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Lang–Trotter and Sato–Tate distributions in single and double parametric families of elliptic curves

Volume 170 / 2015

Min Sha, Igor E. Shparlinski Acta Arithmetica 170 (2015), 299-325 MSC: 11B57, 11G05, 11G20, 14H52. DOI: 10.4064/aa170-4-1

Abstract

We obtain new results concerning the Lang–Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato–Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang–Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of bounded height. Some of the families we consider are much thinner than the ones previously studied.

Authors

  • Min ShaSchool of Mathematics and Statistics
    University of New South Wales
    Sydney, NSW 2052, Australia
    e-mail
  • Igor E. ShparlinskiSchool of Mathematics and Statistics
    University of New South Wales
    Sydney, NSW 2052, Australia
    e-mail

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