Malle’s conjecture for nonic Heisenberg extensions

We prove Malle’s conjecture for nonic Heisenberg extensions over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> . Our main algebraic result shows that the number of nonic Heisenberg extensions over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> with discriminant bounded by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is given by a character sum. We then extract the main term from this sum by exploiting oscillation of characters.