Tag - Class groups

We determine the average size of the 3-torsion in class groups of G-extensions of a number field when G is any transitive 2-group containing a transposition, for example, D₄. It follows from the Cohen-Lenstra-Martinet heuristics that the average size of the p-torsion in class groups of G-extensions of a number field is conjecturally finite for any G and most p. Previously this conjecture had only been proved in the cases of G=S₂ with p=3 and G=S₃ with p=2. We also show that the average 3-torsion in a certain relative class group for these G-extensions is as predicted by Cohen and Martinet, proving new cases of the Cohen-Lenstra-Martinet heuristics. Our new method also works for many other permutation groups G that are not 2-groups.

In this talk, we prove an upper bound on the average number of 2-torsion elements in the class group of monogenised fields of any degree n≥3 and, conditional on a widely expected tail estimate, compute this average exactly. As an application, we show the existence of infinitely many number fields with odd class number in almost every even degree and signature. Time permitting, we will also discuss extensions of these results to orders (joint with Shankar, Swaminathan and Varma) and the relative setting.