Rational points on elliptic curves that are primitive modulo a positive proportion of primes
In 1977, Lang and Trotter conjectured that, subject to certain necessary arithmetic restrictions, a rational point $P$ of infinite order on an elliptic curve $E$ over $\Q$ should generate its reduction modulo infinitely many primes $p$---and indeed should do so for a specified positive density of primes. However, to date, even this infinitude was not known unconditionally for any explicit pair $(E,P)$. For the curve $E:y^2=x^3-2x$ and point $P=(2,2)$, we prove the existence of infinitely many---and indeed a positive lower density of---primes for which the reduction of $P$ generates…
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