On primes represented by $a{X}^2+b{Y}^3$

Let $a,b>0$ be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form $a{x}^2+b{y}^3$ with $x\le X^{1\mathrm{/}2}$ and $y\le X^{1\mathrm{/}3}$. The proof combines sieve methods with the theory of real quadratic fields/indefinite binary quadratic forms, the Weil bound for exponential sums, and spectral methods of GL(2) automorphic forms. We also discuss applications to elliptic curves.