Primes of the form x²+dy² and interactions of class groups

The form x²+y² covers ½ of the primes, while the forms x²+y², x²+2y² cover ¾ of them. In this talk, we will show that the proportion of primes covered by the forms x²+dy², 1 ≤ d ≤ Δ, is1 - exp((α(Δ) + o(1)) √Δ / log Δ)for some 7π/12 ≤ α(Δ) ≤ 7π/12 + log 4. Furthermore, inspired by a result of Kaplansky, we prove that any prime covered by two of the formsx²+17y², x²+65y², x²+1105y²is actually covered by all three. We establish these results by studying how class groups of quadratic fields interact.