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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Akbary, D. Ghioca, and V. Kumar Murty, Reductions of points on elliptic curves, Math. Annalen, 347(2010), no. 2, 365–394.
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.
R. Balasubramanian and V. Kumar Murty, Zeros of Dirichlet \(L\)-functions, Ann. Scient. Ecole Norm. Sup.25(1992), 567–615.
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.
A. Chow, Applications of Fourier coefficients of modular forms, PhD Thesis, University of Toronto, 2015.
S. Chowla, Improvement of a theorem of Linnik and Walfisz, Proc. London Math. Soc. (2)50(1949), 423–429.
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.
A. Dixit, On the Lindelöf class of \(L\)-functions, PhD Thesis, University of Toronto, 2018.
A. Dixit and V. Kumar Murty, The Lindelöf class of \(L\)-functions II, preprint 2017.
P. D. T. A. Elliott, The distribution of the quadratic class number, Litovsk. Mat. Sb.10(1970), 189–197.
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.
R. Foote and V. Kumar Murty, Zeros and poles of Artin \(L\)-functions, Math. Proc. Cambridge Phil. Soc.105(1989), 5–11.
R. Foote, H. Ginsburg and V. Kumar Murty, On Heilbronn characters, Bull. Amer. Math. Soc.52(2015), 465–496.
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.
S. Gun and V. Kumar Murty, A variant of Lehmer’s conjecture II: The CM case, Canadian J. Math. 63(2011), 298–326.
R. Gupta and M. Ram Murty, Primitive points on elliptic curves, Composito Math. 58(1986), 13–44.
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.
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.
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.
Y. Ihara, V. Kumar Murty, M. Shimura, On the logarithmic derivatives of Dirichlet \(L\)-functions at \(s=1\), Acta Arith. 137(2009), 253–276.
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.
K. Khuri-Makdisi, Linear algebra algorithms for divisors on an algebraic cuve, Math. Comp. 73(2003), 333–357.
V. Kumar Murty and Pramathanath Sastry, Explicit arithmetic on Abelian varieties, this volume.
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.
V. Kumar Murty and V. Patankar, Tate cycles on Abelian varieties with complex multiplication, Canadian J. Math. 67(2015), 198–213.
V. Kumar Murty and D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89(1987), 319–345.
V. Kumar Murty and Y. Zong, Elliptic minuscule pairs and splitting of Abelian varieties, Asian J. Math. 21(2017), 287–336.
V. Kumar Murty and Y. Zong, Splitting of Abelian varieties, Math. of Communication, 8(2014), 511–519.
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.
V. Kumar Murty, Algebraic cycles on Abelian varieties, Duke Math. J. 50(1983), 487–504.
V. Kumar Murty, Exceptional Hodge classes on certain Abelian varieties, Math. Annalen, 268(1984), 197–206.
V. Kumar Murty, Stark zeros in certain towers of fields, Math. Res. Letters, 6(1999), 511–520.
V. Kumar Murty, Class numbers of CM-fields with solvable normal closure, Compositio Math. 127(2001), 273–287.
V. Kumar Murty, A variant of Lehmer’s conjecture, J. Number Theory, 123(2007), 80–91.
V. Kumar Murty, The Lindelöf class of \(L\)-functions, in: \(L\)-functions, eds. L. Weng and M. Kaneko, pp. 165–174, World Scientific, 2007.
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.
V. Kumar Murty, The Euler-Kronecker constant of a cyclotomic field, Ann. Sci. Math. Québec35(2011), 239–247.
Y. Lamzouri, The distribution of Euler-Kronecker constants of quadratic fields, J. Math. Anal. Appl. 432(2015), 632–653.
N. Laptyeva, A variant of Lehmer’s conjecture in the CM case, Ph. D Thesis, University of Toronto, 2013.
J. Milne, Lefschetz classes on abelian varieties, Duke Math. J. 96(1999), 639–675.
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.
M. Mourtada and V. Kumar Murty, Distribution of values of \(L^{\prime }/L(\sigma ,\chi \)\(_D)\), Mosc. Math. J. 15(2015) 497–509.
M. Mourtada and V. Kumar Murty, Omega theorems for \(\frac{L^{\prime }}{L}(1,\chi \)\(_D)\), Int. J. Number Theory9(2013), 561–581.
T. Oda, Periods of Hilbert modular surfaces, Progress in Mathematics Volume 19, Birkähauser, Boston, 1982.
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
A. Ogg, A remark on the Sato-Tate conjecture, Invent. Math. 9(1969/1970), 198–200.
M. Ram Murty, Selberg’s conjectures and Artin \(L\)-functions, Bull. Amer. Math. Soc. 31(1994), 1–14.
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.
M. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular \(L\)-series, Annals of Math. 133(1991), 447–475.
M. Ram Murty and V. Kumar Murty, Non-vanishing of\(L\)- functions and applications, Progress in Mathematics Volume 157, Birkhauser, Basel, 1997.
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.
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.
M. Ram Murty, V. Kumar Murty and N. Saradha, Modular forms and the Chebotarev density theorem, Amer. J. Math. 110(1988), 253–281.
M. Ram Murty, V. Kumar Murty and P. J. Wong, Pair correlation and the Chebotarev density theorem, J. Ramanujan Math. Soc. to appear.
K. Ribet, Hodge classes on certain types of abelian varieties, Amer. J. Math.105(1983), 523–538.
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.
J.-P. Serre, Abelian\(\ell \)-adic representations and elliptic curves, Benjamin, New York-Amsterdam, 1968.
K. Soundararajan, Nonvanishing of quadratic Dirichlet \(L\)-functions at \(s=1/2\), Annals of Math. 152(2000), 447–488.
H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math.23(1974), 135–152.
A. Weil, Abelian varieties and the Hodge ring, in: Collected Papers, Volume III, pp. 421–429, Springer, Berlin, 1980.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
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
DOI: https://doi.org/10.1007/978-3-319-97379-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-97378-4
Online ISBN: 978-3-319-97379-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)