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.