Number Theory Seminar | Jerry Wang | Counting A_4-quartics by discriminant

An \(A_4\)-quartic is a degree 4 extension of Q whose Galois closure has Galois group \(A_4\). They are the last remaining quartic fields for which we didn't have an asymptotic counting formula. In this joint work with Stanley Xiao, we prove that the the number of \(A_4\)-quartics of discriminant at most \(X\) is \(\kappa_{A_4}X^{1/2}\log X + O_\epsilon(X^{1/2}(\log\log X)^{3+\epsilon})\)) where \(\kappa_{A_4}\) is an explicit constant with an Euler product expansion.

MC 5501