Gauss conjectured there are exactly nine imaginary quadratic fields with class number one (i.e., unique factorization). This was proven by independent work of Heegner, Baker, and Stark. By now, there are three methods: transcendence theory (Baker), modular functions (Heegner, Stark, and multiple others), and high-order zeros of L-functions (Goldfeld). This third method in fact can show the effective divergence of the class number upon being given a suitable L-function of analytic rank 3, which was subsequently provided by work of Gross and Zagier.
We show a result in a different direction, namely that suitable L-functions of analytic rank 2 already suffice to prove the class number one result. We do this by showing that the L-series coefficients when restricted to the principal form exhibit sufficient cancellation. One method to show this is via spectral theory, and another by adapting a result of Hooley's on equi-distribution of roots of quadratic congruences to varying moduli.
University of Sydney
Wed, 27/02/2019 - 3:00pm
RC-4082, The Red Centre, UNSW