#### Alladi Ramakrishnan Hall

#### Counting D_4-quartic fields ordered by conductor

#### Ila Varma

##### Columbia University

*We consider the family of D_4-quartic fields ordered by the Artin*

conductors of the corresponding 2-dimensional irreducible Galois

representations. In this talk, I will describe ways to compute the number

of such D_4 fields with bounded conductor. Traditionally, there have been

two approaches to counting quartic fields, using arithmetic invariant

theory in combination of geometry-of-number techniques, and applying Kummer

theory together with L-function methods. Both of these strategies fall

short in the case of D_4 fields since counting quartic fields containing a

quadratic subfield of large discriminant is difficult. However, when

ordering by conductor, these techniques can be utilized due to additional

algebraic structure that the Galois closures of such quartic fields have,

arising from the outer automorphism of D_4. This result is joint work with

Ali Altug, Arul Shankar, and Kevin Wilson.

Done