Article  MathSciNet  Google Scholar 

  2. S. Ali Miri and V. Kumar Murty, An application of sieve methods to elliptic curves, in: INDOCRYPT 2001, eds. C. Pandu Rangan and C. Ding, pp. 91–98, Lecture Notes in Computer Science 2247, Springer, Berlin, 2001.

    Google Scholar 

  3. R. Balasubramanian and V. Kumar Murty, Zeros of Dirichlet \(L\)-functions, Ann. Scient. Ecole Norm. Sup.25(1992), 567–615.

    Google Scholar 

  4. D. Bump, S. Friedberg and J. Hoffstein, Non-vanishing theorems for \(L\)-functions of modular forms and their derivatives, Invent. Math. 102(1990), 543–618.

    Google Scholar 

  5. A. Chow, Applications of Fourier coefficients of modular forms, PhD Thesis, University of Toronto, 2015.

    Google Scholar 

  6. S. Chowla, Improvement of a theorem of Linnik and Walfisz, Proc. London Math. Soc. (2)50(1949), 423–429.

    Article  MathSciNet  Google Scholar 

  7. S. Chowla and P. Erdös, A theorem on the distribution of the values of \(L\)-functions, J. Indian Math. Soc. (N.S.)15(1951), 11–18.

    Google Scholar 

  8. A. Dixit, On the Lindelöf class of \(L\)-functions, PhD Thesis, University of Toronto, 2018.

    Google Scholar 

  9. A. Dixit and V. Kumar Murty, The Lindelöf class of \(L\)-functions II, preprint 2017.

    Google Scholar 

  10. P. D. T. A. Elliott, The distribution of the quadratic class number, Litovsk. Mat. Sb.10(1970), 189–197.

    Google Scholar 

  11. P. Erdös and M. Ram Murty, On the order of a (mod p). Number theory (Ottawa, ON, 1996), 87–97, CRM Proc. Lecture Notes, 19, Amer. Math. Soc. Providence, RI, 1999.

    Google Scholar 

  12. R. Foote and V. Kumar Murty, Zeros and poles of Artin \(L\)-functions, Math. Proc. Cambridge Phil. Soc.105(1989), 5–11.

    Google Scholar 

  13. R. Foote, H. Ginsburg and V. Kumar Murty, On Heilbronn characters, Bull. Amer. Math. Soc.52(2015), 465–496.

    Article  MathSciNet  Google Scholar 

  14. K. Ford, F. Luca, and P. Moree, Values of the Euler \(\phi \)-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, Math. Comp. 83(2014), 1447–1476.

    Article  MathSciNet  Google Scholar 

  15. S. Gun and V. Kumar Murty, A variant of Lehmer’s conjecture II: The CM case, Canadian J. Math63(2011), 298–326.

    Article  MathSciNet  Google Scholar 

  16. R. Gupta and M. Ram Murty, Primitive points on elliptic curves, Composito Math. 58(1986), 13–44.

    Google Scholar 

  17. G. Harder, R. Langlands and M. Rapaport, Algebraische Zyklen auf Hilbert-Blumenthal-Flchen. (German) [Algebraic cycles on Hilbert-Blumenthal surfaces] J. Reine Angew. Math. 366(1986), 53–120.

    Google Scholar 

  18. Y. Ihara, On the Euler-Kronecker constants of global fields and primes with small norms, Algebraic geometry and number theory, Progr. Math. 253(2006), 407–451.

    Google Scholar 

  19. Y. Ihara, The Euler-Kronecker invariants in various families of global fields, Arithmetics, geometry, and coding theory (AGCT 2005), Sémin. Congr. 21(2010), 79–102.

    Google Scholar 

  20. Y. Ihara, V. Kumar Murty, M. Shimura, On the logarithmic derivatives of Dirichlet \(L\)-functions at \(s=1\), Acta Arith. 137(2009), 253–276.

    Google Scholar 

  21. H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros, Israel J. Math. 120(2000), 155–177.

    Google Scholar 

  22. K. Khuri-Makdisi, Linear algebra algorithms for divisors on an algebraic cuve, Math. Comp. 73(2003), 333–357.

    Article  MathSciNet  Google Scholar 

  23. V. Kumar Murty and Pramathanath Sastry, Explicit arithmetic on Abelian varieties, this volume.

    Google Scholar 

  24. V. Kumar Murty and V. Patankar, Splitting of Abelian varieties, Intl. Math. Res. Not. 2008(2008), 27 pages, https://doi.org/10.1093/imrn/rnn033, published May 6, 2008.

  25. V. Kumar Murty and V. Patankar, Tate cycles on Abelian varieties with complex multiplication, Canadian J. Math. 67(2015), 198–213.

    Article  MathSciNet  Google Scholar 

  26. V. Kumar Murty and D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89(1987), 319–345.

    Article  MathSciNet  Google Scholar 

  27. V. Kumar Murty and Y. Zong, Elliptic minuscule pairs and splitting of Abelian varieties, Asian J. Math21(2017), 287–336.

    Google Scholar 

  28. V. Kumar Murty and Y. Zong, Splitting of Abelian varieties, Math. of Communication, 8(2014), 511–519.

    Article  MathSciNet  Google Scholar 

  29. V. Kumar Murty, On the Sato-Tate conjecture, in: Number Theory related to Fermat’s Last Theorem, ed. N. Koblitz, pp. 195–205, Birkhauser-Verlag, Boston, 1982.

    Chapter  Google Scholar 

  30. V. Kumar Murty, Algebraic cycles on Abelian varieties, Duke Math. J. 50(1983), 487–504.

    Article  MathSciNet  Google Scholar 

  31. V. Kumar Murty, Exceptional Hodge classes on certain Abelian varieties, Math. Annalen, 268(1984), 197–206.

    Article  MathSciNet  Google Scholar 

  32. V. Kumar Murty, Stark zeros in certain towers of fields, Math. Res. Letters, 6(1999), 511–520.

    Google Scholar 

  33. V. Kumar Murty, Class numbers of CM-fields with solvable normal closure, Compositio Math127(2001), 273–287.

    Google Scholar 

  34. V. Kumar Murty, A variant of Lehmer’s conjecture, J. Number Theory, 123(2007), 80–91.

    Article  MathSciNet  Google Scholar 

  35. V. Kumar Murty, The Lindelöf class of \(L\)-functions, in: \(L\)-functions, eds. L. Weng and M. Kaneko, pp. 165–174, World Scientific, 2007.

    Google Scholar 

  36. V. Kumar Murty, On the Sato-Tate conjecture II, in: On Certain L-functions: A volume in honour of F. Shahidi, Clay Mathematics Proceedings Volume 43, pp. 471–482, Amer. Math. Soc. Providence, 2011.

    Google Scholar 

  37. V. Kumar Murty, The Euler-Kronecker constant of a cyclotomic field, Ann. Sci. Math. Québec35(2011), 239–247.

    Google Scholar 

  38. Y. Lamzouri, The distribution of Euler-Kronecker constants of quadratic fields, J. Math. Anal. Appl. 432(2015), 632–653.

    Article  MathSciNet  Google Scholar 

  39. N. Laptyeva, A variant of Lehmer’s conjecture in the CM case, Ph. D Thesis, University of Toronto, 2013.

    Google Scholar 

  40. J. Milne, Lefschetz classes on abelian varieties, Duke Math. J. 96(1999), 639–675.

    Article  MathSciNet  Google Scholar 

  41. M. Mourtada and Kumar Murty, On the Euler Kronecker constant of a cyclotomic field, II, SCHOLAR—a scientific celebration highlighting open lines of arithmetic research, Contemp. Math. 655 (2015), 143–151.

    Google Scholar 

  42. M. Mourtada and V. Kumar Murty, Distribution of values of \(L^{\prime }/L(\sigma ,\chi \)\(_D)\), Mosc. Math. J. 15(2015) 497–509.

    Google Scholar 

  43. M. Mourtada and V. Kumar Murty, Omega theorems for \(\frac{L^{\prime }}{L}(1,\chi \)\(_D)\), Int. J. Number Theory9(2013), 561–581.

    Google Scholar 

  44. T. Oda, Periods of Hilbert modular surfaces, Progress in Mathematics Volume 19, Birkähauser, Boston, 1982.

    Google Scholar 

  45. A. Odlyzko, Discrete logarithms over finite fields, in: Handbook of Finite Fields, eds. G. L. Mullen and D. Panario, pp. 393-401, CRC Press, Boca Raton, USA, 2013

    Google Scholar 

  46. A. Ogg, A remark on the Sato-Tate conjecture, Invent. Math. 9(1969/1970), 198–200.

    Article  MathSciNet  Google Scholar 

  47. M. Ram Murty, Selberg’s conjectures and Artin \(L\)-functions, Bull. Amer. Math. Soc. 31(1994), 1–14.

    Article  MathSciNet  Google Scholar 

  48. M. Ram Murty and V. Kumar Murty, A variant of the Bombieri-Vinogradov theorem, in: Number Theory, Volume 7, CMS Conference Proceedings, ed. H. Kisilevsky et al. pp 243–272, Amer. Math. Soc, Providence, 1987.

    Google Scholar 

  49. M. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular \(L\)-series, Annals of Math. 133(1991), 447–475.

    Google Scholar 

  50. M. Ram Murty and V. Kumar Murty, Non-vanishing of\(L\)- functions and applications, Progress in Mathematics Volume 157, Birkhauser, Basel, 1997.

    Google Scholar 

  51. M. Ram Murty and A. Perelli, The pair correlation of zeros of functions in the Selberg class, Internat. Math. Res. Notices, 1999, pp. 531–545.

    Article  Google Scholar 

  52. M. Ram Murty and V. Kumar Murty, Some remarks on automorphy and the Sato-Tate conjecture, in: Advances in the theory of numbers, eds. A. Alaca et al. pp. 159–168, Fields Institute Communications Volume 77, Springer, New York, 2015.

    Google Scholar 

  53. M. Ram Murty, V. Kumar Murty and N. Saradha, Modular forms and the Chebotarev density theorem, Amer. J. Math. 110(1988), 253–281.

    Article  MathSciNet  Google Scholar 

  54. M. Ram Murty, V. Kumar Murty and P. J. Wong, Pair correlation and the Chebotarev density theorem, J. Ramanujan Math. Soc. to appear.

    Google Scholar 

  55. K. Ribet, Hodge classes on certain types of abelian varieties, Amer. J. Math.105(1983), 523–538.

    Article  MathSciNet  Google Scholar 

  56. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in: Collected Papers Volume 2, pp. 57–63, Springer, New York, 1991.

    Google Scholar 

  57. J.-P. Serre, Abelian\(\ell \)-adic representations and elliptic curves, Benjamin, New York-Amsterdam, 1968.

    Google Scholar 

  58. K. Soundararajan, Nonvanishing of quadratic Dirichlet \(L\)-functions at \(s=1/2\), Annals of Math. 152(2000), 447–488.

    Google Scholar 

  59. H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math.23(1974), 135–152.

    Article  MathSciNet  Google Scholar 

  60. A. Weil, Abelian varieties and the Hodge ring, in: Collected Papers, Volume III, pp. 421–429, Springer, Berlin, 1980.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Ram Murty .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Akbary, A., Gun, S., Murty, M.R. (2018). Overview of the Work of Kumar Murty. In: Akbary, A., Gun, S. (eds) Geometry, Algebra, Number Theory, and Their Information Technology Applications. GANITA 2016. Springer Proceedings in Mathematics & Statistics, vol 251. Springer, Cham. https://doi.org/10.1007/978-3-319-97379-1_1

Download citation

Publish with us

Policies and ethics