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Tuesday, September 15, 2020

**Abstract:** For a number field F of degree d \geq 2 over the rationals, let D_F be the absolute discriminant. In 1956, Ankeny, Brauer, and Chowla proved that for a given degree d, there exist infinitely many number fields of degree d such that for any fixed \epsilon > 0, the class group of F has size at least (D_F)^{1/2-\epsilon}. This was conditionally refined by Duke in 2003: assuming Artin's holomorphy conjecture and the generalized Riemann hypothesis, there exist infinitely many number fields F of degree d such that the class group of F has size \asymp (D_F)^{1/2} (\log\log D_F / \log D_F)^{d-1}. In particular, given d \geq 2, there are (conditionally) infinitely many number fields of degree d whose class group has maximal asymptotic order. In 2014, Cho showed that Artin's holomorphy conjecture and the generalized Riemann hypothesis can be replaced with the single assumption that Artin representations are automorphic (which implies Artin's holomorphy conjecture), unconditionally establishing Duke's conclusion for d \leq 5. I will discuss joint work with Robert Lemke Oliver and Asif Zaman in which we unconditionally establish Duke's conclusion for all d \geq 2 (among many other things).