Journal of the Ramanujan Mathematical Society

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The Chebotarev Density Theorem and the Pair Correlation Conjecture


Affiliations
1 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
     

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In this note, we formulate pair correlation conjectures and refine the effective version of the Chebotarev density theorem established by the first two authors. Also, we apply our result to study Artin’s primitive ischolar_main conjecture and the Lang-Trotter conjectures and obtain shaper error terms.
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  • A. C. Cojocaru, Cyclicity of elliptic curves modulo p, Ph.D. Thesis, Queen’s University (2002).

  • A. C. Cojocaru and C. David, Frobenius fields for elliptic curves, American Journal of Math., 130 (2008) no. 6, 1535–1560.

  • A. C. Cojocaru, E. Fouvry and M. R. Murty, The square sieve and the Lang-Trotter conjecture, Canadian Journal of Math., 57, (2005) no. 6, 1155–1177.

  • A. C. Cojocaru and M. R. Murty, Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem, Math. Annalen, 330 (2004) no. 3, 601–625.

  • P. Deligne, Formes modulaires et representations l-adiques, Sem. Bourbaki 355, Lecture Notes in Mathematics, Springer Verlag, Heidelberg, 179 (1971) 139–172.

  • R. Gupta and M. R. Murty, A remark on Artin’s conjecture, Inventiones Math., 78 (1984) 127–130.

  • R. Gupta and M. R. Murty, Cyclicity and generation of points mod p on elliptic curves, Inventiones Math., 101 (1990) 225–235.

  • D. R. Heath-Brown, Gaps between primes, and the pair correlation of zeros of the zeta-function, Acta Arith., 41 (1982) 85–99.

  • C. Hooley, On Artin’s conjecture, J. Reine Angew. Math., 225 (1967) 209–220.

  • J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: ζ -functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London (1977) 409–464.

  • S. Lang and H. Trotter, Frobenius distributions in GL2-extensions, Lecture Notes in Mathematics, Springer-Verlag, 504 (1976).

  • F. Momose, On the l-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28, (1981) no. 1, 89–109.

  • H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 227 (1971).

  • M. R. Murty, On Artin’s conjecture, Journal of Number Theory, 16 (1983) 147–168.

  • M. R. Murty, V. K. Murty, and N. Saradha,Modular forms and the Chebotarev density theorem, American Journal of Math., 110 (1988) 253–281.

  • M. R. Murty and A. Perelli, The pair correlation of zeros of functions in the Selberg class, International Math. Res. Notices, 10 (1999) 531–545.

  • M. R. Murty and A. Zaharescu, Explicit formulas for the pair correlation of zeros of functions in the Selberg class, Forum Math., 14 (2002) no. 1, 65–83.

  • V. K. Murty, Explicit formulae and the Lang-Trotter conjecture, Rocky Mountain J. Math., 15(2) (1985) 535–551.

  • V. K. Murty, Modular forms and the Chebotarev density theorem II, Analytic Number Theory, Ed. Y. Motohashi, Cambridge University Press (1997) 287–308.

  • K. Ribet, Galois representations attached to eigenforms with Nebentypus, Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 601 (1976) 17–52,

  • K. Ribet, On l-adic representations attached to modular forms II, Glasgow J. Math., 27 (1985) 185–194.

  • J.-P. Serre, Resume des cours de 1977–1978, Annuaire du College de France 1978, 67–70. (See also Collected Papers, Volume III, Springer-Verlag (1985).)

  • J.-P. Serre, Quelques applications du theoreme de densit´e de Chebotarev, Publ. Math. IHES, 54 (1981) 123–201.


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  • The Chebotarev Density Theorem and the Pair Correlation Conjecture

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Authors

M. Ram Murty
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
V. Kumar Murty
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
Peng-Jie Wong
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada

Abstract


In this note, we formulate pair correlation conjectures and refine the effective version of the Chebotarev density theorem established by the first two authors. Also, we apply our result to study Artin’s primitive ischolar_main conjecture and the Lang-Trotter conjectures and obtain shaper error terms.

References