On a Linnik problem for elliptic curves

Abstract:

Let $S(Q,B)$ denote the number of moduli $q\leq Q$ for which a primitive character $\chi$ mod $q$ exists such that $n_{\chi }>B$, where $n_{\chi }$ denotes the smallest natural number such that $\chi (n) \not =1$. Baier showed that for any $\beta >2$ we have $S(Q,(\log Q)^{\beta }) \ll Q^{\frac {1}{\beta -1}+\varepsilon }$ and asked for an analogue of this result for elliptic curves. It is the aim of this note to establish such an analogue.