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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-4-1,
author = {Min Sha and Igor E. Shparlinski},
title = {Lang-Trotter and Sato-Tate distributions in single and double parametric families of elliptic curves},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {299-325},
zbl = {06480401},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-4-1}
}
Min Sha; Igor E. Shparlinski. Lang-Trotter and Sato-Tate distributions in single and double parametric families of elliptic curves. Acta Arithmetica, Tome 168 (2015) pp. 299-325. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-4-1/