Counting biquadratic number fields that admit quaternionic or dihedral extensions

Abstract: Many interesting problems in arithmetic statistics involve counting number fields (ordered by their discriminants, say) with certain properties. In joint work with Siman Wong (UMass Amherst), we establish asymptotic formulae for the number of biquadratic extensions of $\mathbb{Q}$ that admit a degree-2 extension with Galois group $G$, where $G$ is either the quaternion group or the dihedral group (of order 8). We will discuss these results and how they are proved, and we will discuss their significance with regard to a theorem of Tate on lifts of projective Galois representations.

number theory

Audience: researchers in the topic

( paper )

Five College Number Theory Seminar