Infinitely many primes with a fixed Frobenius field for an elliptic curve over $\mathbb{Q}$
In 1987, Elkies proved the striking result that every elliptic curve over $\mathbb{Q}$ has infinitely many supersingular primes. Motivated by this theorem and its connection with the Lang-Trotter conjecture, we study the analogous problem for Frobenius fields of elliptic curves. For certain families of non-CM elliptic curves $E/\mathbb{Q}$ and imaginary quadratic fields $K$, we prove that there exist infinitely many primes $p$ for which the Frobenius field of $E$ at $p$ equals $K$. More precisely, letting $\pi_E(x,K)$ denote the number of such primes with $p\leq x$, we establish the unconditional bound $\pi_E(x, K)\gg_{E, K, \epsilon} (\log\log x)^{1-\epsilon}$ for every $\epsilon>0$. We also prove unconditional power-saving upper bounds for a restricted counting function associated with $\pi_E(x,K)$. The approach combines Deuring's theory of complex multiplication, properties of singular moduli, and arithmetic intersection theory on modular curves.