On the Lang-Trotter and Sato-Tate Conjectures on Average for Polynomial Families of Elliptic Curves

TL;DR

This paper demonstrates that reductions of certain polynomial families of elliptic curves modulo primes follow the predictions of the Lang-Trotter and Sato-Tate conjectures on average, extending previous results to more general polynomial cases.

Contribution

It introduces a new technique to analyze polynomial families of elliptic curves, generalizing prior results from linear to polynomial cases for the Lang-Trotter and Sato-Tate conjectures.

Findings

Reductions behave as predicted by conjectures on average

Results extend to non-linear polynomial families

New method differs from previous approaches

Abstract

We show that the reductions modulo primes $p\le x$ of the elliptic curve $$ Y^2 = X^3 + f(a)X + g(b), $$ behave as predicted by the Lang-Trotter and Sato-Tate conjectures, on average over integers $a \in [-A,A]$ and $b \in [-B,B]$ for $A$ and $B$ reasonably small compared to $x$, provided that $f(T), g(T) \in \Z[T]$ are not powers of another polynomial over $\Q$. For $f(T) = g(T) = T$ first results of this kind are due to E. Fouvry and M. R. Murty and have been further extended by other authors. Our technique is different from that of E. Fouvry and M. R. Murty which does not seem to work in the case of general polynomials $f$ and $g$.