Strong Malle's conjecture for extensions with Galois group G

At least 1 year old · documented by

Let KK be a number field, let G⊂SnG\subset S_n act transitively on {1,…,n}\{1,\dots,n\}, and let M(K,G;B)M(K,G;B) count degree-nn extensions L/KL/K with Galois closure of group GG and discriminant norm at most BB. Let a(G)a(G) be the reciprocal of the minimal index of a nonidentity element of GG, and let b(K,G)b(K,G) be the number of cyclotomic Galois orbits of conjugacy classes having minimal index. Strong Malle's conjecture. Under these assumptions, there is a positive constant c(K,G)c(K,G) such that

M(K,G;B)∼c(K,G)Ba(G)(log⁡B)b(K,G)−1.M(K,G;B)\sim c(K,G)B^{a(G)}(\log B)^{b(K,G)-1}.

This is the asymptotic refinement of the weak bounds, predicting both the main power of BB and the logarithmic factor. The source does not specify the resolution status here.

References

Primary source

Shabnam Akhtari, Jennifer Park, Marta Pieropan and Soumya Sankar, “On rational points on classifying stacks and Malle's conjecture”, arXiv:2402.10355 (2024).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.