Critical zeros of class group L-functions

Abstract

For an imaginary quadratic field, we define and study L-functions associated to the characters of the ideal class group of that field. After proving the main properties of these L-functions, we analyze them together as a family, with the goal of counting their zeros on the critical line. Our main result shows that any positive proportion of L-functions in this family has critical zeros of height bounded by an absolute constant times a factor that depends only slightly on the discriminant of the field. The main tools we make use of are the approximate functional equation and the equidistribution of Heegner points, which we use to obtain explicit formulas for the average of the square of the L-functions on the critical line.