Arithmetic Geometry and Number Theory RTG Seminar: 8-class ranks of imaginary quadratic fields

Seminar | February 10 | 3:10-5 p.m. | 740 Evans Hall

 Alexander Smith, Harvard

 Department of Mathematics

We prove that the two-primary subgroups of the class groups of imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra-Gerth heuristic. In this talk, we will detail our method for proving the 8-class rank portion of this theorem and will compare our approach to one that uses the governing fields predicted by Cohn and Lagraias.

Pre-talk Title: 2-Selmer ranks and 4-class ranks in non-standard families.

Pre-talk Abstract: Taking $D$ to be a set of integers, and fixing an elliptic curve $E: y^2 = x^3 + ax + b$, the starting question of this talk is how the $2$-Selmer groups in the quadratic twist family $\{ E^d\,:\,\, d \in D\}$ are distributed, or how the $4$-class ranks in the family $\{ Q(\sqrt {-d})\,:\,\, d \in D\}$ are distributed. These questions are particularly natural if there are sets of primes $X_1, \dots , X_k$ so that $D$ is defined as the set \[\left \{ p_1 \cdot \dots p_k \text { squarefree }\,:\,\, p_1 \in X_1, \dots , p_k \in X_k\right \}.\] From the Chebotarev Density theorem, we can understand how these groups vary in this family if $X_{i+1}$ is vastly larger than $X_i$ for each $i < k$ (the heterogeneous case). From the averaged Chebotarev results of Jutila, we can also sometimes understand how the groups vary in this family if the $X_i$ are all roughly of the same size (the homogeneous case). We generalize Jutila's result and explain how it increases our understanding of the distribution of 2-Selmer groups in the homogeneous case. We also indicate how our techniques may be adapted to the standard twist set $D_N = \{ d\,:\, \, 0 < d < N\}$.