Towards Lang-Trotter for elliptic curves over function fields
The conjecture of Lang and Trotter on ``Primitive points on elliptic curves'', originally formulated over number fields, asserts that a point of infinite order on an elliptic curve over a global field generates the points of the reduced curve modulo \(v\) for a positive density of places \(v\). More generally, there are precise predictions for the density, when the point of infinite order is replaced by any free subgroup of the Mordell-Weil group of rank \(r\). The present paper proves that for elliptic curves over function fields with non-constant \(j\)-invariant (the case of constant \(j\)-invariants was treated by the second author in [\textit{J. F. Voloch}, Bol. Soc. Bras. Mat., Nova Sér. 21, No.1, 91-94 (1990; Zbl 0779.14009)], such a density exists when \(r\geq 6\). Concretely, let \(E/K\) be an elliptic curve over a global field of characteristic \(p>0\) and assume that \(j(E)\) is not constant. Assume that \(E(K)\) has a torsion free subgroup \(\Sigma\) of rank \(r\geq 6\). It is shown that, for a positive proportion of places \(v\) of \(K\), the reduction of \(\Sigma\) modulo \(v\) accounts for all points on the reduction of \(E\) modulo \(v\). Moreover, this is true for a non-zero proportion of places of any given degree \(n\), and these proportions are bounded above and below independently of \(n\). The strategy of proof is an adaptation to the function field case of the approach of \textit{R. Gupta} and \textit{M. Ram Murty} [Compos. Math. 58, 13--44 (1986; Zbl 0598.14018)], which deals with the number field version of the Lang-Trotter conjecture, to prove the theorem when \(r\geq 10\), together with a novel idea for bringing down the bound to \(r\geq 6\). The authors remark that the latter might also be useful for the number field case.
scientific article; zbMATH DE number 5510989