Unconditional class group tabulation of imaginary quadratic fields to \(| \Delta| < 2^{40}\) (Q2796029)

The authors use efficient algorithms to tabulate class groups of imaginary quadratic fields of discriminant \(\Delta\) with \(|\Delta|<2\exp(40)\). This is a wonderful interesting report for all those who are interested in the class groups of imaginary quadratic fields, in particular when it comes to conjectures concerning the class groups and to the Cohen-Lenstra heuristics. Some pieces of information concerning the so-called exotic groups are also given. This work is achieved with the use of classical quadratic forms, some Jacobi theta series, and certain class number generating functions à la Watson when \(\Delta\ncong 1\pmod 8\). The group structure is obtained by the use of the factorization of the class number. The case \(\Delta\equiv 1\pmod 8\) is also dealt with, but this is slower than the case \(\Delta\not\equiv 1\pmod 8\). The authors use the methods of Buchmann, Jacobson, Teske, Ramachandran, Sutherland; some techniques based on the Eichler-Selberg trace formula allow them to remove ERH. The tables provide very interesting statistical data. For instance, we come across the number of non-cyclic odd parts of class groups, and across the cardinalities of class numbers divisible by the prime number \(q\) when \(q\) is 3, 5, 7, 11, 13. The smallest \(|\Delta|\) for which the 17-rank is 3 is exhibited. We learn about the first occurrences of doubly and trebly non-cyclic class groups. We also learn about the counts of class groups with \(q\)-rank equal to \(r\) for \(q=3,5,7\) and \(r=2,3\). And there are other interesting pieces of information.