On the probabilities of local behaviors in abelian field extensions

Abstract

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev's density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.

The weak form of Malle’s conjecture and solvable groups

“…The author is also aware of upcoming results for D4 Ă S8 [SV19] and a large family of imprimitive groups (including many wreath products of the form T ≀ B for T abelian or S3) [LOWW19]. Similar results are known when the extensions L{K are ordered by other invariants, for example Wood [Woo09] proves the analogous results for abelian groups ordered by conductor. The constant cpK, Gq, while known to be positive in all of the above cases, is known explicitly in far fewer cases, such as for abelian extensions over Q by Mäki [Ma85] and cyclic quartic extensions over general K by Cohen-Diaz y Diaz-Olivier [CDyDO05].…”

Mass formulas for local Galois representations to wreath products and cross products

“…Ellenberg and Venkatesh [2005, Section 4.2] suggest that we can try to count extensions of global fields by quite general invariants of Galois representations. In [Wood 2008], it is shown that when counting by certain invariants of abelian global fields, such as conductor, Bhargava's question can be answered affirmatively. It is also shown in [Wood 2008] that when counting abelian global fields by discriminant, the analogous conjectures fail in at least some cases.…”

“…In [Wood 2008], it is shown that when counting by certain invariants of abelian global fields, such as conductor, Bhargava's question can be answered affirmatively. It is also shown in [Wood 2008] that when counting abelian global fields by discriminant, the analogous conjectures fail in at least some cases. In light of the fact that Bhargava's conjectures for the asymptotics of the number of S n -number fields arise from his mass formula (1-1) for counting by discriminant, one naturally looks for mass formulas that use other ways of counting, such as Theorem 1.1, which might inspire conjectures for the asymptotics of counting global fields with other Galois groups.…”

“…Equation (1-1) is at the heart of [Bhargava 2007, Conjecture 1] for the asymptotics of the number of S n -number fields with discriminant ≤ X , and also [Bhargava 2007, Conjectures 2-3] for the relative asymptotics of S n -number fields with certain local behaviors specified. These conjectures are theorems for n ≤ 5 [Davenport and Heilbronn 1971;Bhargava 2005;≥ 2008]. Kedlaya [2007, Section 3] has translated Bhargava's formula into the language of Galois representations so that the sum in (1-1) becomes a sum over Galois representations to S n as follows:…”

Artin L-functions of small conductor

“…This phenomenon of different orderings on the same set of number fields plays a prominent role in asymptotic studies [32]. Here we are interested instead in initial segments and how they depend on χ.…”