The age conjecture for counting G-fields

Let KK be a number field, let GG be a finite group, and let VV be a finite-dimensional faithful representation of GG over KK. Let NG,V,K(B)N_{G,V,K}(B) count isomorphism classes of GG-fields L/KL/K with VV-discriminant DLVBD_L^V\le B. Let age(G)\operatorname{age}(G) be the minimum of the age function on G{1}G\setminus\{1\}, and let υ(G)\upsilon(G) be the number of nontrivial KK-conjugacy classes with minimum age. The age conjecture. If KK is sufficiently large, then

NG,V,K(B)CB1/age(G)(logB)υ(G).N_{G,V,K}(B)\sim CB^{1/\operatorname{age}(G)}(\log B)^{\upsilon(G)}.

This is proposed as a representation-theoretic analogue of Malle's conjecture, with age replacing index; the source gives no resolution status.

Sources & referencesView supporting material

Primary source

Takehiko Yasuda, “Manin's conjecture vs. Malle's conjecture”, arXiv:1505.04555 (2015).

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